diff --git a/04-model-description.Rmd b/04-model-description.Rmd index 8d609fc..9e95356 100644 --- a/04-model-description.Rmd +++ b/04-model-description.Rmd @@ -292,7 +292,7 @@ h_t = \text{LSTM}\big(\text{temperature}_{jt}, \ \text{precipitation}_{jt}, \ \t In this formulation, $h_t$ represents the hidden state generated by the LSTM network based on input variables such as temperature, precipitation, and ENSO conditions, while $b_h$ is a bias term added to the output of the hidden state transformation. -The deep learning LSTM model consists of three stacked LSTM-RNN layers. The first LSTM layer has 500 units and the second and third LSTM layers have 250 and 200 units respectively. The architecture the LSTM model is configured to pass node values to subsequent LSTM layers allowing deep learning of more the complex interactions among the climate variable over time. We enforced model sparsity for each LSTM layer using L2 regularization (penalty = 0.001) and used a dropout rate of 0.5 to further prevent overfitting on the limited amount of data. The final output layer was a dense layer with a single unit and a sigmoid activation function to produce a probability for binary classification, i.e. a prediction of environmental suitability $\psi_{jt}$ on a scale of 0 to 1. +The deep learning LSTM model consists of three stacked LSTM-RNN layers. The first LSTM layer has 500 units and the second and third LSTM layers have 250 and 100 units respectively. The architecture the LSTM model is configured to pass node values to subsequent LSTM layers allowing deep learning of more the complex interactions among the climate variable over time. We enforced model sparsity for each LSTM layer using L2 regularization (penalty = 0.001) and used a dropout rate of 0.5 for each LSTM layer to further prevent overfitting on the limited amount of data. The final output layer was a dense layer with a single unit and a sigmoid activation function to produce a probability value for binary classification, i.e. a prediction of environmental suitability $\psi_{jt}$ on a scale of 0 to 1. To fit the LSTM model to data, we modified the learning rate by applying an exponential decay schedule that started at 0.001 and decayed by a factor of 0.9 every 10,000 steps to enable smoother convergence. The model was compiled using the Adam optimizer with this learning rate schedule, along with binary cross-entropy as the loss function and accuracy as the evaluation metric. The model was trained for a maximum of 200 epochs with a batch size of 1024. We allowed model fitting to stop early with a patience parameter of 10 which halts training if no improvement is observed in validation accuracy for 10 consecutive epochs. To train the model we set aside 20% of the observed data for validation and also used 20% of the training data for model fitting. The training history, including loss and accuracy, was monitored over the course of training and gave a final test accuracy of 0.73 and a final test loss of 0.56 (see Figure \@ref(fig:lstm-model-fit)). @@ -300,7 +300,7 @@ To fit the LSTM model to data, we modified the learning rate by applying an expo knitr::include_graphics("figures/suitability_LSTM_fit.png") ``` -After model training was completed, we predicted the values of environmental suitability $\psi_{jt}$ across all time steps for each location. Predictions start in January 1970 and go up to 5 months past the present date (currently February 2025). Given the amount of noise in the model predictions, we added a simple LOESS spline with logit transformation to smooth model predictions over time and give a more stable value of $\psi_{jt}$ when incorporating it into other model features (e.g. Equations \@ref(eq:beta2) and \@ref(eq:delta)). The resulting model predicitons are shown for an example country such as Mozambique in Figure \@ref(fig:psi-prediction-data) which compares model predictions to the original case counts and the binary classification. Predicitons for all model locations are shown in a simplified view in Figure \@ref(fig:psi-prediction-countries). +After model training was completed, we predicted the values of environmental suitability $\psi_{jt}$ across all time steps for each location. Predictions start in January 1970 and go up to 5 months past the present date (currently February 2025). Given the amount of noise in the model predictions, we added a simple LOESS spline with logit transformation to smooth model predictions over time and give a more stable value of $\psi_{jt}$ when incorporating it into other model features (e.g. Equations \@ref(eq:beta2) and \@ref(eq:delta)). The resulting model predictions are shown for an example country such as Mozambique in Figure \@ref(fig:psi-prediction-data) which compares model predictions to the original case counts and the binary classification. Predicitons for all model locations are shown in a simplified view in Figure \@ref(fig:psi-prediction-countries). *Also, please note that this initial version of the model is fitted to a rather small amount of data. Model hyper parameters were specifically chosen to reduce overfitting. Therefore, we recommend to not over-interpret the time series predictions of the model at this early stage since they are likely to change and improve as more historical incidence data is included in future versions.* @@ -356,6 +356,32 @@ knitr::include_graphics("figures/wash_index_by_country.png") ## Immune dynamics +Aside from the current number of infections, population susceptibility is one of the key factors influencing the spread of cholera. Further, since immunity from both vaccination and natural infection provides long-lasting protection, it's crucial to quantify not only the incidence of cholera but also the number of past vaccinations. Additionally, we need to estimate how many individuals with immunity remain in the population at any given time step in the model. + +To achieve this, we estimate the vaccination rate over time ($\nu_{jt}$) based on historical vaccination campaigns and incorporate a model of vaccine effectiveness ($\phi$) and immune decay post-vaccination ($\omega$) to estimate the current number of individuals with vaccine-derived immunity. We also account for the immune decay rate from natural infection ($\varepsilon$), which is generally considered to last longer than immunity from vaccination. + +### Estimating Vaccination Rates + +To estimate the past and current vaccination rates, we sourced data on reported OCV vaccinations from the WHO [International Coordinating Group](https://www.who.int/groups/icg) (ICG) [Cholera vaccine dashboard](https://app.powerbi.com/view?r=eyJrIjoiYmFmZTBmM2EtYWM3Mi00NWYwLTg3YjgtN2Q0MjM5ZmE1ZjFkIiwidCI6ImY2MTBjMGI3LWJkMjQtNGIzOS04MTBiLTNkYzI4MGFmYjU5MCIsImMiOjh9). This resource lists all reactive OCV campaigns conducted from 2016 to the present, with approximately 103 million OCV doses shipped to sub-Saharan African (SSA) countries as of October 9, 2024. However, these data only capture reactive vaccinations in emergency settings and do not include the many preventive campaigns organized by GAVI and in-country partners. + +*As a result, our current estimates of the OCV vaccination rate likely underestimate total OCV coverage. We are working to expand our data sources to better reflect the full number of OCV doses distributed in SSA.* + +To translate the reported number of OCV doses into the model parameter $\nu_{jt}$, we take the number of doses shipped and the reported start date of the vaccination campaign, distributing the doses over subsequent days according to a maximum daily vaccination rate. See Figure \@ref(fig:vaccination-example) for an example of OCV distribution using a maximum daily vaccination rate of 100,000. The resulting time series for each country is shown in Figure \@ref(fig:vaccination-countries), with current totals based on the WHO ICG data displayed in Figure \@ref(fig:vaccination-maps). + +```{r vaccination-example, echo=FALSE, fig.align='center', out.width="100%", fig.cap="Example of the estimated vaccination rate during an OCV campaign."} +knitr::include_graphics("figures/vaccination_example_ZMB.png") +``` + +```{r vaccination-countries, echo=FALSE, fig.align='center', out.width="100%", fig.cap="The estimated vaccination coverage across all countries with reported vaccination data one the WHO ICG dashboard."} +knitr::include_graphics("figures/vaccination_by_country.png") +``` + +```{r vaccination-maps, echo=FALSE, fig.align='center', out.width="100%", fig.cap="The total cumulative number of OCV doses distributed through the WHO ICG from 2016 to present day."} +knitr::include_graphics("figures/vaccination_maps.png") +``` + + + ### Immunity from vaccination The impacts of Oral Cholera Vaccine (OCV) campaigns is incorporated into the model through the Vaccinated compartment (V). The rate that individuals are effectively vaccinated is defined as $\phi\nu_tS_{jt}$, where $S_{jt}$ are the available number of susceptible individuals in location $j$ at time $t$, $\nu_t$ is the number of OCV doses administered at time $t$ and $\phi$ is the estimated vaccine effectiveness. Note that there is just one vaccinated compartment at this time, though future model versions may include $V_1$ an $V_2$ compartments to explore two dose vaccination strategies or to emulate more complex waning patterns. diff --git a/MOSAIC-docs.aux b/MOSAIC-docs.aux index 7ede20d..9621027 100644 --- a/MOSAIC-docs.aux +++ b/MOSAIC-docs.aux @@ -113,129 +113,137 @@ \newlabel{tab:wash-weights}{{4.3}{35}{Table of optimized weights used to calculate the single mean WASH index for all countries}{table.4.3}{}} \@writefile{toc}{\contentsline {section}{\numberline {4.4}Immune dynamics}{35}{section.4.4}\protected@file@percent } \newlabel{immune-dynamics}{{4.4}{35}{Immune dynamics}{section.4.4}{}} -\@writefile{toc}{\contentsline {subsection}{\numberline {4.4.1}Immunity from vaccination}{35}{subsection.4.4.1}\protected@file@percent } -\newlabel{immunity-from-vaccination}{{4.4.1}{35}{Immunity from vaccination}{subsection.4.4.1}{}} +\@writefile{toc}{\contentsline {subsection}{\numberline {4.4.1}Estimating Vaccination Rates}{35}{subsection.4.4.1}\protected@file@percent } +\newlabel{estimating-vaccination-rates}{{4.4.1}{35}{Estimating Vaccination Rates}{subsection.4.4.1}{}} \@writefile{lof}{\contentsline {figure}{\numberline {4.14}{\ignorespaces Relationship between WASH variables and cholera incidences.}}{36}{figure.4.14}\protected@file@percent } \newlabel{fig:wash-incidence}{{4.14}{36}{Relationship between WASH variables and cholera incidences}{figure.4.14}{}} \@writefile{lof}{\contentsline {figure}{\numberline {4.15}{\ignorespaces The optimized weighted mean of WASH variables for AFRO countries. 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--- a/MOSAIC-docs.toc +++ b/MOSAIC-docs.toc @@ -29,35 +29,36 @@ \contentsline {subsection}{\numberline {4.3.3}Shedding}{34}{subsection.4.3.3}% \contentsline {subsection}{\numberline {4.3.4}WAter, Sanitation, and Hygiene (WASH)}{34}{subsection.4.3.4}% \contentsline {section}{\numberline {4.4}Immune dynamics}{35}{section.4.4}% -\contentsline {subsection}{\numberline {4.4.1}Immunity from vaccination}{35}{subsection.4.4.1}% -\contentsline {subsection}{\numberline {4.4.2}Immunity from natural infection}{39}{subsection.4.4.2}% -\contentsline {section}{\numberline {4.5}Spatial dynamics}{39}{section.4.5}% -\contentsline {subsection}{\numberline {4.5.1}Human mobility model}{40}{subsection.4.5.1}% -\contentsline {subsection}{\numberline {4.5.2}Estimating the departure process}{43}{subsection.4.5.2}% -\contentsline {subsection}{\numberline {4.5.3}Estimating the diffusion process}{43}{subsection.4.5.3}% -\contentsline {subsection}{\numberline {4.5.4}The probability of spatial transmission}{43}{subsection.4.5.4}% -\contentsline {subsection}{\numberline {4.5.5}The spatial hazard}{46}{subsection.4.5.5}% -\contentsline {subsection}{\numberline {4.5.6}Coupling among locations}{46}{subsection.4.5.6}% -\contentsline {section}{\numberline {4.6}The observation process}{46}{section.4.6}% -\contentsline {subsection}{\numberline {4.6.1}Rate of symptomatic infection}{46}{subsection.4.6.1}% -\contentsline {subsection}{\numberline {4.6.2}Suspected cases}{49}{subsection.4.6.2}% -\contentsline {subsection}{\numberline {4.6.3}Case fatality rate}{50}{subsection.4.6.3}% -\contentsline {section}{\numberline {4.7}Demographics}{50}{section.4.7}% -\contentsline {section}{\numberline {4.8}The reproductive number}{54}{section.4.8}% -\contentsline {subsection}{\numberline {4.8.1}The generation time distribution}{54}{subsection.4.8.1}% -\contentsline {section}{\numberline {4.9}Initial conditions}{54}{section.4.9}% -\contentsline {section}{\numberline {4.10}Model calibration}{57}{section.4.10}% -\contentsline {section}{\numberline {4.11}Caveats}{57}{section.4.11}% -\contentsline {section}{\numberline {4.12}Table of parameters}{57}{section.4.12}% -\contentsline {section}{\numberline {4.13}References}{57}{section.4.13}% -\contentsline {chapter}{\numberline {5}Scenarios}{59}{chapter.5}% -\contentsline {section}{\numberline {5.1}Vaccination}{59}{section.5.1}% -\contentsline {subsection}{\numberline {5.1.1}Spatial and Temporal Strategies}{59}{subsection.5.1.1}% -\contentsline {subsection}{\numberline {5.1.2}Reactive Vaccination}{59}{subsection.5.1.2}% -\contentsline {section}{\numberline {5.2}Impacts of Climate Change}{60}{section.5.2}% -\contentsline {subsection}{\numberline {5.2.1}Severe Weather Events}{60}{subsection.5.2.1}% -\contentsline {subsection}{\numberline {5.2.2}Long-Term Trends}{60}{subsection.5.2.2}% -\contentsline {chapter}{\numberline {6}Usage}{61}{chapter.6}% -\contentsline {chapter}{\numberline {7}News}{63}{chapter.7}% -\contentsline {section}{\numberline {7.1}Past versions of MOSAIC}{63}{section.7.1}% -\contentsline {chapter}{\numberline {8}References}{65}{chapter.8}% +\contentsline {subsection}{\numberline {4.4.1}Estimating Vaccination Rates}{35}{subsection.4.4.1}% +\contentsline {subsection}{\numberline {4.4.2}Immunity from vaccination}{38}{subsection.4.4.2}% +\contentsline {subsection}{\numberline {4.4.3}Immunity from natural infection}{42}{subsection.4.4.3}% +\contentsline {section}{\numberline {4.5}Spatial dynamics}{43}{section.4.5}% +\contentsline {subsection}{\numberline {4.5.1}Human mobility model}{44}{subsection.4.5.1}% +\contentsline {subsection}{\numberline {4.5.2}Estimating the departure process}{46}{subsection.4.5.2}% +\contentsline {subsection}{\numberline {4.5.3}Estimating the diffusion process}{46}{subsection.4.5.3}% +\contentsline {subsection}{\numberline {4.5.4}The probability of spatial transmission}{47}{subsection.4.5.4}% +\contentsline {subsection}{\numberline {4.5.5}The spatial hazard}{47}{subsection.4.5.5}% +\contentsline {subsection}{\numberline {4.5.6}Coupling among locations}{47}{subsection.4.5.6}% +\contentsline {section}{\numberline {4.6}The observation process}{50}{section.4.6}% +\contentsline {subsection}{\numberline {4.6.1}Rate of symptomatic infection}{50}{subsection.4.6.1}% +\contentsline {subsection}{\numberline {4.6.2}Suspected cases}{50}{subsection.4.6.2}% +\contentsline {subsection}{\numberline {4.6.3}Case fatality rate}{52}{subsection.4.6.3}% +\contentsline {section}{\numberline {4.7}Demographics}{57}{section.4.7}% +\contentsline {section}{\numberline {4.8}The reproductive number}{57}{section.4.8}% +\contentsline {subsection}{\numberline {4.8.1}The generation time distribution}{57}{subsection.4.8.1}% +\contentsline {section}{\numberline {4.9}Initial conditions}{57}{section.4.9}% +\contentsline {section}{\numberline {4.10}Model calibration}{60}{section.4.10}% +\contentsline {section}{\numberline {4.11}Caveats}{60}{section.4.11}% +\contentsline {section}{\numberline {4.12}Table of parameters}{60}{section.4.12}% +\contentsline {section}{\numberline {4.13}References}{60}{section.4.13}% +\contentsline {chapter}{\numberline {5}Scenarios}{63}{chapter.5}% +\contentsline {section}{\numberline {5.1}Vaccination}{63}{section.5.1}% +\contentsline {subsection}{\numberline {5.1.1}Spatial and Temporal Strategies}{63}{subsection.5.1.1}% +\contentsline {subsection}{\numberline {5.1.2}Reactive Vaccination}{63}{subsection.5.1.2}% +\contentsline {section}{\numberline {5.2}Impacts of Climate Change}{64}{section.5.2}% +\contentsline {subsection}{\numberline {5.2.1}Severe Weather Events}{64}{subsection.5.2.1}% +\contentsline {subsection}{\numberline {5.2.2}Long-Term Trends}{64}{subsection.5.2.2}% +\contentsline {chapter}{\numberline {6}Usage}{65}{chapter.6}% +\contentsline {chapter}{\numberline {7}News}{67}{chapter.7}% +\contentsline {section}{\numberline {7.1}Past versions of MOSAIC}{67}{section.7.1}% +\contentsline {chapter}{\numberline {8}References}{69}{chapter.8}% diff --git a/_output.yml b/_output.yml index c3b0c5e..7cb9b77 100644 --- a/_output.yml +++ b/_output.yml @@ -3,7 +3,7 @@ bookdown::bs4_book: theme: primary: "#0167af" border-width: "0px" - repo: https://github.com/gilesjohnr/MOSAIC-docs + repo: https://github.com/InstituteforDiseaseModeling/MOSAIC-docs bookdown::pdf_book: includes: in_header: preamble.tex diff --git a/docs/04-model-description.md b/docs/04-model-description.md index 03ad811..01981ea 100644 --- a/docs/04-model-description.md +++ b/docs/04-model-description.md @@ -339,7 +339,7 @@ h_t = \text{LSTM}\big(\text{temperature}_{jt}, \ \text{precipitation}_{jt}, \ \t In this formulation, $h_t$ represents the hidden state generated by the LSTM network based on input variables such as temperature, precipitation, and ENSO conditions, while $b_h$ is a bias term added to the output of the hidden state transformation. -The deep learning LSTM model consists of three stacked LSTM-RNN layers. The first LSTM layer has 500 units and the second and third LSTM layers have 250 and 200 units respectively. The architecture the LSTM model is configured to pass node values to subsequent LSTM layers allowing deep learning of more the complex interactions among the climate variable over time. We enforced model sparsity for each LSTM layer using L2 regularization (penalty = 0.001) and used a dropout rate of 0.5 to further prevent overfitting on the limited amount of data. The final output layer was a dense layer with a single unit and a sigmoid activation function to produce a probability for binary classification, i.e. a prediction of environmental suitability $\psi_{jt}$ on a scale of 0 to 1. +The deep learning LSTM model consists of three stacked LSTM-RNN layers. The first LSTM layer has 500 units and the second and third LSTM layers have 250 and 100 units respectively. The architecture the LSTM model is configured to pass node values to subsequent LSTM layers allowing deep learning of more the complex interactions among the climate variable over time. We enforced model sparsity for each LSTM layer using L2 regularization (penalty = 0.001) and used a dropout rate of 0.5 for each LSTM layer to further prevent overfitting on the limited amount of data. The final output layer was a dense layer with a single unit and a sigmoid activation function to produce a probability value for binary classification, i.e. a prediction of environmental suitability $\psi_{jt}$ on a scale of 0 to 1. To fit the LSTM model to data, we modified the learning rate by applying an exponential decay schedule that started at 0.001 and decayed by a factor of 0.9 every 10,000 steps to enable smoother convergence. The model was compiled using the Adam optimizer with this learning rate schedule, along with binary cross-entropy as the loss function and accuracy as the evaluation metric. The model was trained for a maximum of 200 epochs with a batch size of 1024. We allowed model fitting to stop early with a patience parameter of 10 which halts training if no improvement is observed in validation accuracy for 10 consecutive epochs. To train the model we set aside 20% of the observed data for validation and also used 20% of the training data for model fitting. The training history, including loss and accuracy, was monitored over the course of training and gave a final test accuracy of 0.73 and a final test loss of 0.56 (see Figure \@ref(fig:lstm-model-fit)). @@ -348,7 +348,7 @@ To fit the LSTM model to data, we modified the learning rate by applying an expo

(\#fig:lstm-model-fit)Model performance on training and validation data.

-After model training was completed, we predicted the values of environmental suitability $\psi_{jt}$ across all time steps for each location. Predictions start in January 1970 and go up to 5 months past the present date (currently February 2025). Given the amount of noise in the model predictions, we added a simple LOESS spline with logit transformation to smooth model predictions over time and give a more stable value of $\psi_{jt}$ when incorporating it into other model features (e.g. Equations \@ref(eq:beta2) and \@ref(eq:delta)). The resulting model predicitons are shown for an example country such as Mozambique in Figure \@ref(fig:psi-prediction-data) which compares model predictions to the original case counts and the binary classification. Predicitons for all model locations are shown in a simplified view in Figure \@ref(fig:psi-prediction-countries). +After model training was completed, we predicted the values of environmental suitability $\psi_{jt}$ across all time steps for each location. Predictions start in January 1970 and go up to 5 months past the present date (currently February 2025). Given the amount of noise in the model predictions, we added a simple LOESS spline with logit transformation to smooth model predictions over time and give a more stable value of $\psi_{jt}$ when incorporating it into other model features (e.g. Equations \@ref(eq:beta2) and \@ref(eq:delta)). The resulting model predictions are shown for an example country such as Mozambique in Figure \@ref(fig:psi-prediction-data) which compares model predictions to the original case counts and the binary classification. Predicitons for all model locations are shown in a simplified view in Figure \@ref(fig:psi-prediction-countries). *Also, please note that this initial version of the model is fitted to a rather small amount of data. Model hyper parameters were specifically chosen to reduce overfitting. Therefore, we recommend to not over-interpret the time series predictions of the model at this early stage since they are likely to change and improve as more historical incidence data is included in future versions.* @@ -440,6 +440,35 @@ To parameterize $\theta_j$, we calculated a weighted mean of the 8 WASH variable ## Immune dynamics +Aside from the current number of infections, population susceptibility is one of the key factors influencing the spread of cholera. Further, since immunity from both vaccination and natural infection provides long-lasting protection, it's crucial to quantify not only the incidence of cholera but also the number of past vaccinations. Additionally, we need to estimate how many individuals with immunity remain in the population at any given time step in the model. + +To achieve this, we estimate the vaccination rate over time ($\nu_{jt}$) based on historical vaccination campaigns and incorporate a model of vaccine effectiveness ($\phi$) and immune decay post-vaccination ($\omega$) to estimate the current number of individuals with vaccine-derived immunity. We also account for the immune decay rate from natural infection ($\varepsilon$), which is generally considered to last longer than immunity from vaccination. + +### Estimating Vaccination Rates + +To estimate the past and current vaccination rates, we sourced data on reported OCV vaccinations from the WHO [International Coordinating Group](https://www.who.int/groups/icg) (ICG) [Cholera vaccine dashboard](https://app.powerbi.com/view?r=eyJrIjoiYmFmZTBmM2EtYWM3Mi00NWYwLTg3YjgtN2Q0MjM5ZmE1ZjFkIiwidCI6ImY2MTBjMGI3LWJkMjQtNGIzOS04MTBiLTNkYzI4MGFmYjU5MCIsImMiOjh9). This resource lists all reactive OCV campaigns conducted from 2016 to the present, with approximately 103 million OCV doses shipped to sub-Saharan African (SSA) countries as of October 9, 2024. However, these data only capture reactive vaccinations in emergency settings and do not include the many preventive campaigns organized by GAVI and in-country partners. + +*As a result, our current estimates of the OCV vaccination rate likely underestimate total OCV coverage. We are working to expand our data sources to better reflect the full number of OCV doses distributed in SSA.* + +To translate the reported number of OCV doses into the model parameter $\nu_{jt}$, we take the number of doses shipped and the reported start date of the vaccination campaign, distributing the doses over subsequent days according to a maximum daily vaccination rate. See Figure \@ref(fig:vaccination-example) for an example of OCV distribution using a maximum daily vaccination rate of 100,000. The resulting time series for each country is shown in Figure \@ref(fig:vaccination-countries), with current totals based on the WHO ICG data displayed in Figure \@ref(fig:vaccination-maps). + +
+Example of the estimated vaccination rate during an OCV campaign. +

(\#fig:vaccination-example)Example of the estimated vaccination rate during an OCV campaign.

+
+ +
+The estimated vaccination coverage across all countries with reported vaccination data one the WHO ICG dashboard. +

(\#fig:vaccination-countries)The estimated vaccination coverage across all countries with reported vaccination data one the WHO ICG dashboard.

+
+ +
+The total cumulative number of OCV doses distributed through the WHO ICG from 2016 to present day. +

(\#fig:vaccination-maps)The total cumulative number of OCV doses distributed through the WHO ICG from 2016 to present day.

+
+ + + ### Immunity from vaccination The impacts of Oral Cholera Vaccine (OCV) campaigns is incorporated into the model through the Vaccinated compartment (V). The rate that individuals are effectively vaccinated is defined as $\phi\nu_tS_{jt}$, where $S_{jt}$ are the available number of susceptible individuals in location $j$ at time $t$, $\nu_t$ is the number of OCV doses administered at time $t$ and $\phi$ is the estimated vaccine effectiveness. Note that there is just one vaccinated compartment at this time, though future model versions may include $V_1$ an $V_2$ compartments to explore two dose vaccination strategies or to emulate more complex waning patterns. diff --git a/docs/MOSAIC-docs.epub b/docs/MOSAIC-docs.epub index a58d6c3..1092571 100644 Binary files a/docs/MOSAIC-docs.epub and b/docs/MOSAIC-docs.epub differ diff --git a/docs/MOSAIC-docs.pdf b/docs/MOSAIC-docs.pdf index 0f9b4df..d94625b 100644 Binary files a/docs/MOSAIC-docs.pdf and b/docs/MOSAIC-docs.pdf differ diff --git a/docs/MOSAIC-docs.tex b/docs/MOSAIC-docs.tex index 21dc8e9..93eba0d 100644 --- a/docs/MOSAIC-docs.tex +++ b/docs/MOSAIC-docs.tex @@ -114,7 +114,7 @@ \chapter*{}\label{section} \hfill\break {\emph{ -Website under development. Last compiled on 2024-10-05 at 05:56 PM PDT. +Website under development. Last compiled on 2024-10-21 at 03:55 PM PDT. }} \section*{Welcome}\label{welcome} @@ -561,7 +561,7 @@ \subsubsection{Deep learning neural network model}\label{deep-learning-neural-ne In this formulation, \(h_t\) represents the hidden state generated by the LSTM network based on input variables such as temperature, precipitation, and ENSO conditions, while \(b_h\) is a bias term added to the output of the hidden state transformation. -The deep learning LSTM model consists of three stacked LSTM-RNN layers. The first LSTM layer has 500 units and the second and third LSTM layers have 250 and 200 units respectively. The architecture the LSTM model is configured to pass node values to subsequent LSTM layers allowing deep learning of more the complex interactions among the climate variable over time. We enforced model sparsity for each LSTM layer using L2 regularization (penalty = 0.001) and used a dropout rate of 0.5 to further prevent overfitting on the limited amount of data. The final output layer was a dense layer with a single unit and a sigmoid activation function to produce a probability for binary classification, i.e.~a prediction of environmental suitability \(\psi_{jt}\) on a scale of 0 to 1. +The deep learning LSTM model consists of three stacked LSTM-RNN layers. The first LSTM layer has 500 units and the second and third LSTM layers have 250 and 100 units respectively. The architecture the LSTM model is configured to pass node values to subsequent LSTM layers allowing deep learning of more the complex interactions among the climate variable over time. We enforced model sparsity for each LSTM layer using L2 regularization (penalty = 0.001) and used a dropout rate of 0.5 for each LSTM layer to further prevent overfitting on the limited amount of data. The final output layer was a dense layer with a single unit and a sigmoid activation function to produce a probability value for binary classification, i.e.~a prediction of environmental suitability \(\psi_{jt}\) on a scale of 0 to 1. To fit the LSTM model to data, we modified the learning rate by applying an exponential decay schedule that started at 0.001 and decayed by a factor of 0.9 every 10,000 steps to enable smoother convergence. The model was compiled using the Adam optimizer with this learning rate schedule, along with binary cross-entropy as the loss function and accuracy as the evaluation metric. The model was trained for a maximum of 200 epochs with a batch size of 1024. We allowed model fitting to stop early with a patience parameter of 10 which halts training if no improvement is observed in validation accuracy for 10 consecutive epochs. To train the model we set aside 20\% of the observed data for validation and also used 20\% of the training data for model fitting. The training history, including loss and accuracy, was monitored over the course of training and gave a final test accuracy of 0.73 and a final test loss of 0.56 (see Figure \ref{fig:lstm-model-fit}). @@ -574,7 +574,7 @@ \subsubsection{Deep learning neural network model}\label{deep-learning-neural-ne \caption{Model performance on training and validation data.}\label{fig:lstm-model-fit} \end{figure} -After model training was completed, we predicted the values of environmental suitability \(\psi_{jt}\) across all time steps for each location. Predictions start in January 1970 and go up to 5 months past the present date (currently February 2025). Given the amount of noise in the model predictions, we added a simple LOESS spline with logit transformation to smooth model predictions over time and give a more stable value of \(\psi_{jt}\) when incorporating it into other model features (e.g.~Equations \eqref{eq:beta2} and \eqref{eq:delta}). The resulting model predicitons are shown for an example country such as Mozambique in Figure \ref{fig:psi-prediction-data} which compares model predictions to the original case counts and the binary classification. Predicitons for all model locations are shown in a simplified view in Figure \ref{fig:psi-prediction-countries}. +After model training was completed, we predicted the values of environmental suitability \(\psi_{jt}\) across all time steps for each location. Predictions start in January 1970 and go up to 5 months past the present date (currently February 2025). Given the amount of noise in the model predictions, we added a simple LOESS spline with logit transformation to smooth model predictions over time and give a more stable value of \(\psi_{jt}\) when incorporating it into other model features (e.g.~Equations \eqref{eq:beta2} and \eqref{eq:delta}). The resulting model predictions are shown for an example country such as Mozambique in Figure \ref{fig:psi-prediction-data} which compares model predictions to the original case counts and the binary classification. Predicitons for all model locations are shown in a simplified view in Figure \ref{fig:psi-prediction-countries}. \emph{Also, please note that this initial version of the model is fitted to a rather small amount of data. Model hyper parameters were specifically chosen to reduce overfitting. Therefore, we recommend to not over-interpret the time series predictions of the model at this early stage since they are likely to change and improve as more historical incidence data is included in future versions.} @@ -659,6 +659,45 @@ \subsection{WAter, Sanitation, and Hygiene (WASH)}\label{water-sanitation-and-hy \section{Immune dynamics}\label{immune-dynamics} +Aside from the current number of infections, population susceptibility is one of the key factors influencing the spread of cholera. Further, since immunity from both vaccination and natural infection provides long-lasting protection, it's crucial to quantify not only the incidence of cholera but also the number of past vaccinations. Additionally, we need to estimate how many individuals with immunity remain in the population at any given time step in the model. + +To achieve this, we estimate the vaccination rate over time (\(\nu_{jt}\)) based on historical vaccination campaigns and incorporate a model of vaccine effectiveness (\(\phi\)) and immune decay post-vaccination (\(\omega\)) to estimate the current number of individuals with vaccine-derived immunity. We also account for the immune decay rate from natural infection (\(\varepsilon\)), which is generally considered to last longer than immunity from vaccination. + +\subsection{Estimating Vaccination Rates}\label{estimating-vaccination-rates} + +To estimate the past and current vaccination rates, we sourced data on reported OCV vaccinations from the WHO \href{https://www.who.int/groups/icg}{International Coordinating Group} (ICG) \href{https://app.powerbi.com/view?r=eyJrIjoiYmFmZTBmM2EtYWM3Mi00NWYwLTg3YjgtN2Q0MjM5ZmE1ZjFkIiwidCI6ImY2MTBjMGI3LWJkMjQtNGIzOS04MTBiLTNkYzI4MGFmYjU5MCIsImMiOjh9}{Cholera vaccine dashboard}. This resource lists all reactive OCV campaigns conducted from 2016 to the present, with approximately 103 million OCV doses shipped to sub-Saharan African (SSA) countries as of October 9, 2024. However, these data only capture reactive vaccinations in emergency settings and do not include the many preventive campaigns organized by GAVI and in-country partners. + +\emph{As a result, our current estimates of the OCV vaccination rate likely underestimate total OCV coverage. We are working to expand our data sources to better reflect the full number of OCV doses distributed in SSA.} + +To translate the reported number of OCV doses into the model parameter \(\nu_{jt}\), we take the number of doses shipped and the reported start date of the vaccination campaign, distributing the doses over subsequent days according to a maximum daily vaccination rate. See Figure \ref{fig:vaccination-example} for an example of OCV distribution using a maximum daily vaccination rate of 100,000. The resulting time series for each country is shown in Figure \ref{fig:vaccination-countries}, with current totals based on the WHO ICG data displayed in Figure \ref{fig:vaccination-maps}. + +\begin{figure} + +{\centering \includegraphics[width=1\linewidth]{figures/vaccination_example_ZMB} + +} + +\caption{Example of the estimated vaccination rate during an OCV campaign.}\label{fig:vaccination-example} +\end{figure} + +\begin{figure} + +{\centering \includegraphics[width=1\linewidth]{figures/vaccination_by_country} + +} + +\caption{The estimated vaccination coverage across all countries with reported vaccination data one the WHO ICG dashboard.}\label{fig:vaccination-countries} +\end{figure} + +\begin{figure} + +{\centering \includegraphics[width=1\linewidth]{figures/vaccination_maps} + +} + +\caption{The total cumulative number of OCV doses distributed through the WHO ICG from 2016 to present day.}\label{fig:vaccination-maps} +\end{figure} + \subsection{Immunity from vaccination}\label{immunity-from-vaccination} The impacts of Oral Cholera Vaccine (OCV) campaigns is incorporated into the model through the Vaccinated compartment (V). The rate that individuals are effectively vaccinated is defined as \(\phi\nu_tS_{jt}\), where \(S_{jt}\) are the available number of susceptible individuals in location \(j\) at time \(t\), \(\nu_t\) is the number of OCV doses administered at time \(t\) and \(\phi\) is the estimated vaccine effectiveness. Note that there is just one vaccinated compartment at this time, though future model versions may include \(V_1\) an \(V_2\) compartments to explore two dose vaccination strategies or to emulate more complex waning patterns. diff --git a/docs/data.html b/docs/data.html index 01ae5ff..3fa35d0 100644 --- a/docs/data.html +++ b/docs/data.html @@ -79,7 +79,7 @@

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diff --git a/docs/estimating-vaccination-rates.html b/docs/estimating-vaccination-rates.html new file mode 100644 index 0000000..ad83a49 --- /dev/null +++ b/docs/estimating-vaccination-rates.html @@ -0,0 +1,2658 @@ + + + + + + + 6 Estimating Vaccination Rates | MOSAIC: a spatial model of endemic cholera + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+
+
+ Skip to main content + +
+

+ MOSAIC: a spatial model of endemic cholera +

+ +
+ + +
+ +
+
+

6 Estimating Vaccination Rates

+

To estimate the past and current vaccination rates, we sourced data on reported OCV vaccinations from the WHO International Coordinating Group (ICG) Cholera vaccine dashboard. This resource lists all reactive OCV campaigns conducted from 2016 to the present, with approximately 103 million OCV doses shipped to sub-Saharan African (SSA) countries as of October 9, 2024. However, these data only capture reactive vaccinations in emergency settings and do not include the many preventive campaigns organized by GAVI and in-country partners.

+

As a result, our current estimates of the OCV vaccination rate likely underestimate total OCV coverage. We are working to expand our data sources to better reflect the full number of OCV doses distributed in SSA.

+

To translate the reported number of OCV doses into the model parameter \(\nu_{jt}\), we take the number of doses shipped and the reported start date of the vaccination campaign, distributing the doses over subsequent days according to a maximum daily vaccination rate. See Figure 6.1 for an example of OCV distribution using a maximum daily vaccination rate of 100,000. The resulting time series is shown in Figure 6.2, with current totals based on WHO ICG data displayed in Figure 6.3.

+
+

6.0.1 Vaccination rate

+

Aside from the number of current infections, population susceptibility is one of the most important factors that determine disease spread. Since immunity from both vaccination and natural infection confer long lasting immunity, it is therefore important to quantify the number of past vaccinations—in addition to estimates of cholera incidence—and how many vaccination individuals are estimated to be in the population at a given time step in the model. Therefore, we estimate the vaccination rate over time (\(\nu_{jt}\)) based on past vaccination campaigns and use a model of immune decay after vaccination (\(\omega\)) to estimate the current number of individuals with vaccine derived immunity. We also estimate the rate of immune decay from natural infection (\(\varepsilon\)) which is considered to be longer that immunity from vaccination.

+

To estimate the current and past vaccination rate, we sourced reported OCV vaccinations from the WHO International Coordinating Group (ICG) Cholera vaccine dashboard. This provides all reactive OCV campaigns undertaken from 2016 to present which we estimate be approximately 103 million OCV doses shipped to SSA countries as of October 09, 2024. Note that these data give reactive vaccinations in emergency settings and do include the many preventative campaigns that have occurred through GAVI and in country partners. Therefore, our estimates of OCV vaccination rate underestimate total OCV coverage. However, we are working to expand our data sources here to better represent the total number of OCV doses that have been distributed in SSA.

+

To convert the reported number of OCV doses into the model parameter \(\nu_{jt}\), we take the number of OCV doses shipped and the reported start date of the vaccination campaign and distribute the shipped doses across subsequent days based a maximum rate of vaccinations per day. See Figure 6.1 for an example of OCV distribution with a maximum daily vaccination rate of 100,000. The resulting time series can be seen in Figure 6.2 with the current totals based on the WHO ICG data shown in Figure 6.3.

+
+Example of the estimated vaccination rate during an OCV campaign. +

+Figure 6.1: Example of the estimated vaccination rate during an OCV campaign. +

+
+
+The estimated vaccination coverage across all countries with reported vaccination data one the WHO ICG dashboard. +

+Figure 6.2: The estimated vaccination coverage across all countries with reported vaccination data one the WHO ICG dashboard. +

+
+
+The total cumulative number of OCV doses distributed through the WHO ICG from 2016 to present day. +

+Figure 6.3: The total cumulative number of OCV doses distributed through the WHO ICG from 2016 to present day. +

+
+
+
+

6.0.2 Immunity from vaccination

+

The impacts of Oral Cholera Vaccine (OCV) campaigns is incorporated into the model through the Vaccinated compartment (V). The rate that individuals are effectively vaccinated is defined as \(\phi\nu_tS_{jt}\), where \(S_{jt}\) are the available number of susceptible individuals in location \(j\) at time \(t\), \(\nu_t\) is the number of OCV doses administered at time \(t\) and \(\phi\) is the estimated vaccine effectiveness. Note that there is just one vaccinated compartment at this time, though future model versions may include \(V_1\) an \(V_2\) compartments to explore two dose vaccination strategies or to emulate more complex waning patterns.

+

The vaccination rate \(\nu_t\) is not an estimated quantity. Rather, it is directly defined by the reported number of OCV doses administered on the WHO OCV dashboard here: https://www.who.int/groups/icg/cholera.

+

\[ +\nu_t := \text{Reported rate of OCV administration} +\]

+

The evidence for waning immunity comes from 4 cohort studies (Table 6.1) from Bangladesh (Qadri et al 2016 and 2018), South Sudan (Azman et al 2016), and Democratic Republic of Congo (Malembaka et al 2024).

+ + +++++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Table 6.1: Summary of Effectiveness Data
EffectivenessUpper CILower CIDay (midpoint)Day (min)Day (max)Source
60.00.8730.9900.702NANAAzman et al (2016)
93.50.4000.6000.1107180Qadri et al (2016)
368.50.3900.5200.2307730Qadri et al (2018)
435.00.5270.6740.314360510Malembaka et al (2024)
900.00.4470.5940.2487201080Malembaka et al (2024)
+

We estimated vaccine effectiveness and waning immunity by fitting an exponential decay model to the reported effectiveness of one dose OCV in these studies using the following formulation:

+

\[\begin{equation} +\text{Proportion immune}\ t \ \text{days after vaccination} = \phi \times (1 - \omega) ^ {t-t_{\text{vaccination}}} +\tag{6.1} +\end{equation}\]

+

Where \(\phi\) is the effectiveness of one dose OCV, and the based on this specification, it is also the initial proportion immune directly after vaccination. The decay rate parameter \(\omega\) is the rate at which initial vaccine derived immunity decays per day post vaccination, and \(t\) and \(t_{\text{vaccination}}\) are the time (in days) the function is evaluated at and the time of vaccination respectively. When we fitted the model to the data from the cohort studies shown in Table (6.1) we found that \(\omega = 0.00057\) (\(0-0.0019\) 95% CI), which gives a mean estimate of 4.8 years for vaccine derived immune duration with unreasonably large confidence intervals (1.4 years to infinite immunity). However, the point estimate of 4.8 years is consistent with anecdotes that one dose OCV is effective for up to at least 3 years.

+

The wide confidence intervals are likely due to the wide range of reported estimates for proportion immune after a short duration in the 7–90 days range (Azman et al 2016 and Qadri et al 2016). Therefore, we chose to use the point estimate of \(\omega\) and incorporate uncertainty based on the initial proportion immune (i.e. vaccine effectiveness \(\phi\)) shortly after vaccination. Using the decay model in Equation (6.1) we estimated \(\phi\) to be \(0.64\) (\(0.32-0.96\) 95% CI). We then fit a Beta distribution to the quantiles of \(\phi\) by minimizing the sums of squares using the Nelder-Mead optimization algorithm to render the following distribution (shown in Figure 6.4B):

+

\[\begin{equation} +\phi \sim \text{Beta}(4.57, 2.41). +\tag{6.2} +\end{equation}\]

+
+This is vaccine effectiveness +

+Figure 6.4: This is vaccine effectiveness +

+
+
+
+

6.0.3 Immunity from natural infection

+

The duration of immunity after a natural infection is likely to be longer lasting than that from vaccination with OCV (especially given the current one dose strategy). As in most SIR-type models, the rate at which individuals leave the Recovered compartment is governed by the immune decay parameter \(\varepsilon\). We estimated the durability of immunity from natural infection based on two cohort studies and fit the following exponential decay model to estimate the rate of immunity decay over time:

+

\[ +\text{Proportion immune}\ t \ \text{days after infection} = 0.99 \times (1 - \varepsilon) ^ {t-t_{\text{infection}}} +\] +Where we make the necessary and simplifying assumption that within 0–90 days after natural infection with V. cholerae, individuals are 95–99% immune. We fit this model to reported data from Ali et al (2011) and Clemens et al (1991) (see Table 6.2).

+ + +++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Table 6.2: Sources for the duration of immunity fro natural infection.
DayEffectivenessUpper CILower CISource
900.950.950.95Assumption
10800.650.810.37Ali et al (2011)
12600.610.810.21Clemens et al (1991)
+

We estimated the mean immune decay to be \(\bar\varepsilon = 3.9 \times 10^{-4}\) (\(1.7 \times 10^{-4}-1.03 \times 10^{-3}\) 95% CI) which is equivalent to an immune duration of \(7.21\) years (\(2.66-16.1\) years 95% CI) as shown in Figure 6.5A. This is slightly longer than previous modeling work estimating the duration of immunity to be ~5 years (King et al 2008). Uncertainty around \(\varepsilon\) in the model is then represented by a Log-Normal distribution as shown in Figure 6.5B:

+

\[ +\varepsilon \sim \text{Lognormal}(\bar\varepsilon+\frac{\sigma^2}{2}, 0.25) +\]

+
+The duration of immunity after natural infection with *V. cholerae*. +

+Figure 6.5: The duration of immunity after natural infection with V. cholerae. +

+
+
+
+

6.1 Spatial dynamics

+

The parameters in the model diagram in Figure 5.2 that have a \(jt\) subscript denote the spatial structure of the model. Each country is modeled as an independent metapopulation that is connected to all others via the spatial force of infection \(\Lambda_{jt}\) which moves contagion among metapopulations according to the connectivity provided by parameters \(\tau_i\) (the probability departure) and \(\pi_{ij}\) (the probability of diffusion to destination \(j\)). Both parameters are estimated using the departure-diffusion model below which is fitted to average weekly air traffic volume between all of the 41 countries included in the MOSAIC framework (Figure 6.6).

+
+The average number of air passengers per week in 2017 among all countries. +

+Figure 6.6: The average number of air passengers per week in 2017 among all countries. +

+
+
+A network map showing the average number of air passengers per week in 2017. +

+Figure 6.7: A network map showing the average number of air passengers per week in 2017. +

+
+
+

6.1.1 Human mobility model

+

The departure-diffusion model estimates diagonal and off-diagonal elements in the mobility matrix (\(M\)) separately and combines them using conditional probability rules. The model first estimates the probability of travel outside the origin location \(i\)—the departure process—and then the distribution of travel from the origin location \(i\) by normalizing connectivity values across all \(j\) destinations—the diffusion process. The values of \(\pi_{ij}\) sum to unity along each row, but the diagonal is not included, indicating that this is a relative quantity. That is to say, \(\pi_{ij}\) gives the probability of going from \(i\) to \(j\) given that travel outside origin \(i\) occurs. Therefore, we can use basic conditional probability rules to define the travel routes in the diagonal elements (trips made within the origin \(i\)) as +\[ +\Pr( \neg \text{depart}_i ) = 1 - \tau_i +\] +and the off-diagonal elements (trips made outside origin \(i\)) as +\[ +\Pr( \text{depart}_i, \text{diffuse}_{i \rightarrow j}) = \Pr( \text{diffuse}_{i \rightarrow j} \mid \text{depart}_i ) \Pr(\text{depart}_i ) = \pi_{ij} \tau_i. +\] +The expected mean number of trips for route \(i \rightarrow j\) is then:

+

\[\begin{equation} +M_{ij} = +\begin{cases} +\theta N_i (1-\tau_i) \ & \text{if} \ i = j \\ +\theta N_i \tau_i \pi_{ij} \ & \text{if} \ i \ne j. +\end{cases} +\tag{6.3} +\end{equation}\]

+

Where, \(\theta\) is a proportionality constant representing the overall number of trips per person in an origin population of size \(N_i\), \(\tau_i\) is the probability of leaving origin \(i\), and \(\pi_{ij}\) is the probability of travel to destination \(j\) given that travel outside origin \(i\) occurs.

+
+
+

6.1.2 Estimating the departure process

+

The probability of travel outside the origin is estimated for each location \(i\) to give the location-specific departure probability \(\tau_i\). +\[ +\tau_i \sim \text{Beta}(1+s, 1+r) +\] +Binomial probabilities for each origin \(\tau_i\) are drawn from a Beta distributed prior with shape (\(s\)) and rate (\(r\)) parameters. +\[ +\begin{aligned} +s &\sim \text{Gamma}(0.01, 0.01)\\ +r &\sim \text{Gamma}(0.01, 0.01) +\end{aligned} +\]

+
+
+

6.1.3 Estimating the diffusion process

+

We use a normalized formulation of the power law gravity model to defined the diffusion process, the probability of travelling to destination \(j\) given travel outside origin \(i\) (\(\pi_{ij}\)) which is defined as:

+

\[\begin{equation} +\pi_{ij} = \frac{ +N_j^\omega d_{ij}^{-\gamma} +}{ +\sum\limits_{\forall j \ne i} N_j^\omega d_{ij}^{-\gamma} +} +\tag{6.4} +\end{equation}\]

+

Where, \(\omega\) scales the attractive force of each \(j\) destination based on its population size \(N_j\). The kernel function \(d_{ij}^{-\gamma}\) serves as a penalty on the proportion of travel from \(i\) to \(j\) based on distance. Prior distributions of diffusion model parameters are defined as: +\[ +\begin{aligned} +\omega &\sim \text{Gamma}(1, 1)\\ +\gamma &\sim \text{Gamma}(1, 1) +\end{aligned} +\]

+

The models for \(\tau_i\) and \(\pi_{ij}\) were fitted to air traffic data from OAG using the mobility R package (Giles 2020). Estimates for mobility model parameters are shown in Figures 6.8 and 6.9.

+
+The estimated weekly probability of travel outside of each origin location $\tau_i$ and 95% confidence intervals is shown in panel A with the population mean indicated as a red dashed line. Panel B shows the estimated total number of travelers leaving origin $i$ each week. +

+Figure 6.8: The estimated weekly probability of travel outside of each origin location \(\tau_i\) and 95% confidence intervals is shown in panel A with the population mean indicated as a red dashed line. Panel B shows the estimated total number of travelers leaving origin \(i\) each week. +

+
+
+The diffusion process $\pi_{ij}$ which gives the estimated probability of travel from origin $i$ to destination $j$ given that travel outside of origin $i$ has occurred. +

+Figure 6.9: The diffusion process \(\pi_{ij}\) which gives the estimated probability of travel from origin \(i\) to destination \(j\) given that travel outside of origin \(i\) has occurred. +

+
+
+
+

6.1.4 The probability of spatial transmission

+

The likelihood of introductions of cholera from disparate locations is a major concern during cholera outbreaks. However, this can be difficult to characterize given the endemic dynamics and patterns of human movement. We include a few measures of spatial heterogeneity here and the first is a simple importation probability based on connectivity and the possibility of incoming infections. The basic probability of transmission from an origin \(i\) to a particular destination \(j\) and time \(t\) is defined as:

+

\[\begin{equation} +p(i,j,t) = 1 - e^{-\beta_{jt}^{\text{hum}} (((1-\tau_j)S_{jt})/N_{jt}) \pi_{ij}\tau_iI_{it}} +\tag{6.5} +\end{equation}\]

+
+
+

6.1.5 The spatial hazard

+

Although we are more concerned with endemic dynamics here, there are likely to be periods of time early in the rainy season where cholera cases and the rate of transmission is low enough for spatial spread to resemble epidemic dynamics for a time. During such times periods, we can estimate the arrival time of contagion for any location where cases are yet to be reported. We do this be estimating the spatial hazard of transmission:

+

\[\begin{equation} +h(j,t) = \frac{ +\beta_{jt}^{\text{hum}} \Big(1 - \exp\big(-((1-\tau_j)S_{jt}/N_{jt}) \sum_{\forall i \not= j} \pi_{ij}\tau_i (I_{it}/N_{it}) \big) \Big) +}{ +1/\big(1 + \beta_{jt}^{\text{hum}} (1-\tau_j)S_{jt}\big) +}. +\tag{6.6} +\end{equation}\]

+

And then normalizing to give the waiting time distribution for all locations:

+

\[\begin{equation} +w(j,t) = h(j,T) \prod_{t=1}^{T-1}1-h(j,t). +\tag{6.7} +\end{equation}\]

+
+
+

6.1.6 Coupling among locations

+

Another measure of spatial heterogeneity is to quantify the coupling of disease dynamics among metapopulations using a correlation coefficient. Here, we use the definition of spatial correlation between locations \(i\) and \(j\) as \(C_{ij}\) described in Keeling and Rohani (2002), which gives a measure of how similar infection dynamics are between locations.

+

\[\begin{equation} +C_{ij} = \frac{ +( y_{it} - \bar{y}_i )( y_{jt} - \bar{y}_j ) +}{ +\sqrt{\text{var}(y_i) \text{var}(y_j)} +} +\tag{6.8} +\end{equation}\] +Where \(y_{it} = I_{it}/N_i\) and \(y_{jt} = I_{jt}/N_j\). Mean prevalence in each location is \(\bar{y_i} = \frac{1}{T} \sum_{t=1}^{T} y_{it}\) and \(\bar{y_j} = \frac{1}{T} \sum_{t=1}^{T} y_{jt}\).

+
+
+
+

6.2 The observation process

+
+

6.2.1 Rate of symptomatic infection

+

The presentation of infection with V. cholerae can be extremely variable. The severity of infection depends many factors such as the amount of the infectious dose, the age of the host, the level of immunity of the host either through vaccination or previous infection, and naivety to the particular strain of V. cholerae. Additional circumstantial factors such as nutritional status and overall pathogen burden may also impact infection severity. At the population level, the observed proportion of infections that are symptomatic is also dependent on the endemicity of cholera in the region. Highly endemic areas (e.g. parts of Bangladesh; Hegde et al 2024) may have a very low proportion of symptomatic infections due to many previous exposures. Inversely, populations that are largely naive to V. cholerae will exhibit a relatively higher proportion of symptomatic infections (e.g. Haiti; Finger et al 2024).

+

Accounting for all of these nuances in the first version of this model not possible, but we can past studies do contain some information that can help to set some sensible bounds on our definition for the proportion of infections that are symptomatic (\(\sigma\)). So we have compiled a short list of studies that have done sero-surveys and cohort studies to assess the likelihood of symptomatic infections in different locations and displayed those results in Table (6.3).

+

To provide a reasonably informed prior for the proportion of infections that are symptomatic, we calculated the combine mean and confidence intervals of all studies in Table 6.3 and fit a Beta distribution that corresponds to these quantiles using least-squares and a Nelder-Mead algorithm. The resulting prior distribution for the symptomatic proportion \(\sigma\) is:

+

\[\begin{equation} +\sigma \sim \text{Beta}(4.30, 13.51) +\end{equation}\]

+ + ++++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Table 6.3: Summary of Studies on Cholera Immunity
MeanLow CIHigh CILocationSourceNote
0.570NANANANelson et al (2009)Review
NA1.0000.250NALueng & Matrajt (2021)Review
NA0.6000.200Endemic regionsHarris et al (2012)Review
0.2380.2500.227HaitiFinger et al (2024)Sero-survey and clinical data
0.2130.2310.194HaitiJackson et al (2013)Cross-sectional sero-survey
0.204NANAPakistanBart et al (1970)Sero-survey during epidemic; El Tor Ogawa strain
0.371NANAPakistanBart et al (1970)Sero-survey during epidemic; Inaba strain
0.1840.2560.112BangladeshHarris et al (2008)Household cohort; mean of all age groups
0.0010.0000.001BangladeshHegde et al (2024)Sero-survey and clinical data
+

The prior distribution for \(\sigma\) is plotted in Figure 6.10A with the reported values of the proportion symptomatic from previous studies shown in 6.10B.

+
+Proportion of infections that are symptomatic. +

+Figure 6.10: Proportion of infections that are symptomatic. +

+
+
+
+

6.2.2 Suspected cases

+

The clinical presentation of diarrheal diseases is often similar across various pathogens, which can lead to systematic biases in the reported number of cholera cases. It is anticipated that the number of suspected cholera cases is related to the actual number of infections by a factor of \(1/\rho\), where \(\rho\) represents the proportion of suspected cases that are true infections. To adjust for this bias, we use estimates from the meta-analysis by Weins et al. (2023), which suggests that suspected cholera cases outnumber true infections by approximately 2 to 1, with a mean across studies indicating that 52% (24-80% 95% CI) of suspected cases are actual cholera infections. A higher estimate was reported for ourbreak settings (78%, 40-99% 95% CI). To account for the variability in this estimate, we fit a Beta distribution to the reported quantiles using a least squares approach and the Nelder-Mead algorithm, resulting in the prior distribution shown in Figure 6.11B:

+

\[\begin{equation} +\rho \sim \text{Beta}(4.79, 1.53). +\end{equation}\]

+
+Proportion of suspected cholera cases that are true infections. Panel A shows the 'low' assumption which estimates across all settings: $\rho \sim \text{Beta}(5.43, 5.01)$. Panel B shows the 'high' assumption where the estimate reflects high-quality studies during outbreaks: $\rho \sim \text{Beta}(4.79, 1.53)$ +

+Figure 6.11: Proportion of suspected cholera cases that are true infections. Panel A shows the ‘low’ assumption which estimates across all settings: \(\rho \sim \text{Beta}(5.43, 5.01)\). Panel B shows the ‘high’ assumption where the estimate reflects high-quality studies during outbreaks: \(\rho \sim \text{Beta}(4.79, 1.53)\) +

+
+
+
+

6.2.3 Case fatality rate

+

The Case Fatality Rate (CFR) among symptomatic infections was calculated using reported cases and deaths data from January 2021 to August 2024. The data were collated from various issues of the WHO Weekly Epidemiological Record the Global Cholera and Acute Watery Diarrhea (AWD) Dashboard (see Data section) which provide annual aggregations of reported cholera cases and deaths. We then used the Binomial exact test (binom.test in R) to calculate the mean probability for the number of deaths (successes) given the number of reported cases (sample size), and the Clopper-Pearson method for calculating the binomial confidence intervals. We then fit Beta distributions to the mean CFR and 95% confidence intervals calculated for each country using least squares and the Nelder-Mead algorithm to give the distributional uncertainty around the CFR estimate for each country (\(\mu_j\)).

+

\[ +\mu_j \sim \text{Beta}(s_{1,j}, s_{2,j}) +\]

+

Where \(s_{1,i}\) and \(s_{2,j}\) are the two positive shape parameters of the Beta distribution estimated for destination \(j\). By definition \(\mu_j\) is the CFR for reported cases which are a subset of the total number of infections. Therefore, to infer the total number of deaths attributable to cholera infection, we assume that the CFR of observed cases is proportionally equivalent to the CFR of all cases and then calculate total deaths \(D\) as follows:

+

\[\begin{equation} +\begin{aligned} +\text{CFR}_{\text{observed}} &= \text{CFR}_{\text{total}}\\ +\\[3pt] +\frac{[\text{observed deaths}]}{[\text{observed cases}]} &= +\frac{[\text{total deaths}]}{[\text{all infections}]}\\ +\\[3pt] +\text{total deaths} &= \frac{[\text{observed deaths}] \times [\text{true infections}]}{[\text{observed cases}]}\\ +\\[3pt] +D_{jt} &= \frac{ [\sigma\rho\mu_j I_{jt}] \times [I_{jt}] }{ [\sigma\rho I_{jt}] } +\end{aligned} +\end{equation}\]

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+Table 6.4: Table 6.5: CFR Values and Beta Shape Parameters for AFRO Countries +
+Country + +Cases (2014-2024) + +Deaths (2014-2024) + +CFR + +CFR Lower + +CFR Upper + +Beta Shape1 + +Beta Shape2 +
+AFRO Region + +1219386 + +23215 + +0.019 + +0.019 + +0.019 + +0.008 + +1.914 +
+Angola + +2665 + +74 + +0.028 + +0.022 + +0.035 + +0.010 + +1.918 +
+Burundi + +5553 + +41 + +0.007 + +0.005 + +0.010 + +0.007 + +1.908 +
+Benin + +3617 + +56 + +0.015 + +0.012 + +0.020 + +0.008 + +1.906 +
+Burkina Faso + +7 + +0 + +0.019 + +0.019 + +0.019 + +0.008 + +1.914 +
+Cote d’Ivoire + +446 + +18 + +0.040 + +0.024 + +0.063 + +0.013 + +1.863 +
+Cameroon + +29946 + +925 + +0.031 + +0.029 + +0.033 + +0.010 + +1.929 +
+Democratic Republic of Congo + +314256 + +5705 + +0.018 + +0.018 + +0.019 + +0.008 + +1.903 +
+Congo + +144 + +10 + +0.019 + +0.019 + +0.019 + +0.008 + +1.914 +
+Comoros + +10342 + +149 + +0.014 + +0.012 + +0.017 + +0.008 + +1.913 +
+Ethiopia + +70755 + +877 + +0.012 + +0.012 + +0.013 + +0.007 + +1.921 +
+Ghana + +29816 + +251 + +0.008 + +0.007 + +0.010 + +0.007 + +1.913 +
+Guinea + +1 + +0 + +0.019 + +0.019 + +0.019 + +0.008 + +1.914 +
+Guinea-Bissau + +11 + +2 + +0.019 + +0.019 + +0.019 + +0.008 + +1.914 +
+Kenya + +47956 + +683 + +0.014 + +0.013 + +0.015 + +0.008 + +1.925 +
+Liberia + +580 + +0 + +0.000 + +0.000 + +0.006 + +0.006 + +1.938 +
+Mali + +12 + +4 + +0.019 + +0.019 + +0.019 + +0.008 + +1.914 +
+Mozambique + +85191 + +306 + +0.004 + +0.003 + +0.004 + +0.006 + +1.882 +
+Malawi + +62654 + +1846 + +0.029 + +0.028 + +0.031 + +0.010 + +1.881 +
+Namibia + +485 + +13 + +0.027 + +0.014 + +0.045 + +0.012 + +2.021 +
+Niger + +11959 + +344 + +0.029 + +0.026 + +0.032 + +0.010 + +1.882 +
+Nigeria + +249406 + +6774 + +0.027 + +0.027 + +0.028 + +0.009 + +1.891 +
+Rwanda + +453 + +0 + +0.000 + +0.000 + +0.008 + +0.007 + +1.926 +
+Sudan + +362 + +11 + +0.030 + +0.015 + +0.054 + +0.012 + +1.855 +
+Somalia + +134839 + +1849 + +0.014 + +0.013 + +0.014 + +0.008 + +1.906 +
+South Sudan + +30517 + +652 + +0.021 + +0.020 + +0.023 + +0.009 + +1.913 +
+Eswatini + +2 + +0 + +0.019 + +0.019 + +0.019 + +0.008 + +1.914 +
+Chad + +1359 + +90 + +0.066 + +0.054 + +0.081 + +0.015 + +1.857 +
+Togo + +410 + +20 + +0.049 + +0.030 + +0.074 + +0.014 + +1.855 +
+Tanzania + +38992 + +604 + +0.015 + +0.014 + +0.017 + +0.008 + +1.917 +
+Uganda + +9199 + +181 + +0.020 + +0.017 + +0.023 + +0.009 + +1.912 +
+South Africa + +1403 + +47 + +0.033 + +0.025 + +0.044 + +0.012 + +2.008 +
+Zambia + +30671 + +894 + +0.029 + +0.027 + +0.031 + +0.010 + +1.905 +
+Zimbabwe + +45377 + +789 + +0.017 + +0.016 + +0.019 + +0.008 + +1.903 +
+
+Case Fatality Rate (CFR) and Total Cases by Country in the AFRO Region from 2014 to 2024. Panel A: Case Fatality Ratio (CFR) with 95% confidence intervals. Panel B: total number of cholera cases. The AFRO Region is highlighted in black, all countries with less than 3/0.2 = 150 total reported cases are assigned the mean CFR for AFRO. +

+Figure 6.12: Case Fatality Rate (CFR) and Total Cases by Country in the AFRO Region from 2014 to 2024. Panel A: Case Fatality Ratio (CFR) with 95% confidence intervals. Panel B: total number of cholera cases. The AFRO Region is highlighted in black, all countries with less than 3/0.2 = 150 total reported cases are assigned the mean CFR for AFRO. +

+
+
+Beta distributions of the overall Case Fatality Rate (CFR) from 2014 to 2024. Examples show the overall CFR for the AFRO region (2%) in black, Congo with the highest CFR (7%) in red, and South Sudan with the lowest CFR (0.1%) in blue. +

+Figure 6.13: Beta distributions of the overall Case Fatality Rate (CFR) from 2014 to 2024. Examples show the overall CFR for the AFRO region (2%) in black, Congo with the highest CFR (7%) in red, and South Sudan with the lowest CFR (0.1%) in blue. +

+
+
+
+
+

6.3 Demographics

+

The model includes basic demographic change by using reported birth and death rates for each of the \(j\) countries, \(b_j\) and \(d_j\) respectively. These rates are static and defined by the United Nations Department of Economic and Social Affairs Population Division World Population Prospects 2024. Values for \(b_j\) and \(d_j\) are derived from crude rates and converted to birth rate per day and death rate per day (shown in Table 6.6).

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+Table 6.6: Table 6.7: Demographic for AFRO countries in 2023. Data include: total population as of January 1, 2023, daily birth rate, and daily death rate. Values are calculate from crude birth and death rates from UN World Population Prospects 2024. +
+Country + +Population + +Birth rate + +Death rate +
+Algeria + +45831343 + +0.0000542 + +1.28e-05 +
+Angola + +36186956 + +0.0001046 + +1.93e-05 +
+Benin + +13934166 + +0.0000940 + +2.44e-05 +
+Botswana + +2459937 + +0.0000683 + +1.58e-05 +
+Burkina Faso + +22765636 + +0.0000877 + +2.21e-05 +
+Burundi + +13503998 + +0.0000935 + +1.87e-05 +
+Cameroon + +27997833 + +0.0000937 + +1.99e-05 +
+Cape Verde + +521047 + +0.0000339 + +1.39e-05 +
+Central African Republic + +5064592 + +0.0001292 + +2.63e-05 +
+Chad + +18767684 + +0.0001196 + +3.11e-05 +
+Comoros + +842267 + +0.0000793 + +1.99e-05 +
+Congo + +6108142 + +0.0000849 + +1.74e-05 +
+Côte d’Ivoire + +30783520 + +0.0000887 + +2.12e-05 +
+Democratic Republic of Congo + +104063312 + +0.0001150 + +2.37e-05 +
+Equatorial Guinea + +1825480 + +0.0000821 + +2.18e-05 +
+Eritrea + +3438999 + +0.0000789 + +1.67e-05 +
+Eswatini + +1224706 + +0.0000663 + +2.12e-05 +
+Ethiopia + +127028360 + +0.0000886 + +1.65e-05 +
+Gabon + +2457715 + +0.0000766 + +1.74e-05 +
+Gambia + +2666786 + +0.0000843 + +1.74e-05 +
+Ghana + +33467371 + +0.0000728 + +1.95e-05 +
+Guinea + +14229395 + +0.0000939 + +2.53e-05 +
+Guinea-Bissau + +2129290 + +0.0000832 + +1.95e-05 +
+Kenya + +54793511 + +0.0000750 + +2.00e-05 +
+Lesotho + +2298496 + +0.0000664 + +2.93e-05 +
+Liberia + +5432670 + +0.0000858 + +2.24e-05 +
+Madagascar + +30813475 + +0.0000890 + +2.09e-05 +
+Malawi + +20832833 + +0.0000871 + +1.49e-05 +
+Mali + +23415909 + +0.0001113 + +2.40e-05 +
+Mauritania + +4948362 + +0.0000957 + +1.54e-05 +
+Mauritius + +1274659 + +0.0000254 + +2.39e-05 +
+Mozambique + +33140626 + +0.0001042 + +1.95e-05 +
+Namibia + +2928037 + +0.0000718 + +1.71e-05 +
+Niger + +25727295 + +0.0001167 + +2.47e-05 +
+Nigeria + +225494749 + +0.0000912 + +3.25e-05 +
+Rwanda + +13802596 + +0.0000785 + +1.64e-05 +
+São Tomé & Príncipe + +228558 + +0.0000780 + +1.54e-05 +
+Senegal + +17867073 + +0.0000816 + +1.55e-05 +
+Seychelles + +126694 + +0.0000377 + +2.27e-05 +
+Sierra Leone + +8368119 + +0.0000848 + +2.30e-05 +
+Somalia + +18031404 + +0.0001198 + +2.74e-05 +
+South Africa + +62796883 + +0.0000518 + +2.55e-05 +
+South Sudan + +11146895 + +0.0000807 + +2.71e-05 +
+Tanzania + +65657004 + +0.0000979 + +1.61e-05 +
+Togo + +9196283 + +0.0000863 + +2.13e-05 +
+Uganda + +47981110 + +0.0000978 + +1.35e-05 +
+Zambia + +20430382 + +0.0000919 + +1.45e-05 +
+Zimbabwe + +16203259 + +0.0000840 + +2.10e-05 +
+
+
+

6.4 The reproductive number

+

The reproductive number is a common metric of epidemic growth that represents the average number of secondary cases generated by a primary case at a specific time during an epidemic. We track how \(R\) changes over time by estimating the instantaneous reproductive number \(R_t\) as described in Cori et al 2013. We track \(R_t\) across all metapopulations in the model to give \(R_{jt}\) using the following formula:

+

\[\begin{equation} +R_{jt} = \frac{I_{jt}}{\sum_{\Delta t=1}^{t} g(\Delta t) I_{j,t-\Delta t}} +\tag{6.9} +\end{equation}\]

+

Where \(I_{jt}\) is the number of new infections in destination \(j\) at time \(t\), and +\(g(\Delta t)\) represents the probability value from the generation time distribution of cholera. This is accomplished by using the weighed sum in the denominator which is highly influenced by the generation time distribution.

+
+

6.4.1 The generation time distribution

+

The generation time distribution gives the time between when an individual is infected and when they infect subsequent individuals. We parameterized this quantity using a Gamma distribution with a mean of 5 days:

+

\[\begin{equation} +g(\cdot) \sim \text{Gamma}(0.5, 0.1). +\tag{6.10} +\end{equation}\]

+

Here, shape=0.5, rate=0.1, and the mean if given by shape/rate. Previous studies use a mean of 5 days (Kahn et al 2020 and Azman 2016), however a mean of 3, 5, 7, or 10 days may be admissible (Azman 2012).

+
+This is generation time +

+Figure 6.14: This is generation time +

+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+Table 6.8: Table 6.9: Generation Time in Weeks +
+Interval + +Week + +Probability +
+[0,7] + +1 + +0.704 +
+(7,14] + +2 + +0.178 +
+(14,21] + +3 + +0.068 +
+(21,28] + +4 + +0.028 +
+(28,35] + +5 + +0.012 +
+(35,42] + +6 + +0.006 +
+(42,49] + +7 + +0.003 +
+(49,56] + +8 + +0.001 +
+
+
+
+

6.5 Initial conditions

+

Since this first version of the model will begin on Jan 2023 (to take advantage of available weekly data), the initial conditions surrounding population immunity must be estimated. To set these initial conditions, we use historical data to find the total number of reported cases for a location over the previous X years, multiply by \(1/\sigma\) to estimate total infections from those symptomatic cases that are reported, and then adjust based on waning immunity. We also sum the total number of vaccinations over the past X years and adjust for vaccine efficacy \(\phi\) and waning immunity from vaccination \(\omega\).

+
    +

  • total number infected? From reported cases… back out symptomatic and asymptomatic

    • +

  • Total number immune due to natural infections in the past X years

    • +

  • total number immune due to past vaccinations in the X years

    • +
+

Use deconvolution based on immune decay estimated in vaccine section

+
+
+

6.6 Model calibration

+
    +

  • The model will be calibrated using Latin hypercube sampling for hyper-parameters and model likelihoods fit to incidence and deaths.

    • +

  • An important challenge is flexibly fitting to data that are often missing or only available in aggregated forms.

    • +
+

[Fig: different spatial and temporal scales of available data]

+
+
+

6.7 Caveats

+
    +
  • Simplest model to start. Easier for initial spatial structure but with minimum additional compartments to calibrate to available data (vaccination, cases, deaths).
    • +
  • Country level aggregations. First generation data is 2023/24…
    • +
  • Assumes vaccinating susceptible only individuals.
    • +
  • For climate, summarizing for whole country.
    • +
+
+
+

6.8 Table of parameters

+ + ++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Table 6.10: Descriptions of model parameters along with prior distributions and sources where applicable.
ParameterDescriptionDistributionSource
\(i\)Index \(i\) represents the origin metapopulation.
\(j\)Index \(j\) represents the destination metapopulation.
\(t\)Index \(t\) is the time step which is one week (7 days).
\(b_j\)Birth rate of population \(j\).
\(d_j\)Overall mortality rate of population \(j\).
\(N_{jt}\)Total population size of destination \(j\) at time \(t\).
\(S_{jt}\)Number of susceptible individuals in destination \(j\) at time \(t\).
\(V_{jt}\)Number of effectively vaccinated individuals in destination \(j\) at time \(t\).
\(I_{jt}\)Number of infected individuals in destination \(j\) at time \(t\).
\(W_{jt}\)Total amount of V. cholerae in the environment in destination \(j\) at time \(t\).
\(R_{jt}\)Number of recovered (and therefore immune) individuals in destination \(j\) at time \(t\).
\(\Lambda_{j,t+1}\)The force of infection due to human-to-human transmission in destination \(j\) at time step \(t+1\).
\(\Psi_{j,t+1}\)The force of infection due to environment-to-human transmission in destination \(j\) at time step \(t+1\).
\(\phi\)The effectiveness of Oral Cholera Vaccine (OCV).
\(\nu_{jt}\)The reported rate of vaccination with OCV in destination \(j\) at time \(t\).
\(\omega\)Rate of waning immunity of vaccinated individuals.
\(\varepsilon\)Rate of waning immunity of recovered individuals.
\(\gamma\)Recovery rate of infected individuals.
\(\mu\)Mortality rate due to infection with V. cholerae.
\(\sigma\)Proportion of V. cholerae infections that are symptomatic.
\(\rho\)The proportion of suspected cholera cases that are true infections.
\(\zeta\)Rate that infected individuals shed V. cholerae into the environment.\(0.1\mbox{-}10\)
\(\delta\)The environmental suitability dependent decay rate of V. cholerae in the environment.
\(\delta_{\text{min}}\)The minimum decay rate of V. cholerae in the environment obtained when environmental suitability (\(\psi_{jt}\)) = 0.
\(\delta_{\text{max}}\)The maximum decay rate of V. cholerae in the environment obtained when environmental suitability (\(\psi_{jt}\)) = 1.
\(\psi_{jt}\)The climatically driven environmental suitability of V. cholerae in destination \(j\) at time \(t\).
\(\beta_{j0}^{\text{hum}}\)The baseline rate of human-to-human transmission in destination \(j\).
\(\beta_{jt}^{\text{hum}}\)The seasonal rate of human-to-human transmission in destination \(j\) at time \(t\).
\(\beta_{j0}^{\text{env}}\)The baseline rate of environment-to-human transmission in destination \(j\).
\(\beta_{jt}^{\text{env}}\)The rate of environment-to-human transmission in destination \(j\) at time \(t\).
\(\alpha\)The overall population mixing parameter.0.95Glass et al 2003
\(\tau_i\)The probability that an individual departs origin \(i\).
\(\pi_{ij}\)The probability that an individual travels from origin \(i\) to destination \(j\) given that they have departed origin \(i\).
\(\theta_{j}\)The proportion of the population that have adequate Water, Sanitation and Hygiene (WASH).
\(\kappa\)The concentration (number of cells per mL) of V. cholerae required for a 50% probability of infection.\(10^5\mbox{-}10^6\)Fung 2014
+
+
+

6.9 References

+ +
+
+
+ +
+ +
+ +
+
+ +
+
+ +
+

"MOSAIC: a spatial model of endemic cholera" was written by John R Giles.

+
+ +
+

This book was built by the bookdown R package.

+
+ +
+
+ + + + + + + + diff --git a/docs/figures/vaccination_by_country.png b/docs/figures/vaccination_by_country.png new file mode 100644 index 0000000..ba1aa44 Binary files /dev/null and b/docs/figures/vaccination_by_country.png differ diff --git a/docs/figures/vaccination_example_ZMB.png b/docs/figures/vaccination_example_ZMB.png new file mode 100644 index 0000000..3d3f5d7 Binary files /dev/null and b/docs/figures/vaccination_example_ZMB.png differ diff --git a/docs/figures/vaccination_maps.png b/docs/figures/vaccination_maps.png new file mode 100644 index 0000000..eee448c Binary files /dev/null and b/docs/figures/vaccination_maps.png differ diff --git a/docs/index.html b/docs/index.html index af2f42b..97be521 100644 --- a/docs/index.html +++ b/docs/index.html @@ -85,7 +85,7 @@

-

View book source

+

View book source

@@ -97,7 +97,7 @@


-Website under development. Last compiled on 2024-10-05 at 05:56 PM PDT. +Website under development. Last compiled on 2024-10-21 at 03:55 PM PDT.
diff --git a/docs/index.md b/docs/index.md index 7826f5b..62de734 100644 --- a/docs/index.md +++ b/docs/index.md @@ -33,7 +33,7 @@ nocite: '@*' \
* -Website under development. Last compiled on 2024-10-05 at 05:57 PM PDT. +Website under development. Last compiled on 2024-10-21 at 03:56 PM PDT. *
## Welcome {-} diff --git a/docs/introduction.html b/docs/introduction.html index 7fea5fb..4e331d3 100644 --- a/docs/introduction.html +++ b/docs/introduction.html @@ -79,7 +79,7 @@

-

View book source

+

View book source

@@ -145,8 +145,8 @@

diff --git a/docs/model-description.html b/docs/model-description.html index 9f72e94..2912650 100644 --- a/docs/model-description.html +++ b/docs/model-description.html @@ -79,7 +79,7 @@

-

View book source

+

View book source

@@ -673,7 +673,7 @@

h_t = \text{LSTM}\big(\text{temperature}_{jt}, \ \text{precipitation}_{jt}, \ \text{ENSO}_{t}, \dots \big) \end{equation}\]

In this formulation, \(h_t\) represents the hidden state generated by the LSTM network based on input variables such as temperature, precipitation, and ENSO conditions, while \(b_h\) is a bias term added to the output of the hidden state transformation.

-

The deep learning LSTM model consists of three stacked LSTM-RNN layers. The first LSTM layer has 500 units and the second and third LSTM layers have 250 and 200 units respectively. The architecture the LSTM model is configured to pass node values to subsequent LSTM layers allowing deep learning of more the complex interactions among the climate variable over time. We enforced model sparsity for each LSTM layer using L2 regularization (penalty = 0.001) and used a dropout rate of 0.5 to further prevent overfitting on the limited amount of data. The final output layer was a dense layer with a single unit and a sigmoid activation function to produce a probability for binary classification, i.e. a prediction of environmental suitability \(\psi_{jt}\) on a scale of 0 to 1.

+

The deep learning LSTM model consists of three stacked LSTM-RNN layers. The first LSTM layer has 500 units and the second and third LSTM layers have 250 and 100 units respectively. The architecture the LSTM model is configured to pass node values to subsequent LSTM layers allowing deep learning of more the complex interactions among the climate variable over time. We enforced model sparsity for each LSTM layer using L2 regularization (penalty = 0.001) and used a dropout rate of 0.5 for each LSTM layer to further prevent overfitting on the limited amount of data. The final output layer was a dense layer with a single unit and a sigmoid activation function to produce a probability value for binary classification, i.e. a prediction of environmental suitability \(\psi_{jt}\) on a scale of 0 to 1.

To fit the LSTM model to data, we modified the learning rate by applying an exponential decay schedule that started at 0.001 and decayed by a factor of 0.9 every 10,000 steps to enable smoother convergence. The model was compiled using the Adam optimizer with this learning rate schedule, along with binary cross-entropy as the loss function and accuracy as the evaluation metric. The model was trained for a maximum of 200 epochs with a batch size of 1024. We allowed model fitting to stop early with a patience parameter of 10 which halts training if no improvement is observed in validation accuracy for 10 consecutive epochs. To train the model we set aside 20% of the observed data for validation and also used 20% of the training data for model fitting. The training history, including loss and accuracy, was monitored over the course of training and gave a final test accuracy of 0.73 and a final test loss of 0.56 (see Figure 4.11).

@@ -681,7 +681,7 @@

Figure 4.11: Model performance on training and validation data.

-

After model training was completed, we predicted the values of environmental suitability \(\psi_{jt}\) across all time steps for each location. Predictions start in January 1970 and go up to 5 months past the present date (currently February 2025). Given the amount of noise in the model predictions, we added a simple LOESS spline with logit transformation to smooth model predictions over time and give a more stable value of \(\psi_{jt}\) when incorporating it into other model features (e.g. Equations (4.6) and (4.7)). The resulting model predicitons are shown for an example country such as Mozambique in Figure 4.12 which compares model predictions to the original case counts and the binary classification. Predicitons for all model locations are shown in a simplified view in Figure 4.13.

+

After model training was completed, we predicted the values of environmental suitability \(\psi_{jt}\) across all time steps for each location. Predictions start in January 1970 and go up to 5 months past the present date (currently February 2025). Given the amount of noise in the model predictions, we added a simple LOESS spline with logit transformation to smooth model predictions over time and give a more stable value of \(\psi_{jt}\) when incorporating it into other model features (e.g. Equations (4.6) and (4.7)). The resulting model predictions are shown for an example country such as Mozambique in Figure 4.12 which compares model predictions to the original case counts and the binary classification. Predicitons for all model locations are shown in a simplified view in Figure 4.13.

Also, please note that this initial version of the model is fitted to a rather small amount of data. Model hyper parameters were specifically chosen to reduce overfitting. Therefore, we recommend to not over-interpret the time series predictions of the model at this early stage since they are likely to change and improve as more historical incidence data is included in future versions.

@@ -810,9 +810,37 @@

4.4 Immune dynamics

-
+

Aside from the current number of infections, population susceptibility is one of the key factors influencing the spread of cholera. Further, since immunity from both vaccination and natural infection provides long-lasting protection, it’s crucial to quantify not only the incidence of cholera but also the number of past vaccinations. Additionally, we need to estimate how many individuals with immunity remain in the population at any given time step in the model.

+

To achieve this, we estimate the vaccination rate over time (\(\nu_{jt}\)) based on historical vaccination campaigns and incorporate a model of vaccine effectiveness (\(\phi\)) and immune decay post-vaccination (\(\omega\)) to estimate the current number of individuals with vaccine-derived immunity. We also account for the immune decay rate from natural infection (\(\varepsilon\)), which is generally considered to last longer than immunity from vaccination.

+

-4.4.1 Immunity from vaccination +4.4.1 Estimating Vaccination Rates +

+

To estimate the past and current vaccination rates, we sourced data on reported OCV vaccinations from the WHO International Coordinating Group (ICG) Cholera vaccine dashboard. This resource lists all reactive OCV campaigns conducted from 2016 to the present, with approximately 103 million OCV doses shipped to sub-Saharan African (SSA) countries as of October 9, 2024. However, these data only capture reactive vaccinations in emergency settings and do not include the many preventive campaigns organized by GAVI and in-country partners.

+

As a result, our current estimates of the OCV vaccination rate likely underestimate total OCV coverage. We are working to expand our data sources to better reflect the full number of OCV doses distributed in SSA.

+

To translate the reported number of OCV doses into the model parameter \(\nu_{jt}\), we take the number of doses shipped and the reported start date of the vaccination campaign, distributing the doses over subsequent days according to a maximum daily vaccination rate. See Figure 4.16 for an example of OCV distribution using a maximum daily vaccination rate of 100,000. The resulting time series for each country is shown in Figure 4.17, with current totals based on the WHO ICG data displayed in Figure 4.18.

+
+ +Example of the estimated vaccination rate during an OCV campaign.

+Figure 4.16: Example of the estimated vaccination rate during an OCV campaign. +

+
+
+ +The estimated vaccination coverage across all countries with reported vaccination data one the WHO ICG dashboard.

+Figure 4.17: The estimated vaccination coverage across all countries with reported vaccination data one the WHO ICG dashboard. +

+
+
+ +The total cumulative number of OCV doses distributed through the WHO ICG from 2016 to present day.

+Figure 4.18: The total cumulative number of OCV doses distributed through the WHO ICG from 2016 to present day. +

+
+
+
+

+4.4.2 Immunity from vaccination

The impacts of Oral Cholera Vaccine (OCV) campaigns is incorporated into the model through the Vaccinated compartment (V). The rate that individuals are effectively vaccinated is defined as \(\phi\nu_tS_{jt}\), where \(S_{jt}\) are the available number of susceptible individuals in location \(j\) at time \(t\), \(\nu_t\) is the number of OCV doses administered at time \(t\) and \(\phi\) is the estimated vaccine effectiveness. Note that there is just one vaccinated compartment at this time, though future model versions may include \(V_1\) an \(V_2\) compartments to explore two dose vaccination strategies or to emulate more complex waning patterns.

The vaccination rate \(\nu_t\) is not an estimated quantity. Rather, it is directly defined by the reported number of OCV doses administered on the WHO OCV dashboard here: https://www.who.int/groups/icg/cholera.

@@ -895,7 +923,7 @@

\tag{4.9} \end{equation}\]

Where \(\phi\) is the effectiveness of one dose OCV, and the based on this specification, it is also the initial proportion immune directly after vaccination. The decay rate parameter \(\omega\) is the rate at which initial vaccine derived immunity decays per day post vaccination, and \(t\) and \(t_{\text{vaccination}}\) are the time (in days) the function is evaluated at and the time of vaccination respectively. When we fitted the model to the data from the cohort studies shown in Table (4.6) we found that \(\omega = 0.00057\) (\(0-0.0019\) 95% CI), which gives a mean estimate of 4.8 years for vaccine derived immune duration with unreasonably large confidence intervals (1.4 years to infinite immunity). However, the point estimate of 4.8 years is consistent with anecdotes that one dose OCV is effective for up to at least 3 years.

-

The wide confidence intervals are likely due to the wide range of reported estimates for proportion immune after a short duration in the 7–90 days range (Azman et al 2016 and Qadri et al 2016). Therefore, we chose to use the point estimate of \(\omega\) and incorporate uncertainty based on the initial proportion immune (i.e. vaccine effectiveness \(\phi\)) shortly after vaccination. Using the decay model in Equation (4.9) we estimated \(\phi\) to be \(0.64\) (\(0.32-0.96\) 95% CI). We then fit a Beta distribution to the quantiles of \(\phi\) by minimizing the sums of squares using the Nelder-Mead optimization algorithm to render the following distribution (shown in Figure 4.16B):

+

The wide confidence intervals are likely due to the wide range of reported estimates for proportion immune after a short duration in the 7–90 days range (Azman et al 2016 and Qadri et al 2016). Therefore, we chose to use the point estimate of \(\omega\) and incorporate uncertainty based on the initial proportion immune (i.e. vaccine effectiveness \(\phi\)) shortly after vaccination. Using the decay model in Equation (4.9) we estimated \(\phi\) to be \(0.64\) (\(0.32-0.96\) 95% CI). We then fit a Beta distribution to the quantiles of \(\phi\) by minimizing the sums of squares using the Nelder-Mead optimization algorithm to render the following distribution (shown in Figure 4.19B):

\[\begin{equation} \phi \sim \text{Beta}(4.57, 2.41). \tag{4.10} @@ -903,13 +931,13 @@

This is vaccine effectiveness

-Figure 4.16: This is vaccine effectiveness +Figure 4.19: This is vaccine effectiveness

-
+

-4.4.2 Immunity from natural infection +4.4.3 Immunity from natural infection

The duration of immunity after a natural infection is likely to be longer lasting than that from vaccination with OCV (especially given the current one dose strategy). As in most SIR-type models, the rate at which individuals leave the Recovered compartment is governed by the immune decay parameter \(\varepsilon\). We estimated the durability of immunity from natural infection based on two cohort studies and fit the following exponential decay model to estimate the rate of immunity decay over time:

\[ @@ -957,14 +985,14 @@

-

We estimated the mean immune decay to be \(\bar\varepsilon = 3.9 \times 10^{-4}\) (\(1.7 \times 10^{-4}-1.03 \times 10^{-3}\) 95% CI) which is equivalent to an immune duration of \(7.21\) years (\(2.66-16.1\) years 95% CI) as shown in Figure 4.17A. This is slightly longer than previous modeling work estimating the duration of immunity to be ~5 years (King et al 2008). Uncertainty around \(\varepsilon\) in the model is then represented by a Log-Normal distribution as shown in Figure 4.17B:

+

We estimated the mean immune decay to be \(\bar\varepsilon = 3.9 \times 10^{-4}\) (\(1.7 \times 10^{-4}-1.03 \times 10^{-3}\) 95% CI) which is equivalent to an immune duration of \(7.21\) years (\(2.66-16.1\) years 95% CI) as shown in Figure 4.20A. This is slightly longer than previous modeling work estimating the duration of immunity to be ~5 years (King et al 2008). Uncertainty around \(\varepsilon\) in the model is then represented by a Log-Normal distribution as shown in Figure 4.20B:

\[ \varepsilon \sim \text{Lognormal}(\bar\varepsilon+\frac{\sigma^2}{2}, 0.25) \]

The duration of immunity after natural infection with *V. cholerae*.

-Figure 4.17: The duration of immunity after natural infection with V. cholerae. +Figure 4.20: The duration of immunity after natural infection with V. cholerae.

@@ -973,17 +1001,17 @@

4.5 Spatial dynamics

-

The parameters in the model diagram in Figure 4.2 that have a \(jt\) subscript denote the spatial structure of the model. Each country is modeled as an independent metapopulation that is connected to all others via the spatial force of infection \(\Lambda_{jt}\) which moves contagion among metapopulations according to the connectivity provided by parameters \(\tau_i\) (the probability departure) and \(\pi_{ij}\) (the probability of diffusion to destination \(j\)). Both parameters are estimated using the departure-diffusion model below which is fitted to average weekly air traffic volume between all of the 41 countries included in the MOSAIC framework (Figure 4.18).

+

The parameters in the model diagram in Figure 4.2 that have a \(jt\) subscript denote the spatial structure of the model. Each country is modeled as an independent metapopulation that is connected to all others via the spatial force of infection \(\Lambda_{jt}\) which moves contagion among metapopulations according to the connectivity provided by parameters \(\tau_i\) (the probability departure) and \(\pi_{ij}\) (the probability of diffusion to destination \(j\)). Both parameters are estimated using the departure-diffusion model below which is fitted to average weekly air traffic volume between all of the 41 countries included in the MOSAIC framework (Figure 4.21).

The average number of air passengers per week in 2017 among all countries.

-Figure 4.18: The average number of air passengers per week in 2017 among all countries. +Figure 4.21: The average number of air passengers per week in 2017 among all countries.

A network map showing the average number of air passengers per week in 2017.

-Figure 4.19: A network map showing the average number of air passengers per week in 2017. +Figure 4.22: A network map showing the average number of air passengers per week in 2017.

@@ -1045,17 +1073,17 @@

\gamma &\sim \text{Gamma}(1, 1) \end{aligned} \]

-

The models for \(\tau_i\) and \(\pi_{ij}\) were fitted to air traffic data from OAG using the mobility R package (Giles 2020). Estimates for mobility model parameters are shown in Figures 4.20 and 4.21.

+

The models for \(\tau_i\) and \(\pi_{ij}\) were fitted to air traffic data from OAG using the mobility R package (Giles 2020). Estimates for mobility model parameters are shown in Figures 4.23 and 4.24.

The estimated weekly probability of travel outside of each origin location $\tau_i$ and 95% confidence intervals is shown in panel A with the population mean indicated as a red dashed line. Panel B shows the estimated total number of travelers leaving origin $i$ each week.

-Figure 4.20: The estimated weekly probability of travel outside of each origin location \(\tau_i\) and 95% confidence intervals is shown in panel A with the population mean indicated as a red dashed line. Panel B shows the estimated total number of travelers leaving origin \(i\) each week. +Figure 4.23: The estimated weekly probability of travel outside of each origin location \(\tau_i\) and 95% confidence intervals is shown in panel A with the population mean indicated as a red dashed line. Panel B shows the estimated total number of travelers leaving origin \(i\) each week.

The diffusion process $\pi_{ij}$ which gives the estimated probability of travel from origin $i$ to destination $j$ given that travel outside of origin $i$ has occurred.

-Figure 4.21: The diffusion process \(\pi_{ij}\) which gives the estimated probability of travel from origin \(i\) to destination \(j\) given that travel outside of origin \(i\) has occurred. +Figure 4.24: The diffusion process \(\pi_{ij}\) which gives the estimated probability of travel from origin \(i\) to destination \(j\) given that travel outside of origin \(i\) has occurred.

@@ -1212,11 +1240,11 @@

-

The prior distribution for \(\sigma\) is plotted in Figure 4.22A with the reported values of the proportion symptomatic from previous studies shown in 4.22B.

+

The prior distribution for \(\sigma\) is plotted in Figure 4.25A with the reported values of the proportion symptomatic from previous studies shown in 4.25B.

Proportion of infections that are symptomatic.

-Figure 4.22: Proportion of infections that are symptomatic. +Figure 4.25: Proportion of infections that are symptomatic.

@@ -1224,14 +1252,14 @@

4.6.2 Suspected cases

-

The clinical presentation of diarrheal diseases is often similar across various pathogens, which can lead to systematic biases in the reported number of cholera cases. It is anticipated that the number of suspected cholera cases is related to the actual number of infections by a factor of \(1/\rho\), where \(\rho\) represents the proportion of suspected cases that are true infections. To adjust for this bias, we use estimates from the meta-analysis by Weins et al. (2023), which suggests that suspected cholera cases outnumber true infections by approximately 2 to 1, with a mean across studies indicating that 52% (24-80% 95% CI) of suspected cases are actual cholera infections. A higher estimate was reported for ourbreak settings (78%, 40-99% 95% CI). To account for the variability in this estimate, we fit a Beta distribution to the reported quantiles using a least squares approach and the Nelder-Mead algorithm, resulting in the prior distribution shown in Figure 4.23B:

+

The clinical presentation of diarrheal diseases is often similar across various pathogens, which can lead to systematic biases in the reported number of cholera cases. It is anticipated that the number of suspected cholera cases is related to the actual number of infections by a factor of \(1/\rho\), where \(\rho\) represents the proportion of suspected cases that are true infections. To adjust for this bias, we use estimates from the meta-analysis by Weins et al. (2023), which suggests that suspected cholera cases outnumber true infections by approximately 2 to 1, with a mean across studies indicating that 52% (24-80% 95% CI) of suspected cases are actual cholera infections. A higher estimate was reported for ourbreak settings (78%, 40-99% 95% CI). To account for the variability in this estimate, we fit a Beta distribution to the reported quantiles using a least squares approach and the Nelder-Mead algorithm, resulting in the prior distribution shown in Figure 4.26B:

\[\begin{equation} \rho \sim \text{Beta}(4.79, 1.53). \end{equation}\]

Proportion of suspected cholera cases that are true infections. Panel A shows the 'low' assumption which estimates across all settings: $\rho \sim \text{Beta}(5.43, 5.01)$. Panel B shows the 'high' assumption where the estimate reflects high-quality studies during outbreaks: $\rho \sim \text{Beta}(4.79, 1.53)$

-Figure 4.23: Proportion of suspected cholera cases that are true infections. Panel A shows the ‘low’ assumption which estimates across all settings: \(\rho \sim \text{Beta}(5.43, 5.01)\). Panel B shows the ‘high’ assumption where the estimate reflects high-quality studies during outbreaks: \(\rho \sim \text{Beta}(4.79, 1.53)\) +Figure 4.26: Proportion of suspected cholera cases that are true infections. Panel A shows the ‘low’ assumption which estimates across all settings: \(\rho \sim \text{Beta}(5.43, 5.01)\). Panel B shows the ‘high’ assumption where the estimate reflects high-quality studies during outbreaks: \(\rho \sim \text{Beta}(4.79, 1.53)\)

@@ -2176,13 +2204,13 @@

Case Fatality Rate (CFR) and Total Cases by Country in the AFRO Region from 2014 to 2024. Panel A: Case Fatality Ratio (CFR) with 95% confidence intervals. Panel B: total number of cholera cases. The AFRO Region is highlighted in black, all countries with less than 3/0.2 = 150 total reported cases are assigned the mean CFR for AFRO.

-Figure 4.24: Case Fatality Rate (CFR) and Total Cases by Country in the AFRO Region from 2014 to 2024. Panel A: Case Fatality Ratio (CFR) with 95% confidence intervals. Panel B: total number of cholera cases. The AFRO Region is highlighted in black, all countries with less than 3/0.2 = 150 total reported cases are assigned the mean CFR for AFRO. +Figure 4.27: Case Fatality Rate (CFR) and Total Cases by Country in the AFRO Region from 2014 to 2024. Panel A: Case Fatality Ratio (CFR) with 95% confidence intervals. Panel B: total number of cholera cases. The AFRO Region is highlighted in black, all countries with less than 3/0.2 = 150 total reported cases are assigned the mean CFR for AFRO.

Beta distributions of the overall Case Fatality Rate (CFR) from 2014 to 2024. Examples show the overall CFR for the AFRO region (2%) in black, Congo with the highest CFR (7%) in red, and South Sudan with the lowest CFR (0.1%) in blue.

-Figure 4.25: Beta distributions of the overall Case Fatality Rate (CFR) from 2014 to 2024. Examples show the overall CFR for the AFRO region (2%) in black, Congo with the highest CFR (7%) in red, and South Sudan with the lowest CFR (0.1%) in blue. +Figure 4.28: Beta distributions of the overall Case Fatality Rate (CFR) from 2014 to 2024. Examples show the overall CFR for the AFRO region (2%) in black, Congo with the highest CFR (7%) in red, and South Sudan with the lowest CFR (0.1%) in blue.

@@ -2910,7 +2938,7 @@

This is generation time

-Figure 4.26: This is generation time +Figure 4.29: This is generation time

@@ -3321,8 +3349,9 @@

  • 4.4 Immune dynamics
  • @@ -3355,8 +3384,8 @@

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    diff --git a/docs/reference-keys.txt b/docs/reference-keys.txt index 768ae1f..c215f7e 100644 --- a/docs/reference-keys.txt +++ b/docs/reference-keys.txt @@ -24,6 +24,9 @@ fig:psi-prediction-countries tab:wash-weights fig:wash-incidence fig:wash-country +fig:vaccination-example +fig:vaccination-countries +fig:vaccination-maps tab:effectiveness-papers eq:omega eq:effectiveness @@ -80,6 +83,7 @@ deep-learning-neural-network-model shedding water-sanitation-and-hygiene-wash immune-dynamics +estimating-vaccination-rates immunity-from-vaccination immunity-from-natural-infection spatial-dynamics diff --git a/docs/references-1.html b/docs/references-1.html index d631394..ec7aba8 100644 --- a/docs/references-1.html +++ b/docs/references-1.html @@ -79,7 +79,7 @@

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    diff --git a/docs/search.json b/docs/search.json index 09b057f..89428ef 100644 --- a/docs/search.json +++ b/docs/search.json @@ -1 +1 @@ -[{"path":"index.html","id":"section","chapter":"","heading":"","text":"","code":""},{"path":"index.html","id":"welcome","chapter":"","heading":"Welcome","text":"Welcome Metapopulation Outbreak Simulation Agent-based Implementation Cholera (MOSAIC). MOSAIC framework simulates transmission dynamics cholera Sub-Saharan Africa (SSA) provides tools understand impact interventions, vaccination, well large-scale drivers like climate change. MOSAIC built using Light-agent Spatial Model ERadication (LASER) platform, site serves documentation model’s methods associated analyses. Please note MOSAIC currently development, content may change regularly. sharing increase visibility welcome feedback aspect model.","code":""},{"path":"index.html","id":"contact","chapter":"","heading":"Contact","text":"MOSAIC developed team researchers Institute Disease Modeling (IDM) dedicated developing modeling methods software tools help decision-makers understand respond infectious disease outbreaks. website currently maintained John Giles (@gilesjohnr). general questions, contact John Giles (john.giles@gatesfoundation.org), Jillian Gauld (jillian.gauld@gatesfoundation.org), /Rajiv Sodhi (rajiv.sodhi@gatesfoundation.org).","code":""},{"path":"index.html","id":"funding","chapter":"","heading":"Funding","text":"work developed Institute Disease Modeling support funded research grants made Bill & Melinda Gates Foundation.","code":""},{"path":"introduction.html","id":"introduction","chapter":"1 Introduction","heading":"1 Introduction","text":"","code":""},{"path":"introduction.html","id":"history","chapter":"1 Introduction","heading":"1.1 History","text":"Vibrio cholerae introduced African continent Asia 1970s since become endemic many countries.","code":""},{"path":"introduction.html","id":"recent-surge","chapter":"1 Introduction","heading":"1.2 Recent Surge","text":"Although sporadic cholera outbreaks past five decades, significant surge cases since 2021. increase likely due factors climate change disruptions municipal services COVID-19 pandemic.","code":""},{"path":"introduction.html","id":"gtfcc-goals","chapter":"1 Introduction","heading":"1.3 GTFCC Goals","text":"Global Task Force Cholera Control (GTFCC) aims reduce cholera deaths 90% 2030.","code":""},{"path":"introduction.html","id":"ocv-stockpiles","chapter":"1 Introduction","heading":"1.4 OCV Stockpiles","text":"major concern recent surge cases depletion oral cholera vaccine (OCV) stockpiles. response, officials shifted single-dose OCV strategies. Efforts underway replenish stockpiles, key question best allocate reduce transmission regionally support GTFCC’s goal.","code":""},{"path":"introduction.html","id":"climate-change","chapter":"1 Introduction","heading":"1.5 Climate Change","text":"Environmental factors play significant role cholera outbreaks, warmer wetter conditions creating favorable environment Vibrio cholerae. conditions likely exacerbate already challenging endemic outbreak settings. Models incorporate climatic forcing can provide insights future cholera dynamics due climate change aid achieving GTFCC goal.","code":""},{"path":"rationale.html","id":"rationale","chapter":"2 Rationale","heading":"2 Rationale","text":"significant challenge controlling cholera transmission Sub-Saharan Africa (SSA) lack comprehensive datasets dynamic models designed support ongoing policy-making. persistent endemic nature cholera SSA presents complex quantitative challenge, requiring sophisticated models produce meaningful inferences. Models incorporate necessary natural history disease dynamics, operate adequate spatial temporal scales, crucial providing timely actionable information address ongoing future cholera outbreaks.Although developing data models scales challenging, goal iteratively create landscape-scale transmission model cholera SSA can provide weekly predictions key epidemiological metrics. modeling methods leverage wide array --date data sources, including incidence mortality reports, patterns human movement, vaccination history schedules, environmental factors.Key questions address include administer limited supply oral cholera vaccine (OCV) severe weather events climate change impact future outbreaks. landscape-scale model accounts endemic transmission patterns valuable tool addressing questions.","code":""},{"path":"data.html","id":"data","chapter":"3 Data","heading":"3 Data","text":"MOSAIC model requires diverse set data sources, directly used define model parameters (e.g., birth death rates), others help fit models priori provide informative priors transmission model. additional data sources become available, future versions model adapt incorporate . now, following data sources represent minimum requirements initiate viable first model.","code":""},{"path":"data.html","id":"historical-incidence-and-deaths","chapter":"3 Data","heading":"3.1 Historical Incidence and Deaths","text":"Data historical cholera incidence deaths crucial establishing baseline transmission patterns. compiled annual total reported cases deaths AFRO region countries January 1970 August 2024. data comes several sources include:World Data (1970-2021): Number Reported Cases Cholera (1949-2021) Number Reported Deaths Cholera (1949-2021). World Data group compiled data previously published annual reports.Annual Report 2022: data manually extracted World Health Organization’s Weekly Epidemiological Record 38, 2023, 98, 431–452.Global Cholera Acute Watery Diarrhea Dashboard (2023-2024): Unofficial tallies reported cases deaths 2023 part 2024 available Global Cholera AWD Dashboard.","code":""},{"path":"data.html","id":"recent-incidence-and-deaths","chapter":"3 Data","heading":"3.2 Recent Incidence and Deaths","text":"capture recent cholera trends, retrieved reported cases deaths data Global Cholera Acute Watery Diarrhea Dashboard REST API. data provide weekly incidence deaths January 2023 August 2024 provides --date counts country level.","code":""},{"path":"data.html","id":"vaccinations","chapter":"3 Data","heading":"3.3 Vaccinations","text":"Accurate data oral cholera vaccine (OCV) campaigns vaccination history vital understanding impact vaccination efforts. data come :Cholera Vaccine Dashboard: resource (link) provides detailed information OCV distribution vaccination campaigns 2016 2024.GTFCC OCV Dashboard: Managed Médecins Sans Frontières, dashboard (link) tracks OCV deployments globally, offering granular insights vaccination efforts 2013 2024.","code":""},{"path":"data.html","id":"human-mobility-data","chapter":"3 Data","heading":"3.4 Human Mobility Data","text":"Human mobility patterns significantly influence cholera transmission. Relevant data include:OAG Passenger Booking Data: dataset (link) offers insights air passenger movements, can used model spread cholera across regions.Namibia Call Data Records: additional source Giles et al. (2020) (link) provides detailed mobility data based mobile phone records, useful localized modeling.","code":""},{"path":"data.html","id":"climate-data","chapter":"3 Data","heading":"3.5 Climate Data","text":"Climate conditions, including temperature, precipitation, extreme weather events, play critical role cholera dynamics. captured :OpenMeteo Historical Weather Data API: API (link) offers access historical climate data, essential modeling environmental factors influencing cholera outbreaks.","code":""},{"path":"data.html","id":"storms-and-floods","chapter":"3 Data","heading":"3.5.1 Storms and Floods","text":"Data extreme weather events, specifically storms floods, obtained :EM-DAT International Disaster Database: Maintained Centre Research Epidemiology Disasters (CRED) UCLouvain, database (link) provides comprehensive records disasters 2000 present, including affecting African countries.","code":""},{"path":"data.html","id":"wash-water-sanitation-and-hygiene","chapter":"3 Data","heading":"3.6 WASH (Water, Sanitation, and Hygiene)","text":"Data water, sanitation, hygiene (WASH) critical understanding environmental infrastructural factors influence cholera transmission. data sourced :UNICEF Joint Monitoring Program (JMP) Database: resource (link) offers detailed information household-level access clean water sanitation, integral cholera prevention efforts.","code":""},{"path":"data.html","id":"demographics","chapter":"3 Data","heading":"3.7 Demographics","text":"Demographic data, including population size, birth rates, death rates, foundational accurate disease modeling. data sourced :UN World Population Prospects 2024: database (link) provides probabilistic projections key demographic metrics, essential estimating population-level impacts cholera.","code":""},{"path":"model-description.html","id":"model-description","chapter":"4 Model description","heading":"4 Model description","text":"describe methods MOSAIC beta version 0.1. model version provides starting point understanding cholera transmission Sub-Saharan Africa, incorporating important drivers disease dynamics human mobility, environmental conditions, vaccination schedules. MOSAIC continues evolve, future iterations refine model components based available data improved model mechanisms, hope increase applicability real-world scenarios.model operates weekly time steps January 2023 August 2024 includes 41 countries Sub-Saharan Africa (SSA), see Figure 4.1.\nFigure 4.1: map Sub-Saharan Africa countries experienced cholera outbreak past 5 10 years highlighted green.\n","code":""},{"path":"model-description.html","id":"transmission-dynamics","chapter":"4 Model description","heading":"4.1 Transmission dynamics","text":"model metapopulation structure familiar compartments Susceptible, Infected, Recovered individuals SIRS dynamics. model also contains compartments vaccinated individuals (V) Water & environment based transmission (W) refer SVIWRS.\nFigure 4.2: diagram SVIWRS (Susceptible-Vaccinated-Infected-Water/environmental-Recovered-Susceptible) model shows model compartments circles rate parameters displayed. primary data sources model fit shown square nodes (Vaccination data, reported cases deaths).\nSVIWRS metapopulation model, shown Figure 4.2, governed following difference equations:\\[\\begin{equation}\n\\begin{aligned}\nS_{j,t+1} &= b_j N_{jt} + S_{jt} - \\phi \\nu_{jt} S_{jt} + \\omega V_{jt} - \\Lambda_{j,t+1} - \\Psi_{j,t+1} + \\varepsilon R_{jt} - d_j S_{jt}\\\\[11pt]\nV_{j,t+1} &= V_{jt} + \\phi \\nu_{jt} S_{jt} - \\omega V_{jt} - d_j V_{jt}\\\\[11pt]\nI_{j,t+1} &= I_{jt} + \\Lambda_{j,t+1} + \\Psi_{j,t+1} - \\gamma I_{jt} - \\mu \\sigma I_{jt} - d_j I_{jt}\\\\[11pt]\nW_{j,t+1} &= W_{jt} + \\zeta I_{jt} - \\delta_{jt} W_{jt}\\\\[11pt]\nR_{j,t+1} &= R_{jt} + \\gamma I_{jt} - \\varepsilon R_{jt} - d_j R_{jt}\\\\[11pt]\n\\end{aligned}\n\\tag{4.1}\n\\end{equation}\\]descriptions parameters Equation (4.1), see Table (4.15). Transmission dynamics driven two force infection terms, \\(\\Lambda_{jt}\\) \\(\\Psi_{jt}\\). force infection due human--human (\\(\\Lambda_{jt}\\)) :\\[\\begin{equation}\n\\begin{aligned}\n\\Lambda_{j,t+1} &= \\frac{\n\\beta_{jt}^{\\text{hum}} \\Big(\\big(S_{jt}(1-\\tau_{j})\\big) \\big(I_{jt}(1-\\tau_{j}) + \\sum_{\\forall \\= j} (\\pi_{ij}\\tau_iI_{}) \\big)\\Big)^\\alpha}{N_{jt}}.\\\\[11pt]\n\\end{aligned}\n\\tag{4.2}\n\\end{equation}\\]\\(\\beta_{jt}^{\\text{hum}}\\) rate human--human transmission. Movement within among metapopulations governed \\(\\tau_i\\), probability departing origin location \\(\\), \\(\\pi_{ij}\\) relative probability travel origin \\(\\) destination \\(j\\) (see section spatial dynamics). include environmental effects, force infection due environment--human transmission (\\(\\Psi_{jt}\\)) defined :\\[\\begin{equation}\n\\begin{aligned}\n\\Psi_{j,t+1} &= \\frac{\\beta_{jt}^{\\text{env}} \\big(S_{jt}(1-\\tau_{j})\\big) (1-\\theta_j) W_{jt}}{\\kappa+W_{jt}},\\\\[11pt]\n\\end{aligned}\n\\tag{4.3}\n\\end{equation}\\]\\(\\beta_{jt}^{\\text{env}}\\) rate environment--human transmission \\(\\theta_j\\) proportion population location \\(j\\) least basic access Water, Sanitation, Hygiene (WASH). environmental compartment model also scaled concentration (cells per mL) V. cholerae required 50% probability infection Fung 2014. See section environmental transmission water/environment compartment climatic drivers transmission.Note model processes stochastic. Transition rates converted probabilities commonly used formula \\(p(t) = 1-e^{-rt}\\) (see Ross 2007), integer quantities moved model compartments time step according binomial process like example recovery infected individuals (\\(\\gamma I_{jt}\\)):\\[\\begin{equation}\n\\frac{\\partial R}{\\partial t} \\sim \\text{Binom}(I_{jt}, 1-\\exp(-\\gamma))\n\\tag{4.4}\n\\end{equation}\\]","code":""},{"path":"model-description.html","id":"seasonality","chapter":"4 Model description","heading":"4.2 Seasonality","text":"Cholera transmission seasonal typically associated rainy season, transmission rate terms \\(\\beta_{jt}^{\\text{*}}\\) temporally forced. human--human transmission used truncated sine-cosine form Fourier series two harmonic features flexibility capture seasonal transmission dynamics driven extended rainy seasons /biannual trends:\\[\\begin{equation}\n\\beta_{jt}^{\\text{hum}} = \\beta_{j0}^{\\text{hum}} + a_1 \\cos\\left(\\frac{2\\pi t}{p}\\right) + b_1 \\sin\\left(\\frac{2\\pi t}{p}\\right) + a_2 \\cos\\left(\\frac{4\\pi t}{p}\\right) + b_2 \\sin\\left(\\frac{4\\pi t}{p}\\right)\n\\tag{4.5}\n\\end{equation}\\], \\(\\beta_{j0}^{\\text{hum}}\\) mean human--human transmission rate location \\(j\\) time steps. Seasonal dynamics determined parameters \\(a_1\\), \\(b_1\\) \\(a_2\\), \\(b_2\\) gives amplitude first second waves respectively. periodic cycle \\(p\\) 52, function controls temporal variation \\(\\beta_{jt}^{\\text{hum}}\\) 52 weeks year.\nFigure 4.3: example temporal distribution human--human transmission rate across 52 weeks year given cosine wave function. wave function fitted country designed align rainy season indicated shaded region figure.\nestimated parameters Fourier series (\\(a_1\\), \\(b_1\\), \\(a_2\\), \\(b_2\\)) using Levenberg–Marquardt algorithm minpack.lm R library. Given lack reported cholera case data many countries SSA association cholera transmission rainy season, leveraged seasonal precipitation data help fit Fourier wave function countries. first gathered weekly precipitation values 1994 2024 30 uniformly distributed points within country Open-Meteo Historical Weather Data API. fit Fourier series weekly precipitation data used parameters starting values fitting model sparse cholera case data.\nFigure 4.4: Example grid 30 uniformly distributed points within Mozambique (). scatterplot shows weekly summed precipitation values 30 grid points cholera cases plotted scale Z-Score shows variance around mean terms standard deviation. Fitted Fourier series fucntions shown blue (fit precipitation data) red (fit cholera case data) lines.\ncountries reported case data, inferred seasonal dynamics using fitted wave function neighboring country available case data. selected neighbor chosen cluster countries (grouped hierarchically four clusters based precipitation seasonality using Ward’s method; see Figure 4.5) highest correlation seasonal precipitation country lacking case data. rare event country reported case data found within seasonal cluster, expanded search 10 nearest neighbors continued expanding adding next nearest neighbor match found.\nFigure 4.5: ) Map showing clustering African countries based seasonal precipitation patterns (1994-2024). Countries colored according cluster assignments, identified using hierarchical clustering. B) Fourier series fitted weekly precipitation country. line plot shows seasonal pattern countries within given cluster. Clusteres used infer seasonal transmission dynamics countries reported cholera cases.\nUsing model fitting methods described , cluster-based approach inferring seasonal Fourier series pattern countries without reported cholera cases, modeled seasonal dynamics 41 countries MOSAIC framework. dynamics visualized Figure 4.6, corresponding Fourier model coefficients presented Table 4.1.\nFigure 4.6: Seasonal transmission patterns countries modeled MOSAIC modeled truncated Fourier series Equation (4.5). Blues lines give Fourier series model fits precipitation (1994-2024) red lines give models fits reported cholera cases (2023-2024). countries reported case data available, Fourier model inferred nearest country similar seasonal precipitation patterns determined hierarchical clustering. Countries inferred case data neighboring locations annotated red. X-axis represents weeks year (1-52), Y-axis shows Z-score weekly precipitation cholera cases.\n\nTable 4.1: Table 4.2: Estimated coefficients truncated Fourier model Equation (4.5) fit countries reported cholera cases. Model fits shown Figure 4.6.\n","code":""},{"path":"model-description.html","id":"environmental-transmission","chapter":"4 Model description","heading":"4.3 Environmental transmission","text":"Environmental transmission critical factor cholera spread consists several key components: rate infected individuals shed V. cholerae environment, pathogen’s survival rate environmental conditions, overall suitability environment sustaining bacteria time.","code":""},{"path":"model-description.html","id":"climate-driven-transmission","chapter":"4 Model description","heading":"4.3.1 Climate-driven transmission","text":"capture impacts climate-drivers cholera transmission, included parameter \\(\\psi_{jt}\\), represents current state environmental suitability respect : ) survival time V. cholerae environment , ii) rate environment--human transmission contributes overall force infection.\\[\\begin{equation}\n\\beta_{jt}^{\\text{env}} = \\beta_{j0}^{\\text{env}} \\Bigg(1 + \\frac{\\psi_{jt}-\\bar\\psi_j}{\\bar\\psi_j} \\Bigg) \\quad \\text{} \\quad \\bar\\psi_j = \\frac{1}{T} \\sum_{t=1}^{T} \\psi_{jt}\n\\tag{4.6}\n\\end{equation}\\]formulation effectively scales base environmental transmission rate \\(\\beta_{jt}^{\\text{env}}\\) varies time according climatically driven model suitability. Note , unlike cosine wave function \\(\\beta_{jt}^{\\text{hum}}\\), temporal term can increase decrease time following multi-annual cycles.Environmental suitability (\\(\\psi_{jt}\\)) also impacts survival rate V. cholerae environment (\\(\\delta_{jt}\\)) form:\\[\\begin{equation}\n\\delta_{jt} = \\delta_{\\text{min}} + \\psi_{jt} \\times (\\delta_{\\text{max}} - \\delta_{\\text{min}})\n\\tag{4.7}\n\\end{equation}\\]normalizes variance suitability parameter bounded within minimum (\\(\\delta_{\\text{min}}\\)) maximum (\\(\\delta_{\\text{max}}\\)) survival times V. cholerae.\nFigure 4.7: Relationship environmental suitability (\\(\\psi_{jt}\\)) rate V. cholerae decay environment (\\(\\delta_j\\)). green line shows mildest penalty V. cholerae survival, survival environment \\(1/\\delta_{\\text{min}}\\) = 3 days suitability = 0 \\(1/\\delta_{\\text{max}}\\) = 90 days suitability = 1.\n","code":""},{"path":"model-description.html","id":"modeling-environmental-suitability","chapter":"4 Model description","heading":"4.3.2 Modeling environmental suitability","text":"","code":""},{"path":"model-description.html","id":"environmental-data","chapter":"4 Model description","heading":"4.3.2.1 Environmental data","text":"mechanism environment--human transmission (Equation (4.6)) rate decay V. cholerae environment (Equation (4.7)) driven parameter \\(\\psi_{jt}\\), refer environmental suitability. parameter \\(\\psi_{jt}\\) modeled time series location using Long Short-Term Memory (LSTM) Recurrent Neural Network (RNN) model suite 24 covariates include 19 historical forecasted climate variables MRI-AGCM3-2-S climate model. Covariates also include 4 large-scale climate drivers Indian Ocean Dipole Mode Index (DMI), El Niño Southern Oscillation (ENSO) 3 different Pacific Ocean regions. also included location specific variable giving mean elevation country. See example time series climate variables one country (Mozambique) Figure 4.8 DMI ENSO variables Figure 4.9. list covariates sources can seen Table 4.3.Note 19 climate variables offer forecasts 2030 beyond, forecasts DMI ENSO variables limited 5 months future. , environmental suitability model predictions currently limited 5 month time horizon future iterations may allow longer forecasts. Additional data sources integrated subsequent versions suitability model. instance, flood cyclone data likely incorporated later, though initial version model.\nFigure 4.8: Climate data acquired OpenMeteo data API. Data collected 30 uniformly distributed points across country aggregated give weekly values 17 climate variable 1970 2030.\n\nFigure 4.9: Historical forecasted values Indian Ocean Dipole Mode Index (DMI) El Niño Southern Oscillation (ENSO) 2015 2025. ENSO values come three different regions: Niño3 (central eastern Pacific), Niño3.4 (central Pacific), Niño4 (western-central Pacifi). Data National Oceanic Atmospheric Administration (NOAA) Bureau Meteorology (BOM).\nTable 4.3: full list covariates sources used LSTM RNN model predict environmental suitability V. cholerae (\\(\\psi_{jt}\\)).","code":""},{"path":"model-description.html","id":"deep-learning-neural-network-model","chapter":"4 Model description","heading":"4.3.2.2 Deep learning neural network model","text":"mentioned , model environmental suitability \\(\\psi_{jt}\\) using Long Short-Term Memory (LSTM) Recurrent Neural Network (RNN) model. LSTM model developed using keras tensorflow R predict binary outcomes. Thus modeled quantity \\(\\psi_{jt}\\) proportion implying unsuitable conditions 0 perfectly suitable conditions 1.model fitted reported case counts converted binary variable using threshold 200 reported cases per week. Given delays reporting likely lead times environmental suitability ahead transmission case reporting, also set preceding one week suitable cases two consecutive weeks >200 cases per week, assumed preceding two weeks also suitable. See Figure 4.10 example reported case counts converted binary variable representing presumed environmental suitability V. cholerae.\nFigure 4.10: Reported cases converted binary variable modeling environmental suitability.\nmodel Long Short-Term Memory (LSTM) neural network designed binary classification, environmental suitability, \\(\\psi_{jt}\\), modeled function hidden state \\(h_t\\) hidden bias term \\(b_h\\). Specifically, \\(\\psi_{jt}\\) defined sigmoid activation function applied linear combination hidden state \\(h_t\\) bias \\(b_h\\) given 3 layers LSTM model:\\[\\begin{equation}\n\\psi_{jt} \\sim \\text{Sigmoid}(w_h \\cdot h_t + b_h)\n\\tag{4.8}\n\\end{equation}\\]\\[\\begin{equation}\nh_t = \\text{LSTM}\\big(\\text{temperature}_{jt}, \\ \\text{precipitation}_{jt}, \\ \\text{ENSO}_{t}, \\dots \\big)\n\\end{equation}\\]formulation, \\(h_t\\) represents hidden state generated LSTM network based input variables temperature, precipitation, ENSO conditions, \\(b_h\\) bias term added output hidden state transformation.deep learning LSTM model consists three stacked LSTM-RNN layers. first LSTM layer 500 units second third LSTM layers 250 200 units respectively. architecture LSTM model configured pass node values subsequent LSTM layers allowing deep learning complex interactions among climate variable time. enforced model sparsity LSTM layer using L2 regularization (penalty = 0.001) used dropout rate 0.5 prevent overfitting limited amount data. final output layer dense layer single unit sigmoid activation function produce probability binary classification, .e. prediction environmental suitability \\(\\psi_{jt}\\) scale 0 1.fit LSTM model data, modified learning rate applying exponential decay schedule started 0.001 decayed factor 0.9 every 10,000 steps enable smoother convergence. model compiled using Adam optimizer learning rate schedule, along binary cross-entropy loss function accuracy evaluation metric. model trained maximum 200 epochs batch size 1024. allowed model fitting stop early patience parameter 10 halts training improvement observed validation accuracy 10 consecutive epochs. train model set aside 20% observed data validation also used 20% training data model fitting. training history, including loss accuracy, monitored course training gave final test accuracy 0.73 final test loss 0.56 (see Figure 4.11).\nFigure 4.11: Model performance training validation data.\nmodel training completed, predicted values environmental suitability \\(\\psi_{jt}\\) across time steps location. Predictions start January 1970 go 5 months past present date (currently February 2025). Given amount noise model predictions, added simple LOESS spline logit transformation smooth model predictions time give stable value \\(\\psi_{jt}\\) incorporating model features (e.g. Equations (4.6) (4.7)). resulting model predicitons shown example country Mozambique Figure 4.12 compares model predictions original case counts binary classification. Predicitons model locations shown simplified view Figure 4.13.Also, please note initial version model fitted rather small amount data. Model hyper parameters specifically chosen reduce overfitting. Therefore, recommend -interpret time series predictions model early stage since likely change improve historical incidence data included future versions.\nFigure 4.12: LSTM model predictions time reported cases example country Mozambique. Reported cases shown top panel tje shaded areas show binary classification used characterize environmental suitability. Raw model predicitons shown transparent brown line solid black line showing LOESS smoothing. Forecasted values beyond current time point shown orange limited 5 month time horizon.\n\nFigure 4.13: smoothed LSTM model predictions (lines) binary suitability classification (shaded areas) time countries MOSAIC framework. Orange lines show forecasts beyond current date. ENSO DMI covariates included model, forecasts limited 5 months.\n","code":""},{"path":"model-description.html","id":"shedding","chapter":"4 Model description","heading":"4.3.3 Shedding","text":"rate infected individuals shed V. cholerae environment (\\(\\zeta\\)) critical factor influencing cholera transmission. Shedding rates can vary widely depending severity infection, immune response individual, environmental factors. According Fung 2014, shedding rate estimated range 0.01 10 cells per mL per person per day.studies support findings, indicating shedding rates can indeed fluctuate significantly. instance, Nelson et al (2009) note , depending phase infection, individuals can shed \\(10^3\\) (asymptomatic cases) \\(10^{12}\\) (severe cases) V. cholerae cells per gram stool. Future version model may attempt capture nuances shedding dynamics, make simplifying assumption shedding constant across infected individuals wide range variability prior distributional assumptions:\\[\n\\zeta \\sim \\text{Uniform}(0.01, 10).\n\\]","code":""},{"path":"model-description.html","id":"water-sanitation-and-hygiene-wash","chapter":"4 Model description","heading":"4.3.4 WAter, Sanitation, and Hygiene (WASH)","text":"Since V. cholerae transmitted fecal contamination water consumables, level exposure contaminated substrates significantly impacts transmission rates. Interventions involving Water, Sanitation, Hygiene (WASH) long first line defense reducing cholera transmission, context, WASH variables can serve proxy rate contact environmental risk factors. MOSAIC model, WASH variables incorporated mechanistically, allowing intervention scenarios include changes WASH. However, necessary distill available WASH variables single parameter represents WASH-determined contact rate contaminated substrates location \\(j\\), define \\(\\theta_j\\).parameterize \\(\\theta_j\\), calculated weighted mean 8 WASH variables Sikder et al 2023 originally modeled Local Burden Disease WaSH Collaborators 2020. 8 WASH variables (listed Table 4.4) provide population-weighted measures proportion population either: ) access WASH resources (e.g., piped water, septic sewer sanitation), ii) exposed risk factors (e.g. surface water, open defecation). risk associated WASH variables, used complement (\\(1-\\text{value}\\)) give proportion population exposed risk factor. used optim function R L-BFGS-B algorithm estimate set optimal weights (Table 4.4) maximize correlation weighted mean 8 WASH variables reported cholera incidence per 1000 population across 40 SSA countries 2000 2016. optimal weighted mean correlation coefficient \\(r =\\) -0.33 (-0.51 -0.09 95% CI) higher basic mean correlations provided individual WASH variables (see Figure 4.14). weighted mean provides single variable 0 1 represents overall proportion population access WASH /exposed environmental risk factors. Thus, WASH-mediated contact rate sources environmental transmission represented (\\(1-\\theta_j\\)) environment--human force infection (\\(\\Psi_{jt}\\)). Values \\(\\theta_j\\) countries shown Figure 4.15.\nTable 4.4: Table 4.5: Table optimized weights used calculate single mean WASH index countries.\n\nFigure 4.14: Relationship WASH variables cholera incidences.\n\nFigure 4.15: optimized weighted mean WASH variables AFRO countries. Countries labeled orange denote countries imputed weighted mean WASH variable. Imputed values weighted mean 3 similar countries.\n","code":""},{"path":"model-description.html","id":"immune-dynamics","chapter":"4 Model description","heading":"4.4 Immune dynamics","text":"","code":""},{"path":"model-description.html","id":"immunity-from-vaccination","chapter":"4 Model description","heading":"4.4.1 Immunity from vaccination","text":"impacts Oral Cholera Vaccine (OCV) campaigns incorporated model Vaccinated compartment (V). rate individuals effectively vaccinated defined \\(\\phi\\nu_tS_{jt}\\), \\(S_{jt}\\) available number susceptible individuals location \\(j\\) time \\(t\\), \\(\\nu_t\\) number OCV doses administered time \\(t\\) \\(\\phi\\) estimated vaccine effectiveness. Note just one vaccinated compartment time, though future model versions may include \\(V_1\\) \\(V_2\\) compartments explore two dose vaccination strategies emulate complex waning patterns.vaccination rate \\(\\nu_t\\) estimated quantity. Rather, directly defined reported number OCV doses administered OCV dashboard : https://www..int/groups/icg/cholera.\\[\n\\nu_t := \\text{Reported rate OCV administration} \n\\]evidence waning immunity comes 4 cohort studies (Table 4.6) Bangladesh (Qadri et al 2016 2018), South Sudan (Azman et al 2016), Democratic Republic Congo (Malembaka et al 2024).Table 4.6: Summary Effectiveness DataWe estimated vaccine effectiveness waning immunity fitting exponential decay model reported effectiveness one dose OCV studies using following formulation:\\[\\begin{equation}\n\\text{Proportion immune}\\ t \\ \\text{days vaccination} = \\phi \\times (1 - \\omega) ^ {t-t_{\\text{vaccination}}}\n\\tag{4.9}\n\\end{equation}\\]\\(\\phi\\) effectiveness one dose OCV, based specification, also initial proportion immune directly vaccination. decay rate parameter \\(\\omega\\) rate initial vaccine derived immunity decays per day post vaccination, \\(t\\) \\(t_{\\text{vaccination}}\\) time (days) function evaluated time vaccination respectively. fitted model data cohort studies shown Table (4.6) found \\(\\omega = 0.00057\\) (\\(0-0.0019\\) 95% CI), gives mean estimate 4.8 years vaccine derived immune duration unreasonably large confidence intervals (1.4 years infinite immunity). However, point estimate 4.8 years consistent anecdotes one dose OCV effective least 3 years.wide confidence intervals likely due wide range reported estimates proportion immune short duration 7–90 days range (Azman et al 2016 Qadri et al 2016). Therefore, chose use point estimate \\(\\omega\\) incorporate uncertainty based initial proportion immune (.e. vaccine effectiveness \\(\\phi\\)) shortly vaccination. Using decay model Equation (4.9) estimated \\(\\phi\\) \\(0.64\\) (\\(0.32-0.96\\) 95% CI). fit Beta distribution quantiles \\(\\phi\\) minimizing sums squares using Nelder-Mead optimization algorithm render following distribution (shown Figure 4.16B):\\[\\begin{equation}\n\\phi \\sim \\text{Beta}(4.57, 2.41).\n\\tag{4.10}\n\\end{equation}\\]\nFigure 4.16: vaccine effectiveness\n","code":""},{"path":"model-description.html","id":"immunity-from-natural-infection","chapter":"4 Model description","heading":"4.4.2 Immunity from natural infection","text":"duration immunity natural infection likely longer lasting vaccination OCV (especially given current one dose strategy). SIR-type models, rate individuals leave Recovered compartment governed immune decay parameter \\(\\varepsilon\\). estimated durability immunity natural infection based two cohort studies fit following exponential decay model estimate rate immunity decay time:\\[\n\\text{Proportion immune}\\ t \\ \\text{days infection} = 0.99 \\times (1 - \\varepsilon) ^ {t-t_{\\text{infection}}}\n\\]\nmake necessary simplifying assumption within 0–90 days natural infection V. cholerae, individuals 95–99% immune. fit model reported data Ali et al (2011) Clemens et al (1991) (see Table 4.7).Table 4.7: Sources duration immunity fro natural infection.estimated mean immune decay \\(\\bar\\varepsilon = 3.9 \\times 10^{-4}\\) (\\(1.7 \\times 10^{-4}-1.03 \\times 10^{-3}\\) 95% CI) equivalent immune duration \\(7.21\\) years (\\(2.66-16.1\\) years 95% CI) shown Figure 4.17A. slightly longer previous modeling work estimating duration immunity ~5 years (King et al 2008). Uncertainty around \\(\\varepsilon\\) model represented Log-Normal distribution shown Figure 4.17B:\\[\n\\varepsilon \\sim \\text{Lognormal}(\\bar\\varepsilon+\\frac{\\sigma^2}{2}, 0.25)\n\\]\nFigure 4.17: duration immunity natural infection V. cholerae.\n","code":""},{"path":"model-description.html","id":"spatial-dynamics","chapter":"4 Model description","heading":"4.5 Spatial dynamics","text":"parameters model diagram Figure 4.2 \\(jt\\) subscript denote spatial structure model. country modeled independent metapopulation connected others via spatial force infection \\(\\Lambda_{jt}\\) moves contagion among metapopulations according connectivity provided parameters \\(\\tau_i\\) (probability departure) \\(\\pi_{ij}\\) (probability diffusion destination \\(j\\)). parameters estimated using departure-diffusion model fitted average weekly air traffic volume 41 countries included MOSAIC framework (Figure 4.18).\nFigure 4.18: average number air passengers per week 2017 among countries.\n\nFigure 4.19: network map showing average number air passengers per week 2017.\n","code":""},{"path":"model-description.html","id":"human-mobility-model","chapter":"4 Model description","heading":"4.5.1 Human mobility model","text":"departure-diffusion model estimates diagonal -diagonal elements mobility matrix (\\(M\\)) separately combines using conditional probability rules. model first estimates probability travel outside origin location \\(\\)—departure process—distribution travel origin location \\(\\) normalizing connectivity values across \\(j\\) destinations—diffusion process. values \\(\\pi_{ij}\\) sum unity along row, diagonal included, indicating relative quantity. say, \\(\\pi_{ij}\\) gives probability going \\(\\) \\(j\\) given travel outside origin \\(\\) occurs. Therefore, can use basic conditional probability rules define travel routes diagonal elements (trips made within origin \\(\\)) \n\\[\n\\Pr( \\neg \\text{depart}_i ) = 1 - \\tau_i\n\\]\n-diagonal elements (trips made outside origin \\(\\)) \n\\[\n\\Pr( \\text{depart}_i, \\text{diffuse}_{\\rightarrow j}) = \\Pr( \\text{diffuse}_{\\rightarrow j} \\mid \\text{depart}_i ) \\Pr(\\text{depart}_i ) = \\pi_{ij} \\tau_i.\n\\]\nexpected mean number trips route \\(\\rightarrow j\\) :\\[\\begin{equation}\nM_{ij} =\n\\begin{cases}\n\\theta N_i (1-\\tau_i) \\ & \\text{} \\ = j \\\\\n\\theta N_i \\tau_i \\pi_{ij} \\ & \\text{} \\ \\ne j.\n\\end{cases}\n\\tag{4.11}\n\\end{equation}\\], \\(\\theta\\) proportionality constant representing overall number trips per person origin population size \\(N_i\\), \\(\\tau_i\\) probability leaving origin \\(\\), \\(\\pi_{ij}\\) probability travel destination \\(j\\) given travel outside origin \\(\\) occurs.","code":""},{"path":"model-description.html","id":"estimating-the-departure-process","chapter":"4 Model description","heading":"4.5.2 Estimating the departure process","text":"probability travel outside origin estimated location \\(\\) give location-specific departure probability \\(\\tau_i\\).\n\\[\n\\tau_i \\sim \\text{Beta}(1+s, 1+r)\n\\]\nBinomial probabilities origin \\(\\tau_i\\) drawn Beta distributed prior shape (\\(s\\)) rate (\\(r\\)) parameters.\n\\[\n\\begin{aligned}\ns &\\sim \\text{Gamma}(0.01, 0.01)\\\\\nr &\\sim \\text{Gamma}(0.01, 0.01)\n\\end{aligned}\n\\]","code":""},{"path":"model-description.html","id":"estimating-the-diffusion-process","chapter":"4 Model description","heading":"4.5.3 Estimating the diffusion process","text":"use normalized formulation power law gravity model defined diffusion process, probability travelling destination \\(j\\) given travel outside origin \\(\\) (\\(\\pi_{ij}\\)) defined :\\[\\begin{equation}\n\\pi_{ij} = \\frac{\nN_j^\\omega d_{ij}^{-\\gamma}\n}{\n\\sum\\limits_{\\forall j \\ne } N_j^\\omega d_{ij}^{-\\gamma}\n}\n\\tag{4.12}\n\\end{equation}\\], \\(\\omega\\) scales attractive force \\(j\\) destination based population size \\(N_j\\). kernel function \\(d_{ij}^{-\\gamma}\\) serves penalty proportion travel \\(\\) \\(j\\) based distance. Prior distributions diffusion model parameters defined :\n\\[\n\\begin{aligned}\n\\omega &\\sim \\text{Gamma}(1, 1)\\\\\n\\gamma &\\sim \\text{Gamma}(1, 1)\n\\end{aligned}\n\\]models \\(\\tau_i\\) \\(\\pi_{ij}\\) fitted air traffic data OAG using mobility R package (Giles 2020). Estimates mobility model parameters shown Figures 4.20 4.21.\nFigure 4.20: estimated weekly probability travel outside origin location \\(\\tau_i\\) 95% confidence intervals shown panel population mean indicated red dashed line. Panel B shows estimated total number travelers leaving origin \\(\\) week.\n\nFigure 4.21: diffusion process \\(\\pi_{ij}\\) gives estimated probability travel origin \\(\\) destination \\(j\\) given travel outside origin \\(\\) occurred.\n","code":""},{"path":"model-description.html","id":"the-probability-of-spatial-transmission","chapter":"4 Model description","heading":"4.5.4 The probability of spatial transmission","text":"likelihood introductions cholera disparate locations major concern cholera outbreaks. However, can difficult characterize given endemic dynamics patterns human movement. include measures spatial heterogeneity first simple importation probability based connectivity possibility incoming infections. basic probability transmission origin \\(\\) particular destination \\(j\\) time \\(t\\) defined :\\[\\begin{equation}\np(,j,t) = 1 - e^{-\\beta_{jt}^{\\text{hum}} (((1-\\tau_j)S_{jt})/N_{jt}) \\pi_{ij}\\tau_iI_{}}\n\\tag{4.13}\n\\end{equation}\\]","code":""},{"path":"model-description.html","id":"the-spatial-hazard","chapter":"4 Model description","heading":"4.5.5 The spatial hazard","text":"Although concerned endemic dynamics , likely periods time early rainy season cholera cases rate transmission low enough spatial spread resemble epidemic dynamics time. times periods, can estimate arrival time contagion location cases yet reported. estimating spatial hazard transmission:\\[\\begin{equation}\nh(j,t) = \\frac{\n\\beta_{jt}^{\\text{hum}} \\Big(1 - \\exp\\big(-((1-\\tau_j)S_{jt}/N_{jt}) \\sum_{\\forall \\= j} \\pi_{ij}\\tau_i (I_{}/N_{}) \\big) \\Big)\n}{\n1/\\big(1 + \\beta_{jt}^{\\text{hum}} (1-\\tau_j)S_{jt}\\big)\n}.\n\\tag{4.14}\n\\end{equation}\\]normalizing give waiting time distribution locations:\\[\\begin{equation}\nw(j,t) = h(j,T) \\prod_{t=1}^{T-1}1-h(j,t).\n\\tag{4.15}\n\\end{equation}\\]","code":""},{"path":"model-description.html","id":"coupling-among-locations","chapter":"4 Model description","heading":"4.5.6 Coupling among locations","text":"Another measure spatial heterogeneity quantify coupling disease dynamics among metapopulations using correlation coefficient. , use definition spatial correlation locations \\(\\) \\(j\\) \\(C_{ij}\\) described Keeling Rohani (2002), gives measure similar infection dynamics locations.\\[\\begin{equation}\nC_{ij} = \\frac{\n( y_{} - \\bar{y}_i )( y_{jt} - \\bar{y}_j )\n}{\n\\sqrt{\\text{var}(y_i) \\text{var}(y_j)}\n}\n\\tag{4.16}\n\\end{equation}\\]\n\\(y_{} = I_{}/N_i\\) \\(y_{jt} = I_{jt}/N_j\\). Mean prevalence location \\(\\bar{y_i} = \\frac{1}{T} \\sum_{t=1}^{T} y_{}\\) \\(\\bar{y_j} = \\frac{1}{T} \\sum_{t=1}^{T} y_{jt}\\).","code":""},{"path":"model-description.html","id":"the-observation-process","chapter":"4 Model description","heading":"4.6 The observation process","text":"","code":""},{"path":"model-description.html","id":"rate-of-symptomatic-infection","chapter":"4 Model description","heading":"4.6.1 Rate of symptomatic infection","text":"presentation infection V. cholerae can extremely variable. severity infection depends many factors amount infectious dose, age host, level immunity host either vaccination previous infection, naivety particular strain V. cholerae. Additional circumstantial factors nutritional status overall pathogen burden may also impact infection severity. population level, observed proportion infections symptomatic also dependent endemicity cholera region. Highly endemic areas (e.g. parts Bangladesh; Hegde et al 2024) may low proportion symptomatic infections due many previous exposures. Inversely, populations largely naive V. cholerae exhibit relatively higher proportion symptomatic infections (e.g. Haiti; Finger et al 2024).Accounting nuances first version model possible, can past studies contain information can help set sensible bounds definition proportion infections symptomatic (\\(\\sigma\\)). compiled short list studies done sero-surveys cohort studies assess likelihood symptomatic infections different locations displayed results Table (4.8).provide reasonably informed prior proportion infections symptomatic, calculated combine mean confidence intervals studies Table 4.8 fit Beta distribution corresponds quantiles using least-squares Nelder-Mead algorithm. resulting prior distribution symptomatic proportion \\(\\sigma\\) :\\[\\begin{equation}\n\\sigma \\sim \\text{Beta}(4.30, 13.51)\n\\end{equation}\\]Table 4.8: Summary Studies Cholera ImmunityThe prior distribution \\(\\sigma\\) plotted Figure 4.22A reported values proportion symptomatic previous studies shown 4.22B.\nFigure 4.22: Proportion infections symptomatic.\n","code":""},{"path":"model-description.html","id":"suspected-cases","chapter":"4 Model description","heading":"4.6.2 Suspected cases","text":"clinical presentation diarrheal diseases often similar across various pathogens, can lead systematic biases reported number cholera cases. anticipated number suspected cholera cases related actual number infections factor \\(1/\\rho\\), \\(\\rho\\) represents proportion suspected cases true infections. adjust bias, use estimates meta-analysis Weins et al. (2023), suggests suspected cholera cases outnumber true infections approximately 2 1, mean across studies indicating 52% (24-80% 95% CI) suspected cases actual cholera infections. higher estimate reported ourbreak settings (78%, 40-99% 95% CI). account variability estimate, fit Beta distribution reported quantiles using least squares approach Nelder-Mead algorithm, resulting prior distribution shown Figure 4.23B:\\[\\begin{equation}\n\\rho \\sim \\text{Beta}(4.79, 1.53).\n\\end{equation}\\]\nFigure 4.23: Proportion suspected cholera cases true infections. Panel shows ‘low’ assumption estimates across settings: \\(\\rho \\sim \\text{Beta}(5.43, 5.01)\\). Panel B shows ‘high’ assumption estimate reflects high-quality studies outbreaks: \\(\\rho \\sim \\text{Beta}(4.79, 1.53)\\)\n","code":""},{"path":"model-description.html","id":"case-fatality-rate","chapter":"4 Model description","heading":"4.6.3 Case fatality rate","text":"Case Fatality Rate (CFR) among symptomatic infections calculated using reported cases deaths data January 2021 August 2024. data collated various issues Weekly Epidemiological Record Global Cholera Acute Watery Diarrhea (AWD) Dashboard (see Data section) provide annual aggregations reported cholera cases deaths. used Binomial exact test (binom.test R) calculate mean probability number deaths (successes) given number reported cases (sample size), Clopper-Pearson method calculating binomial confidence intervals. fit Beta distributions mean CFR 95% confidence intervals calculated country using least squares Nelder-Mead algorithm give distributional uncertainty around CFR estimate country (\\(\\mu_j\\)).\\[\n\\mu_j \\sim \\text{Beta}(s_{1,j}, s_{2,j})\n\\]\\(s_{1,}\\) \\(s_{2,j}\\) two positive shape parameters Beta distribution estimated destination \\(j\\). definition \\(\\mu_j\\) CFR reported cases subset total number infections. Therefore, infer total number deaths attributable cholera infection, assume CFR observed cases proportionally equivalent CFR cases calculate total deaths \\(D\\) follows:\\[\\begin{equation}\n\\begin{aligned}\n\\text{CFR}_{\\text{observed}} &= \\text{CFR}_{\\text{total}}\\\\\n\\\\[3pt]\n\\frac{[\\text{observed deaths}]}{[\\text{observed cases}]} &=\n\\frac{[\\text{total deaths}]}{[\\text{infections}]}\\\\\n\\\\[3pt]\n\\text{total deaths} &= \\frac{[\\text{observed deaths}] \\times [\\text{true infections}]}{[\\text{observed cases}]}\\\\\n\\\\[3pt]\nD_{jt} &= \\frac{ [\\sigma\\rho\\mu_j I_{jt}] \\times [I_{jt}] }{ [\\sigma\\rho I_{jt}] }\n\\end{aligned}\n\\end{equation}\\]\nTable 4.9: Table 4.10: CFR Values Beta Shape Parameters AFRO Countries\n\nFigure 4.24: Case Fatality Rate (CFR) Total Cases Country AFRO Region 2014 2024. Panel : Case Fatality Ratio (CFR) 95% confidence intervals. Panel B: total number cholera cases. AFRO Region highlighted black, countries less 3/0.2 = 150 total reported cases assigned mean CFR AFRO.\n\nFigure 4.25: Beta distributions overall Case Fatality Rate (CFR) 2014 2024. Examples show overall CFR AFRO region (2%) black, Congo highest CFR (7%) red, South Sudan lowest CFR (0.1%) blue.\n","code":""},{"path":"model-description.html","id":"demographics-1","chapter":"4 Model description","heading":"4.7 Demographics","text":"model includes basic demographic change using reported birth death rates \\(j\\) countries, \\(b_j\\) \\(d_j\\) respectively. rates static defined United Nations Department Economic Social Affairs Population Division World Population Prospects 2024. Values \\(b_j\\) \\(d_j\\) derived crude rates converted birth rate per day death rate per day (shown Table 4.11).\nTable 4.11: Table 4.12: Demographic AFRO countries 2023. Data include: total population January 1, 2023, daily birth rate, daily death rate. Values calculate crude birth death rates UN World Population Prospects 2024.\n","code":""},{"path":"model-description.html","id":"the-reproductive-number","chapter":"4 Model description","heading":"4.8 The reproductive number","text":"reproductive number common metric epidemic growth represents average number secondary cases generated primary case specific time epidemic. track \\(R\\) changes time estimating instantaneous reproductive number \\(R_t\\) described Cori et al 2013. track \\(R_t\\) across metapopulations model give \\(R_{jt}\\) using following formula:\\[\\begin{equation}\nR_{jt} = \\frac{I_{jt}}{\\sum_{\\Delta t=1}^{t} g(\\Delta t) I_{j,t-\\Delta t}}\n\\tag{4.17}\n\\end{equation}\\]\\(I_{jt}\\) number new infections destination \\(j\\) time \\(t\\), \n\\(g(\\Delta t)\\) represents probability value generation time distribution cholera. accomplished using weighed sum denominator highly influenced generation time distribution.","code":""},{"path":"model-description.html","id":"the-generation-time-distribution","chapter":"4 Model description","heading":"4.8.1 The generation time distribution","text":"generation time distribution gives time individual infected infect subsequent individuals. parameterized quantity using Gamma distribution mean 5 days:\\[\\begin{equation}\ng(\\cdot) \\sim \\text{Gamma}(0.5, 0.1).\n\\tag{4.18}\n\\end{equation}\\], shape=0.5, rate=0.1, mean given shape/rate. Previous studies use mean 5 days (Kahn et al 2020 Azman 2016), however mean 3, 5, 7, 10 days may admissible (Azman 2012).\nFigure 4.26: generation time\n\nTable 4.13: Table 4.14: Generation Time Weeks\n","code":""},{"path":"model-description.html","id":"initial-conditions","chapter":"4 Model description","heading":"4.9 Initial conditions","text":"Since first version model begin Jan 2023 (take advantage available weekly data), initial conditions surrounding population immunity must estimated. set initial conditions, use historical data find total number reported cases location previous X years, multiply \\(1/\\sigma\\) estimate total infections symptomatic cases reported, adjust based waning immunity. also sum total number vaccinations past X years adjust vaccine efficacy \\(\\phi\\) waning immunity vaccination \\(\\omega\\).total number infected? reported cases… back symptomatic asymptomatictotal number infected? reported cases… back symptomatic asymptomaticTotal number immune due natural infections past X yearsTotal number immune due natural infections past X yearstotal number immune due past vaccinations X yearstotal number immune due past vaccinations X yearsUse deconvolution based immune decay estimated vaccine section","code":""},{"path":"model-description.html","id":"model-calibration","chapter":"4 Model description","heading":"4.10 Model calibration","text":"model calibrated using Latin hypercube sampling hyper-parameters model likelihoods fit incidence deaths.model calibrated using Latin hypercube sampling hyper-parameters model likelihoods fit incidence deaths.important challenge flexibly fitting data often missing available aggregated forms.important challenge flexibly fitting data often missing available aggregated forms.[Fig: different spatial temporal scales available data]","code":""},{"path":"model-description.html","id":"caveats","chapter":"4 Model description","heading":"4.11 Caveats","text":"Simplest model start. Easier initial spatial structure minimum additional compartments calibrate available data (vaccination, cases, deaths).Country level aggregations. First generation data 2023/24…Assumes vaccinating susceptible individuals.climate, summarizing whole country.","code":""},{"path":"model-description.html","id":"table-of-parameters","chapter":"4 Model description","heading":"4.12 Table of parameters","text":"Table 4.15: Descriptions model parameters along prior distributions sources applicable.","code":""},{"path":"model-description.html","id":"references","chapter":"4 Model description","heading":"4.13 References","text":"","code":""},{"path":"scenarios.html","id":"scenarios","chapter":"5 Scenarios","heading":"5 Scenarios","text":"key aim MOSAIC model provide near-term forecasts cholera transmission Sub-Saharan Africa (SSA) using current data available. However, MOSAIC just forecasting tool; dynamic model designed explore various scenarios influence critical factors vaccination, environmental conditions, Water, Sanitation, Hygiene (WASH) interventions.","code":""},{"path":"scenarios.html","id":"vaccination","chapter":"5 Scenarios","heading":"5.1 Vaccination","text":"","code":""},{"path":"scenarios.html","id":"spatial-and-temporal-strategies","chapter":"5 Scenarios","heading":"5.1.1 Spatial and Temporal Strategies","text":"Understanding spatial temporal distribution cholera vaccination efforts crucial effective outbreak control. Key resources include:Stockpile Status: availability oral cholera vaccine emergency stockpiles can tracked UNICEF’s Emergency Stockpile Availability.OCV Dashboard: dashboard (link) provides insights deployment oral cholera vaccines (OCV) across different regions.","code":""},{"path":"scenarios.html","id":"reactive-vaccination","chapter":"5 Scenarios","heading":"5.1.2 Reactive Vaccination","text":"timing logistics reactive vaccination campaigns critical controlling ongoing outbreaks. Relevant resources include:Recommended Timing: Guidelines recommendations timing reactive OCV campaigns available (link).Requests Delay Time Distributions: Information vaccine request processes distribution delays vaccine deployment can accessed GTFCC OCV Dashboard (link).","code":""},{"path":"scenarios.html","id":"impacts-of-climate-change","chapter":"5 Scenarios","heading":"5.2 Impacts of Climate Change","text":"","code":""},{"path":"scenarios.html","id":"severe-weather-events","chapter":"5 Scenarios","heading":"5.2.1 Severe Weather Events","text":"Projections climate shocks, including frequency severity cyclones floods, essential modeling future impacts climate change cholera transmission. Key references include:Chen Chavas 2020: study cyclone season dynamics climate change scenarios (link).Sparks Toumi 2024: Research projected flood frequencies due climate change (link).Switzer et al. 2023: analysis climate shock impacts cholera outbreaks (link).","code":""},{"path":"scenarios.html","id":"long-term-trends","chapter":"5 Scenarios","heading":"5.2.2 Long-Term Trends","text":"Long-term trends weather variables various climate change scenarios can explored using following resource:Weather Variables Climate Change: OpenMeteo Climate API provides access projected weather data different climate change scenarios (link).","code":""},{"path":"usage.html","id":"usage","chapter":"6 Usage","heading":"6 Usage","text":"open-source code used run MOSAIC currently development presented future.","code":""},{"path":"news.html","id":"news","chapter":"7 News","heading":"7 News","text":"","code":""},{"path":"news.html","id":"past-versions-of-mosaic","chapter":"7 News","heading":"7.1 Past versions of MOSAIC","text":"Table 7.1: Current future planned model versions brief descriptions.","code":""},{"path":"references-1.html","id":"references-1","chapter":"8 References","heading":"8 References","text":"","code":""}] +[{"path":"index.html","id":"section","chapter":"","heading":"","text":"","code":""},{"path":"index.html","id":"welcome","chapter":"","heading":"Welcome","text":"Welcome Metapopulation Outbreak Simulation Agent-based Implementation Cholera (MOSAIC). MOSAIC framework simulates transmission dynamics cholera Sub-Saharan Africa (SSA) provides tools understand impact interventions, vaccination, well large-scale drivers like climate change. MOSAIC built using Light-agent Spatial Model ERadication (LASER) platform, site serves documentation model’s methods associated analyses. Please note MOSAIC currently development, content may change regularly. sharing increase visibility welcome feedback aspect model.","code":""},{"path":"index.html","id":"contact","chapter":"","heading":"Contact","text":"MOSAIC developed team researchers Institute Disease Modeling (IDM) dedicated developing modeling methods software tools help decision-makers understand respond infectious disease outbreaks. website currently maintained John Giles (@gilesjohnr). general questions, contact John Giles (john.giles@gatesfoundation.org), Jillian Gauld (jillian.gauld@gatesfoundation.org), /Rajiv Sodhi (rajiv.sodhi@gatesfoundation.org).","code":""},{"path":"index.html","id":"funding","chapter":"","heading":"Funding","text":"work developed Institute Disease Modeling support funded research grants made Bill & Melinda Gates Foundation.","code":""},{"path":"introduction.html","id":"introduction","chapter":"1 Introduction","heading":"1 Introduction","text":"","code":""},{"path":"introduction.html","id":"history","chapter":"1 Introduction","heading":"1.1 History","text":"Vibrio cholerae introduced African continent Asia 1970s since become endemic many countries.","code":""},{"path":"introduction.html","id":"recent-surge","chapter":"1 Introduction","heading":"1.2 Recent Surge","text":"Although sporadic cholera outbreaks past five decades, significant surge cases since 2021. increase likely due factors climate change disruptions municipal services COVID-19 pandemic.","code":""},{"path":"introduction.html","id":"gtfcc-goals","chapter":"1 Introduction","heading":"1.3 GTFCC Goals","text":"Global Task Force Cholera Control (GTFCC) aims reduce cholera deaths 90% 2030.","code":""},{"path":"introduction.html","id":"ocv-stockpiles","chapter":"1 Introduction","heading":"1.4 OCV Stockpiles","text":"major concern recent surge cases depletion oral cholera vaccine (OCV) stockpiles. response, officials shifted single-dose OCV strategies. Efforts underway replenish stockpiles, key question best allocate reduce transmission regionally support GTFCC’s goal.","code":""},{"path":"introduction.html","id":"climate-change","chapter":"1 Introduction","heading":"1.5 Climate Change","text":"Environmental factors play significant role cholera outbreaks, warmer wetter conditions creating favorable environment Vibrio cholerae. conditions likely exacerbate already challenging endemic outbreak settings. Models incorporate climatic forcing can provide insights future cholera dynamics due climate change aid achieving GTFCC goal.","code":""},{"path":"rationale.html","id":"rationale","chapter":"2 Rationale","heading":"2 Rationale","text":"significant challenge controlling cholera transmission Sub-Saharan Africa (SSA) lack comprehensive datasets dynamic models designed support ongoing policy-making. persistent endemic nature cholera SSA presents complex quantitative challenge, requiring sophisticated models produce meaningful inferences. Models incorporate necessary natural history disease dynamics, operate adequate spatial temporal scales, crucial providing timely actionable information address ongoing future cholera outbreaks.Although developing data models scales challenging, goal iteratively create landscape-scale transmission model cholera SSA can provide weekly predictions key epidemiological metrics. modeling methods leverage wide array --date data sources, including incidence mortality reports, patterns human movement, vaccination history schedules, environmental factors.Key questions address include administer limited supply oral cholera vaccine (OCV) severe weather events climate change impact future outbreaks. landscape-scale model accounts endemic transmission patterns valuable tool addressing questions.","code":""},{"path":"data.html","id":"data","chapter":"3 Data","heading":"3 Data","text":"MOSAIC model requires diverse set data sources, directly used define model parameters (e.g., birth death rates), others help fit models priori provide informative priors transmission model. additional data sources become available, future versions model adapt incorporate . now, following data sources represent minimum requirements initiate viable first model.","code":""},{"path":"data.html","id":"historical-incidence-and-deaths","chapter":"3 Data","heading":"3.1 Historical Incidence and Deaths","text":"Data historical cholera incidence deaths crucial establishing baseline transmission patterns. compiled annual total reported cases deaths AFRO region countries January 1970 August 2024. data comes several sources include:World Data (1970-2021): Number Reported Cases Cholera (1949-2021) Number Reported Deaths Cholera (1949-2021). World Data group compiled data previously published annual reports.Annual Report 2022: data manually extracted World Health Organization’s Weekly Epidemiological Record 38, 2023, 98, 431–452.Global Cholera Acute Watery Diarrhea Dashboard (2023-2024): Unofficial tallies reported cases deaths 2023 part 2024 available Global Cholera AWD Dashboard.","code":""},{"path":"data.html","id":"recent-incidence-and-deaths","chapter":"3 Data","heading":"3.2 Recent Incidence and Deaths","text":"capture recent cholera trends, retrieved reported cases deaths data Global Cholera Acute Watery Diarrhea Dashboard REST API. data provide weekly incidence deaths January 2023 August 2024 provides --date counts country level.","code":""},{"path":"data.html","id":"vaccinations","chapter":"3 Data","heading":"3.3 Vaccinations","text":"Accurate data oral cholera vaccine (OCV) campaigns vaccination history vital understanding impact vaccination efforts. data come :Cholera Vaccine Dashboard: resource (link) provides detailed information OCV distribution vaccination campaigns 2016 2024.GTFCC OCV Dashboard: Managed Médecins Sans Frontières, dashboard (link) tracks OCV deployments globally, offering granular insights vaccination efforts 2013 2024.","code":""},{"path":"data.html","id":"human-mobility-data","chapter":"3 Data","heading":"3.4 Human Mobility Data","text":"Human mobility patterns significantly influence cholera transmission. Relevant data include:OAG Passenger Booking Data: dataset (link) offers insights air passenger movements, can used model spread cholera across regions.Namibia Call Data Records: additional source Giles et al. (2020) (link) provides detailed mobility data based mobile phone records, useful localized modeling.","code":""},{"path":"data.html","id":"climate-data","chapter":"3 Data","heading":"3.5 Climate Data","text":"Climate conditions, including temperature, precipitation, extreme weather events, play critical role cholera dynamics. captured :OpenMeteo Historical Weather Data API: API (link) offers access historical climate data, essential modeling environmental factors influencing cholera outbreaks.","code":""},{"path":"data.html","id":"storms-and-floods","chapter":"3 Data","heading":"3.5.1 Storms and Floods","text":"Data extreme weather events, specifically storms floods, obtained :EM-DAT International Disaster Database: Maintained Centre Research Epidemiology Disasters (CRED) UCLouvain, database (link) provides comprehensive records disasters 2000 present, including affecting African countries.","code":""},{"path":"data.html","id":"wash-water-sanitation-and-hygiene","chapter":"3 Data","heading":"3.6 WASH (Water, Sanitation, and Hygiene)","text":"Data water, sanitation, hygiene (WASH) critical understanding environmental infrastructural factors influence cholera transmission. data sourced :UNICEF Joint Monitoring Program (JMP) Database: resource (link) offers detailed information household-level access clean water sanitation, integral cholera prevention efforts.","code":""},{"path":"data.html","id":"demographics","chapter":"3 Data","heading":"3.7 Demographics","text":"Demographic data, including population size, birth rates, death rates, foundational accurate disease modeling. data sourced :UN World Population Prospects 2024: database (link) provides probabilistic projections key demographic metrics, essential estimating population-level impacts cholera.","code":""},{"path":"model-description.html","id":"model-description","chapter":"4 Model description","heading":"4 Model description","text":"describe methods MOSAIC beta version 0.1. model version provides starting point understanding cholera transmission Sub-Saharan Africa, incorporating important drivers disease dynamics human mobility, environmental conditions, vaccination schedules. MOSAIC continues evolve, future iterations refine model components based available data improved model mechanisms, hope increase applicability real-world scenarios.model operates weekly time steps January 2023 August 2024 includes 41 countries Sub-Saharan Africa (SSA), see Figure 4.1.\nFigure 4.1: map Sub-Saharan Africa countries experienced cholera outbreak past 5 10 years highlighted green.\n","code":""},{"path":"model-description.html","id":"transmission-dynamics","chapter":"4 Model description","heading":"4.1 Transmission dynamics","text":"model metapopulation structure familiar compartments Susceptible, Infected, Recovered individuals SIRS dynamics. model also contains compartments vaccinated individuals (V) Water & environment based transmission (W) refer SVIWRS.\nFigure 4.2: diagram SVIWRS (Susceptible-Vaccinated-Infected-Water/environmental-Recovered-Susceptible) model shows model compartments circles rate parameters displayed. primary data sources model fit shown square nodes (Vaccination data, reported cases deaths).\nSVIWRS metapopulation model, shown Figure 4.2, governed following difference equations:\\[\\begin{equation}\n\\begin{aligned}\nS_{j,t+1} &= b_j N_{jt} + S_{jt} - \\phi \\nu_{jt} S_{jt} + \\omega V_{jt} - \\Lambda_{j,t+1} - \\Psi_{j,t+1} + \\varepsilon R_{jt} - d_j S_{jt}\\\\[11pt]\nV_{j,t+1} &= V_{jt} + \\phi \\nu_{jt} S_{jt} - \\omega V_{jt} - d_j V_{jt}\\\\[11pt]\nI_{j,t+1} &= I_{jt} + \\Lambda_{j,t+1} + \\Psi_{j,t+1} - \\gamma I_{jt} - \\mu \\sigma I_{jt} - d_j I_{jt}\\\\[11pt]\nW_{j,t+1} &= W_{jt} + \\zeta I_{jt} - \\delta_{jt} W_{jt}\\\\[11pt]\nR_{j,t+1} &= R_{jt} + \\gamma I_{jt} - \\varepsilon R_{jt} - d_j R_{jt}\\\\[11pt]\n\\end{aligned}\n\\tag{4.1}\n\\end{equation}\\]descriptions parameters Equation (4.1), see Table (4.15). Transmission dynamics driven two force infection terms, \\(\\Lambda_{jt}\\) \\(\\Psi_{jt}\\). force infection due human--human (\\(\\Lambda_{jt}\\)) :\\[\\begin{equation}\n\\begin{aligned}\n\\Lambda_{j,t+1} &= \\frac{\n\\beta_{jt}^{\\text{hum}} \\Big(\\big(S_{jt}(1-\\tau_{j})\\big) \\big(I_{jt}(1-\\tau_{j}) + \\sum_{\\forall \\= j} (\\pi_{ij}\\tau_iI_{}) \\big)\\Big)^\\alpha}{N_{jt}}.\\\\[11pt]\n\\end{aligned}\n\\tag{4.2}\n\\end{equation}\\]\\(\\beta_{jt}^{\\text{hum}}\\) rate human--human transmission. Movement within among metapopulations governed \\(\\tau_i\\), probability departing origin location \\(\\), \\(\\pi_{ij}\\) relative probability travel origin \\(\\) destination \\(j\\) (see section spatial dynamics). include environmental effects, force infection due environment--human transmission (\\(\\Psi_{jt}\\)) defined :\\[\\begin{equation}\n\\begin{aligned}\n\\Psi_{j,t+1} &= \\frac{\\beta_{jt}^{\\text{env}} \\big(S_{jt}(1-\\tau_{j})\\big) (1-\\theta_j) W_{jt}}{\\kappa+W_{jt}},\\\\[11pt]\n\\end{aligned}\n\\tag{4.3}\n\\end{equation}\\]\\(\\beta_{jt}^{\\text{env}}\\) rate environment--human transmission \\(\\theta_j\\) proportion population location \\(j\\) least basic access Water, Sanitation, Hygiene (WASH). environmental compartment model also scaled concentration (cells per mL) V. cholerae required 50% probability infection Fung 2014. See section environmental transmission water/environment compartment climatic drivers transmission.Note model processes stochastic. Transition rates converted probabilities commonly used formula \\(p(t) = 1-e^{-rt}\\) (see Ross 2007), integer quantities moved model compartments time step according binomial process like example recovery infected individuals (\\(\\gamma I_{jt}\\)):\\[\\begin{equation}\n\\frac{\\partial R}{\\partial t} \\sim \\text{Binom}(I_{jt}, 1-\\exp(-\\gamma))\n\\tag{4.4}\n\\end{equation}\\]","code":""},{"path":"model-description.html","id":"seasonality","chapter":"4 Model description","heading":"4.2 Seasonality","text":"Cholera transmission seasonal typically associated rainy season, transmission rate terms \\(\\beta_{jt}^{\\text{*}}\\) temporally forced. human--human transmission used truncated sine-cosine form Fourier series two harmonic features flexibility capture seasonal transmission dynamics driven extended rainy seasons /biannual trends:\\[\\begin{equation}\n\\beta_{jt}^{\\text{hum}} = \\beta_{j0}^{\\text{hum}} + a_1 \\cos\\left(\\frac{2\\pi t}{p}\\right) + b_1 \\sin\\left(\\frac{2\\pi t}{p}\\right) + a_2 \\cos\\left(\\frac{4\\pi t}{p}\\right) + b_2 \\sin\\left(\\frac{4\\pi t}{p}\\right)\n\\tag{4.5}\n\\end{equation}\\], \\(\\beta_{j0}^{\\text{hum}}\\) mean human--human transmission rate location \\(j\\) time steps. Seasonal dynamics determined parameters \\(a_1\\), \\(b_1\\) \\(a_2\\), \\(b_2\\) gives amplitude first second waves respectively. periodic cycle \\(p\\) 52, function controls temporal variation \\(\\beta_{jt}^{\\text{hum}}\\) 52 weeks year.\nFigure 4.3: example temporal distribution human--human transmission rate across 52 weeks year given cosine wave function. wave function fitted country designed align rainy season indicated shaded region figure.\nestimated parameters Fourier series (\\(a_1\\), \\(b_1\\), \\(a_2\\), \\(b_2\\)) using Levenberg–Marquardt algorithm minpack.lm R library. Given lack reported cholera case data many countries SSA association cholera transmission rainy season, leveraged seasonal precipitation data help fit Fourier wave function countries. first gathered weekly precipitation values 1994 2024 30 uniformly distributed points within country Open-Meteo Historical Weather Data API. fit Fourier series weekly precipitation data used parameters starting values fitting model sparse cholera case data.\nFigure 4.4: Example grid 30 uniformly distributed points within Mozambique (). scatterplot shows weekly summed precipitation values 30 grid points cholera cases plotted scale Z-Score shows variance around mean terms standard deviation. Fitted Fourier series fucntions shown blue (fit precipitation data) red (fit cholera case data) lines.\ncountries reported case data, inferred seasonal dynamics using fitted wave function neighboring country available case data. selected neighbor chosen cluster countries (grouped hierarchically four clusters based precipitation seasonality using Ward’s method; see Figure 4.5) highest correlation seasonal precipitation country lacking case data. rare event country reported case data found within seasonal cluster, expanded search 10 nearest neighbors continued expanding adding next nearest neighbor match found.\nFigure 4.5: ) Map showing clustering African countries based seasonal precipitation patterns (1994-2024). Countries colored according cluster assignments, identified using hierarchical clustering. B) Fourier series fitted weekly precipitation country. line plot shows seasonal pattern countries within given cluster. Clusteres used infer seasonal transmission dynamics countries reported cholera cases.\nUsing model fitting methods described , cluster-based approach inferring seasonal Fourier series pattern countries without reported cholera cases, modeled seasonal dynamics 41 countries MOSAIC framework. dynamics visualized Figure 4.6, corresponding Fourier model coefficients presented Table 4.1.\nFigure 4.6: Seasonal transmission patterns countries modeled MOSAIC modeled truncated Fourier series Equation (4.5). Blues lines give Fourier series model fits precipitation (1994-2024) red lines give models fits reported cholera cases (2023-2024). countries reported case data available, Fourier model inferred nearest country similar seasonal precipitation patterns determined hierarchical clustering. Countries inferred case data neighboring locations annotated red. X-axis represents weeks year (1-52), Y-axis shows Z-score weekly precipitation cholera cases.\n\nTable 4.1: Table 4.2: Estimated coefficients truncated Fourier model Equation (4.5) fit countries reported cholera cases. Model fits shown Figure 4.6.\n","code":""},{"path":"model-description.html","id":"environmental-transmission","chapter":"4 Model description","heading":"4.3 Environmental transmission","text":"Environmental transmission critical factor cholera spread consists several key components: rate infected individuals shed V. cholerae environment, pathogen’s survival rate environmental conditions, overall suitability environment sustaining bacteria time.","code":""},{"path":"model-description.html","id":"climate-driven-transmission","chapter":"4 Model description","heading":"4.3.1 Climate-driven transmission","text":"capture impacts climate-drivers cholera transmission, included parameter \\(\\psi_{jt}\\), represents current state environmental suitability respect : ) survival time V. cholerae environment , ii) rate environment--human transmission contributes overall force infection.\\[\\begin{equation}\n\\beta_{jt}^{\\text{env}} = \\beta_{j0}^{\\text{env}} \\Bigg(1 + \\frac{\\psi_{jt}-\\bar\\psi_j}{\\bar\\psi_j} \\Bigg) \\quad \\text{} \\quad \\bar\\psi_j = \\frac{1}{T} \\sum_{t=1}^{T} \\psi_{jt}\n\\tag{4.6}\n\\end{equation}\\]formulation effectively scales base environmental transmission rate \\(\\beta_{jt}^{\\text{env}}\\) varies time according climatically driven model suitability. Note , unlike cosine wave function \\(\\beta_{jt}^{\\text{hum}}\\), temporal term can increase decrease time following multi-annual cycles.Environmental suitability (\\(\\psi_{jt}\\)) also impacts survival rate V. cholerae environment (\\(\\delta_{jt}\\)) form:\\[\\begin{equation}\n\\delta_{jt} = \\delta_{\\text{min}} + \\psi_{jt} \\times (\\delta_{\\text{max}} - \\delta_{\\text{min}})\n\\tag{4.7}\n\\end{equation}\\]normalizes variance suitability parameter bounded within minimum (\\(\\delta_{\\text{min}}\\)) maximum (\\(\\delta_{\\text{max}}\\)) survival times V. cholerae.\nFigure 4.7: Relationship environmental suitability (\\(\\psi_{jt}\\)) rate V. cholerae decay environment (\\(\\delta_j\\)). green line shows mildest penalty V. cholerae survival, survival environment \\(1/\\delta_{\\text{min}}\\) = 3 days suitability = 0 \\(1/\\delta_{\\text{max}}\\) = 90 days suitability = 1.\n","code":""},{"path":"model-description.html","id":"modeling-environmental-suitability","chapter":"4 Model description","heading":"4.3.2 Modeling environmental suitability","text":"","code":""},{"path":"model-description.html","id":"environmental-data","chapter":"4 Model description","heading":"4.3.2.1 Environmental data","text":"mechanism environment--human transmission (Equation (4.6)) rate decay V. cholerae environment (Equation (4.7)) driven parameter \\(\\psi_{jt}\\), refer environmental suitability. parameter \\(\\psi_{jt}\\) modeled time series location using Long Short-Term Memory (LSTM) Recurrent Neural Network (RNN) model suite 24 covariates include 19 historical forecasted climate variables MRI-AGCM3-2-S climate model. Covariates also include 4 large-scale climate drivers Indian Ocean Dipole Mode Index (DMI), El Niño Southern Oscillation (ENSO) 3 different Pacific Ocean regions. also included location specific variable giving mean elevation country. See example time series climate variables one country (Mozambique) Figure 4.8 DMI ENSO variables Figure 4.9. list covariates sources can seen Table 4.3.Note 19 climate variables offer forecasts 2030 beyond, forecasts DMI ENSO variables limited 5 months future. , environmental suitability model predictions currently limited 5 month time horizon future iterations may allow longer forecasts. Additional data sources integrated subsequent versions suitability model. instance, flood cyclone data likely incorporated later, though initial version model.\nFigure 4.8: Climate data acquired OpenMeteo data API. Data collected 30 uniformly distributed points across country aggregated give weekly values 17 climate variable 1970 2030.\n\nFigure 4.9: Historical forecasted values Indian Ocean Dipole Mode Index (DMI) El Niño Southern Oscillation (ENSO) 2015 2025. ENSO values come three different regions: Niño3 (central eastern Pacific), Niño3.4 (central Pacific), Niño4 (western-central Pacifi). Data National Oceanic Atmospheric Administration (NOAA) Bureau Meteorology (BOM).\nTable 4.3: full list covariates sources used LSTM RNN model predict environmental suitability V. cholerae (\\(\\psi_{jt}\\)).","code":""},{"path":"model-description.html","id":"deep-learning-neural-network-model","chapter":"4 Model description","heading":"4.3.2.2 Deep learning neural network model","text":"mentioned , model environmental suitability \\(\\psi_{jt}\\) using Long Short-Term Memory (LSTM) Recurrent Neural Network (RNN) model. LSTM model developed using keras tensorflow R predict binary outcomes. Thus modeled quantity \\(\\psi_{jt}\\) proportion implying unsuitable conditions 0 perfectly suitable conditions 1.model fitted reported case counts converted binary variable using threshold 200 reported cases per week. Given delays reporting likely lead times environmental suitability ahead transmission case reporting, also set preceding one week suitable cases two consecutive weeks >200 cases per week, assumed preceding two weeks also suitable. See Figure 4.10 example reported case counts converted binary variable representing presumed environmental suitability V. cholerae.\nFigure 4.10: Reported cases converted binary variable modeling environmental suitability.\nmodel Long Short-Term Memory (LSTM) neural network designed binary classification, environmental suitability, \\(\\psi_{jt}\\), modeled function hidden state \\(h_t\\) hidden bias term \\(b_h\\). Specifically, \\(\\psi_{jt}\\) defined sigmoid activation function applied linear combination hidden state \\(h_t\\) bias \\(b_h\\) given 3 layers LSTM model:\\[\\begin{equation}\n\\psi_{jt} \\sim \\text{Sigmoid}(w_h \\cdot h_t + b_h)\n\\tag{4.8}\n\\end{equation}\\]\\[\\begin{equation}\nh_t = \\text{LSTM}\\big(\\text{temperature}_{jt}, \\ \\text{precipitation}_{jt}, \\ \\text{ENSO}_{t}, \\dots \\big)\n\\end{equation}\\]formulation, \\(h_t\\) represents hidden state generated LSTM network based input variables temperature, precipitation, ENSO conditions, \\(b_h\\) bias term added output hidden state transformation.deep learning LSTM model consists three stacked LSTM-RNN layers. first LSTM layer 500 units second third LSTM layers 250 100 units respectively. architecture LSTM model configured pass node values subsequent LSTM layers allowing deep learning complex interactions among climate variable time. enforced model sparsity LSTM layer using L2 regularization (penalty = 0.001) used dropout rate 0.5 LSTM layer prevent overfitting limited amount data. final output layer dense layer single unit sigmoid activation function produce probability value binary classification, .e. prediction environmental suitability \\(\\psi_{jt}\\) scale 0 1.fit LSTM model data, modified learning rate applying exponential decay schedule started 0.001 decayed factor 0.9 every 10,000 steps enable smoother convergence. model compiled using Adam optimizer learning rate schedule, along binary cross-entropy loss function accuracy evaluation metric. model trained maximum 200 epochs batch size 1024. allowed model fitting stop early patience parameter 10 halts training improvement observed validation accuracy 10 consecutive epochs. train model set aside 20% observed data validation also used 20% training data model fitting. training history, including loss accuracy, monitored course training gave final test accuracy 0.73 final test loss 0.56 (see Figure 4.11).\nFigure 4.11: Model performance training validation data.\nmodel training completed, predicted values environmental suitability \\(\\psi_{jt}\\) across time steps location. Predictions start January 1970 go 5 months past present date (currently February 2025). Given amount noise model predictions, added simple LOESS spline logit transformation smooth model predictions time give stable value \\(\\psi_{jt}\\) incorporating model features (e.g. Equations (4.6) (4.7)). resulting model predictions shown example country Mozambique Figure 4.12 compares model predictions original case counts binary classification. Predicitons model locations shown simplified view Figure 4.13.Also, please note initial version model fitted rather small amount data. Model hyper parameters specifically chosen reduce overfitting. Therefore, recommend -interpret time series predictions model early stage since likely change improve historical incidence data included future versions.\nFigure 4.12: LSTM model predictions time reported cases example country Mozambique. Reported cases shown top panel tje shaded areas show binary classification used characterize environmental suitability. Raw model predicitons shown transparent brown line solid black line showing LOESS smoothing. Forecasted values beyond current time point shown orange limited 5 month time horizon.\n\nFigure 4.13: smoothed LSTM model predictions (lines) binary suitability classification (shaded areas) time countries MOSAIC framework. Orange lines show forecasts beyond current date. ENSO DMI covariates included model, forecasts limited 5 months.\n","code":""},{"path":"model-description.html","id":"shedding","chapter":"4 Model description","heading":"4.3.3 Shedding","text":"rate infected individuals shed V. cholerae environment (\\(\\zeta\\)) critical factor influencing cholera transmission. Shedding rates can vary widely depending severity infection, immune response individual, environmental factors. According Fung 2014, shedding rate estimated range 0.01 10 cells per mL per person per day.studies support findings, indicating shedding rates can indeed fluctuate significantly. instance, Nelson et al (2009) note , depending phase infection, individuals can shed \\(10^3\\) (asymptomatic cases) \\(10^{12}\\) (severe cases) V. cholerae cells per gram stool. Future version model may attempt capture nuances shedding dynamics, make simplifying assumption shedding constant across infected individuals wide range variability prior distributional assumptions:\\[\n\\zeta \\sim \\text{Uniform}(0.01, 10).\n\\]","code":""},{"path":"model-description.html","id":"water-sanitation-and-hygiene-wash","chapter":"4 Model description","heading":"4.3.4 WAter, Sanitation, and Hygiene (WASH)","text":"Since V. cholerae transmitted fecal contamination water consumables, level exposure contaminated substrates significantly impacts transmission rates. Interventions involving Water, Sanitation, Hygiene (WASH) long first line defense reducing cholera transmission, context, WASH variables can serve proxy rate contact environmental risk factors. MOSAIC model, WASH variables incorporated mechanistically, allowing intervention scenarios include changes WASH. However, necessary distill available WASH variables single parameter represents WASH-determined contact rate contaminated substrates location \\(j\\), define \\(\\theta_j\\).parameterize \\(\\theta_j\\), calculated weighted mean 8 WASH variables Sikder et al 2023 originally modeled Local Burden Disease WaSH Collaborators 2020. 8 WASH variables (listed Table 4.4) provide population-weighted measures proportion population either: ) access WASH resources (e.g., piped water, septic sewer sanitation), ii) exposed risk factors (e.g. surface water, open defecation). risk associated WASH variables, used complement (\\(1-\\text{value}\\)) give proportion population exposed risk factor. used optim function R L-BFGS-B algorithm estimate set optimal weights (Table 4.4) maximize correlation weighted mean 8 WASH variables reported cholera incidence per 1000 population across 40 SSA countries 2000 2016. optimal weighted mean correlation coefficient \\(r =\\) -0.33 (-0.51 -0.09 95% CI) higher basic mean correlations provided individual WASH variables (see Figure 4.14). weighted mean provides single variable 0 1 represents overall proportion population access WASH /exposed environmental risk factors. Thus, WASH-mediated contact rate sources environmental transmission represented (\\(1-\\theta_j\\)) environment--human force infection (\\(\\Psi_{jt}\\)). Values \\(\\theta_j\\) countries shown Figure 4.15.\nTable 4.4: Table 4.5: Table optimized weights used calculate single mean WASH index countries.\n\nFigure 4.14: Relationship WASH variables cholera incidences.\n\nFigure 4.15: optimized weighted mean WASH variables AFRO countries. Countries labeled orange denote countries imputed weighted mean WASH variable. Imputed values weighted mean 3 similar countries.\n","code":""},{"path":"model-description.html","id":"immune-dynamics","chapter":"4 Model description","heading":"4.4 Immune dynamics","text":"Aside current number infections, population susceptibility one key factors influencing spread cholera. , since immunity vaccination natural infection provides long-lasting protection, ’s crucial quantify incidence cholera also number past vaccinations. Additionally, need estimate many individuals immunity remain population given time step model.achieve , estimate vaccination rate time (\\(\\nu_{jt}\\)) based historical vaccination campaigns incorporate model vaccine effectiveness (\\(\\phi\\)) immune decay post-vaccination (\\(\\omega\\)) estimate current number individuals vaccine-derived immunity. also account immune decay rate natural infection (\\(\\varepsilon\\)), generally considered last longer immunity vaccination.","code":""},{"path":"model-description.html","id":"estimating-vaccination-rates","chapter":"4 Model description","heading":"4.4.1 Estimating Vaccination Rates","text":"estimate past current vaccination rates, sourced data reported OCV vaccinations International Coordinating Group (ICG) Cholera vaccine dashboard. resource lists reactive OCV campaigns conducted 2016 present, approximately 103 million OCV doses shipped sub-Saharan African (SSA) countries October 9, 2024. However, data capture reactive vaccinations emergency settings include many preventive campaigns organized GAVI -country partners.result, current estimates OCV vaccination rate likely underestimate total OCV coverage. working expand data sources better reflect full number OCV doses distributed SSA.translate reported number OCV doses model parameter \\(\\nu_{jt}\\), take number doses shipped reported start date vaccination campaign, distributing doses subsequent days according maximum daily vaccination rate. See Figure 4.16 example OCV distribution using maximum daily vaccination rate 100,000. resulting time series country shown Figure 4.17, current totals based ICG data displayed Figure 4.18.\nFigure 4.16: Example estimated vaccination rate OCV campaign.\n\nFigure 4.17: estimated vaccination coverage across countries reported vaccination data one ICG dashboard.\n\nFigure 4.18: total cumulative number OCV doses distributed ICG 2016 present day.\n","code":""},{"path":"model-description.html","id":"immunity-from-vaccination","chapter":"4 Model description","heading":"4.4.2 Immunity from vaccination","text":"impacts Oral Cholera Vaccine (OCV) campaigns incorporated model Vaccinated compartment (V). rate individuals effectively vaccinated defined \\(\\phi\\nu_tS_{jt}\\), \\(S_{jt}\\) available number susceptible individuals location \\(j\\) time \\(t\\), \\(\\nu_t\\) number OCV doses administered time \\(t\\) \\(\\phi\\) estimated vaccine effectiveness. Note just one vaccinated compartment time, though future model versions may include \\(V_1\\) \\(V_2\\) compartments explore two dose vaccination strategies emulate complex waning patterns.vaccination rate \\(\\nu_t\\) estimated quantity. Rather, directly defined reported number OCV doses administered OCV dashboard : https://www..int/groups/icg/cholera.\\[\n\\nu_t := \\text{Reported rate OCV administration} \n\\]evidence waning immunity comes 4 cohort studies (Table 4.6) Bangladesh (Qadri et al 2016 2018), South Sudan (Azman et al 2016), Democratic Republic Congo (Malembaka et al 2024).Table 4.6: Summary Effectiveness DataWe estimated vaccine effectiveness waning immunity fitting exponential decay model reported effectiveness one dose OCV studies using following formulation:\\[\\begin{equation}\n\\text{Proportion immune}\\ t \\ \\text{days vaccination} = \\phi \\times (1 - \\omega) ^ {t-t_{\\text{vaccination}}}\n\\tag{4.9}\n\\end{equation}\\]\\(\\phi\\) effectiveness one dose OCV, based specification, also initial proportion immune directly vaccination. decay rate parameter \\(\\omega\\) rate initial vaccine derived immunity decays per day post vaccination, \\(t\\) \\(t_{\\text{vaccination}}\\) time (days) function evaluated time vaccination respectively. fitted model data cohort studies shown Table (4.6) found \\(\\omega = 0.00057\\) (\\(0-0.0019\\) 95% CI), gives mean estimate 4.8 years vaccine derived immune duration unreasonably large confidence intervals (1.4 years infinite immunity). However, point estimate 4.8 years consistent anecdotes one dose OCV effective least 3 years.wide confidence intervals likely due wide range reported estimates proportion immune short duration 7–90 days range (Azman et al 2016 Qadri et al 2016). Therefore, chose use point estimate \\(\\omega\\) incorporate uncertainty based initial proportion immune (.e. vaccine effectiveness \\(\\phi\\)) shortly vaccination. Using decay model Equation (4.9) estimated \\(\\phi\\) \\(0.64\\) (\\(0.32-0.96\\) 95% CI). fit Beta distribution quantiles \\(\\phi\\) minimizing sums squares using Nelder-Mead optimization algorithm render following distribution (shown Figure 4.19B):\\[\\begin{equation}\n\\phi \\sim \\text{Beta}(4.57, 2.41).\n\\tag{4.10}\n\\end{equation}\\]\nFigure 4.19: vaccine effectiveness\n","code":""},{"path":"model-description.html","id":"immunity-from-natural-infection","chapter":"4 Model description","heading":"4.4.3 Immunity from natural infection","text":"duration immunity natural infection likely longer lasting vaccination OCV (especially given current one dose strategy). SIR-type models, rate individuals leave Recovered compartment governed immune decay parameter \\(\\varepsilon\\). estimated durability immunity natural infection based two cohort studies fit following exponential decay model estimate rate immunity decay time:\\[\n\\text{Proportion immune}\\ t \\ \\text{days infection} = 0.99 \\times (1 - \\varepsilon) ^ {t-t_{\\text{infection}}}\n\\]\nmake necessary simplifying assumption within 0–90 days natural infection V. cholerae, individuals 95–99% immune. fit model reported data Ali et al (2011) Clemens et al (1991) (see Table 4.7).Table 4.7: Sources duration immunity fro natural infection.estimated mean immune decay \\(\\bar\\varepsilon = 3.9 \\times 10^{-4}\\) (\\(1.7 \\times 10^{-4}-1.03 \\times 10^{-3}\\) 95% CI) equivalent immune duration \\(7.21\\) years (\\(2.66-16.1\\) years 95% CI) shown Figure 4.20A. slightly longer previous modeling work estimating duration immunity ~5 years (King et al 2008). Uncertainty around \\(\\varepsilon\\) model represented Log-Normal distribution shown Figure 4.20B:\\[\n\\varepsilon \\sim \\text{Lognormal}(\\bar\\varepsilon+\\frac{\\sigma^2}{2}, 0.25)\n\\]\nFigure 4.20: duration immunity natural infection V. cholerae.\n","code":""},{"path":"model-description.html","id":"spatial-dynamics","chapter":"4 Model description","heading":"4.5 Spatial dynamics","text":"parameters model diagram Figure 4.2 \\(jt\\) subscript denote spatial structure model. country modeled independent metapopulation connected others via spatial force infection \\(\\Lambda_{jt}\\) moves contagion among metapopulations according connectivity provided parameters \\(\\tau_i\\) (probability departure) \\(\\pi_{ij}\\) (probability diffusion destination \\(j\\)). parameters estimated using departure-diffusion model fitted average weekly air traffic volume 41 countries included MOSAIC framework (Figure 4.21).\nFigure 4.21: average number air passengers per week 2017 among countries.\n\nFigure 4.22: network map showing average number air passengers per week 2017.\n","code":""},{"path":"model-description.html","id":"human-mobility-model","chapter":"4 Model description","heading":"4.5.1 Human mobility model","text":"departure-diffusion model estimates diagonal -diagonal elements mobility matrix (\\(M\\)) separately combines using conditional probability rules. model first estimates probability travel outside origin location \\(\\)—departure process—distribution travel origin location \\(\\) normalizing connectivity values across \\(j\\) destinations—diffusion process. values \\(\\pi_{ij}\\) sum unity along row, diagonal included, indicating relative quantity. say, \\(\\pi_{ij}\\) gives probability going \\(\\) \\(j\\) given travel outside origin \\(\\) occurs. Therefore, can use basic conditional probability rules define travel routes diagonal elements (trips made within origin \\(\\)) \n\\[\n\\Pr( \\neg \\text{depart}_i ) = 1 - \\tau_i\n\\]\n-diagonal elements (trips made outside origin \\(\\)) \n\\[\n\\Pr( \\text{depart}_i, \\text{diffuse}_{\\rightarrow j}) = \\Pr( \\text{diffuse}_{\\rightarrow j} \\mid \\text{depart}_i ) \\Pr(\\text{depart}_i ) = \\pi_{ij} \\tau_i.\n\\]\nexpected mean number trips route \\(\\rightarrow j\\) :\\[\\begin{equation}\nM_{ij} =\n\\begin{cases}\n\\theta N_i (1-\\tau_i) \\ & \\text{} \\ = j \\\\\n\\theta N_i \\tau_i \\pi_{ij} \\ & \\text{} \\ \\ne j.\n\\end{cases}\n\\tag{4.11}\n\\end{equation}\\], \\(\\theta\\) proportionality constant representing overall number trips per person origin population size \\(N_i\\), \\(\\tau_i\\) probability leaving origin \\(\\), \\(\\pi_{ij}\\) probability travel destination \\(j\\) given travel outside origin \\(\\) occurs.","code":""},{"path":"model-description.html","id":"estimating-the-departure-process","chapter":"4 Model description","heading":"4.5.2 Estimating the departure process","text":"probability travel outside origin estimated location \\(\\) give location-specific departure probability \\(\\tau_i\\).\n\\[\n\\tau_i \\sim \\text{Beta}(1+s, 1+r)\n\\]\nBinomial probabilities origin \\(\\tau_i\\) drawn Beta distributed prior shape (\\(s\\)) rate (\\(r\\)) parameters.\n\\[\n\\begin{aligned}\ns &\\sim \\text{Gamma}(0.01, 0.01)\\\\\nr &\\sim \\text{Gamma}(0.01, 0.01)\n\\end{aligned}\n\\]","code":""},{"path":"model-description.html","id":"estimating-the-diffusion-process","chapter":"4 Model description","heading":"4.5.3 Estimating the diffusion process","text":"use normalized formulation power law gravity model defined diffusion process, probability travelling destination \\(j\\) given travel outside origin \\(\\) (\\(\\pi_{ij}\\)) defined :\\[\\begin{equation}\n\\pi_{ij} = \\frac{\nN_j^\\omega d_{ij}^{-\\gamma}\n}{\n\\sum\\limits_{\\forall j \\ne } N_j^\\omega d_{ij}^{-\\gamma}\n}\n\\tag{4.12}\n\\end{equation}\\], \\(\\omega\\) scales attractive force \\(j\\) destination based population size \\(N_j\\). kernel function \\(d_{ij}^{-\\gamma}\\) serves penalty proportion travel \\(\\) \\(j\\) based distance. Prior distributions diffusion model parameters defined :\n\\[\n\\begin{aligned}\n\\omega &\\sim \\text{Gamma}(1, 1)\\\\\n\\gamma &\\sim \\text{Gamma}(1, 1)\n\\end{aligned}\n\\]models \\(\\tau_i\\) \\(\\pi_{ij}\\) fitted air traffic data OAG using mobility R package (Giles 2020). Estimates mobility model parameters shown Figures 4.23 4.24.\nFigure 4.23: estimated weekly probability travel outside origin location \\(\\tau_i\\) 95% confidence intervals shown panel population mean indicated red dashed line. Panel B shows estimated total number travelers leaving origin \\(\\) week.\n\nFigure 4.24: diffusion process \\(\\pi_{ij}\\) gives estimated probability travel origin \\(\\) destination \\(j\\) given travel outside origin \\(\\) occurred.\n","code":""},{"path":"model-description.html","id":"the-probability-of-spatial-transmission","chapter":"4 Model description","heading":"4.5.4 The probability of spatial transmission","text":"likelihood introductions cholera disparate locations major concern cholera outbreaks. However, can difficult characterize given endemic dynamics patterns human movement. include measures spatial heterogeneity first simple importation probability based connectivity possibility incoming infections. basic probability transmission origin \\(\\) particular destination \\(j\\) time \\(t\\) defined :\\[\\begin{equation}\np(,j,t) = 1 - e^{-\\beta_{jt}^{\\text{hum}} (((1-\\tau_j)S_{jt})/N_{jt}) \\pi_{ij}\\tau_iI_{}}\n\\tag{4.13}\n\\end{equation}\\]","code":""},{"path":"model-description.html","id":"the-spatial-hazard","chapter":"4 Model description","heading":"4.5.5 The spatial hazard","text":"Although concerned endemic dynamics , likely periods time early rainy season cholera cases rate transmission low enough spatial spread resemble epidemic dynamics time. times periods, can estimate arrival time contagion location cases yet reported. estimating spatial hazard transmission:\\[\\begin{equation}\nh(j,t) = \\frac{\n\\beta_{jt}^{\\text{hum}} \\Big(1 - \\exp\\big(-((1-\\tau_j)S_{jt}/N_{jt}) \\sum_{\\forall \\= j} \\pi_{ij}\\tau_i (I_{}/N_{}) \\big) \\Big)\n}{\n1/\\big(1 + \\beta_{jt}^{\\text{hum}} (1-\\tau_j)S_{jt}\\big)\n}.\n\\tag{4.14}\n\\end{equation}\\]normalizing give waiting time distribution locations:\\[\\begin{equation}\nw(j,t) = h(j,T) \\prod_{t=1}^{T-1}1-h(j,t).\n\\tag{4.15}\n\\end{equation}\\]","code":""},{"path":"model-description.html","id":"coupling-among-locations","chapter":"4 Model description","heading":"4.5.6 Coupling among locations","text":"Another measure spatial heterogeneity quantify coupling disease dynamics among metapopulations using correlation coefficient. , use definition spatial correlation locations \\(\\) \\(j\\) \\(C_{ij}\\) described Keeling Rohani (2002), gives measure similar infection dynamics locations.\\[\\begin{equation}\nC_{ij} = \\frac{\n( y_{} - \\bar{y}_i )( y_{jt} - \\bar{y}_j )\n}{\n\\sqrt{\\text{var}(y_i) \\text{var}(y_j)}\n}\n\\tag{4.16}\n\\end{equation}\\]\n\\(y_{} = I_{}/N_i\\) \\(y_{jt} = I_{jt}/N_j\\). Mean prevalence location \\(\\bar{y_i} = \\frac{1}{T} \\sum_{t=1}^{T} y_{}\\) \\(\\bar{y_j} = \\frac{1}{T} \\sum_{t=1}^{T} y_{jt}\\).","code":""},{"path":"model-description.html","id":"the-observation-process","chapter":"4 Model description","heading":"4.6 The observation process","text":"","code":""},{"path":"model-description.html","id":"rate-of-symptomatic-infection","chapter":"4 Model description","heading":"4.6.1 Rate of symptomatic infection","text":"presentation infection V. cholerae can extremely variable. severity infection depends many factors amount infectious dose, age host, level immunity host either vaccination previous infection, naivety particular strain V. cholerae. Additional circumstantial factors nutritional status overall pathogen burden may also impact infection severity. population level, observed proportion infections symptomatic also dependent endemicity cholera region. Highly endemic areas (e.g. parts Bangladesh; Hegde et al 2024) may low proportion symptomatic infections due many previous exposures. Inversely, populations largely naive V. cholerae exhibit relatively higher proportion symptomatic infections (e.g. Haiti; Finger et al 2024).Accounting nuances first version model possible, can past studies contain information can help set sensible bounds definition proportion infections symptomatic (\\(\\sigma\\)). compiled short list studies done sero-surveys cohort studies assess likelihood symptomatic infections different locations displayed results Table (4.8).provide reasonably informed prior proportion infections symptomatic, calculated combine mean confidence intervals studies Table 4.8 fit Beta distribution corresponds quantiles using least-squares Nelder-Mead algorithm. resulting prior distribution symptomatic proportion \\(\\sigma\\) :\\[\\begin{equation}\n\\sigma \\sim \\text{Beta}(4.30, 13.51)\n\\end{equation}\\]Table 4.8: Summary Studies Cholera ImmunityThe prior distribution \\(\\sigma\\) plotted Figure 4.25A reported values proportion symptomatic previous studies shown 4.25B.\nFigure 4.25: Proportion infections symptomatic.\n","code":""},{"path":"model-description.html","id":"suspected-cases","chapter":"4 Model description","heading":"4.6.2 Suspected cases","text":"clinical presentation diarrheal diseases often similar across various pathogens, can lead systematic biases reported number cholera cases. anticipated number suspected cholera cases related actual number infections factor \\(1/\\rho\\), \\(\\rho\\) represents proportion suspected cases true infections. adjust bias, use estimates meta-analysis Weins et al. (2023), suggests suspected cholera cases outnumber true infections approximately 2 1, mean across studies indicating 52% (24-80% 95% CI) suspected cases actual cholera infections. higher estimate reported ourbreak settings (78%, 40-99% 95% CI). account variability estimate, fit Beta distribution reported quantiles using least squares approach Nelder-Mead algorithm, resulting prior distribution shown Figure 4.26B:\\[\\begin{equation}\n\\rho \\sim \\text{Beta}(4.79, 1.53).\n\\end{equation}\\]\nFigure 4.26: Proportion suspected cholera cases true infections. Panel shows ‘low’ assumption estimates across settings: \\(\\rho \\sim \\text{Beta}(5.43, 5.01)\\). Panel B shows ‘high’ assumption estimate reflects high-quality studies outbreaks: \\(\\rho \\sim \\text{Beta}(4.79, 1.53)\\)\n","code":""},{"path":"model-description.html","id":"case-fatality-rate","chapter":"4 Model description","heading":"4.6.3 Case fatality rate","text":"Case Fatality Rate (CFR) among symptomatic infections calculated using reported cases deaths data January 2021 August 2024. data collated various issues Weekly Epidemiological Record Global Cholera Acute Watery Diarrhea (AWD) Dashboard (see Data section) provide annual aggregations reported cholera cases deaths. used Binomial exact test (binom.test R) calculate mean probability number deaths (successes) given number reported cases (sample size), Clopper-Pearson method calculating binomial confidence intervals. fit Beta distributions mean CFR 95% confidence intervals calculated country using least squares Nelder-Mead algorithm give distributional uncertainty around CFR estimate country (\\(\\mu_j\\)).\\[\n\\mu_j \\sim \\text{Beta}(s_{1,j}, s_{2,j})\n\\]\\(s_{1,}\\) \\(s_{2,j}\\) two positive shape parameters Beta distribution estimated destination \\(j\\). definition \\(\\mu_j\\) CFR reported cases subset total number infections. Therefore, infer total number deaths attributable cholera infection, assume CFR observed cases proportionally equivalent CFR cases calculate total deaths \\(D\\) follows:\\[\\begin{equation}\n\\begin{aligned}\n\\text{CFR}_{\\text{observed}} &= \\text{CFR}_{\\text{total}}\\\\\n\\\\[3pt]\n\\frac{[\\text{observed deaths}]}{[\\text{observed cases}]} &=\n\\frac{[\\text{total deaths}]}{[\\text{infections}]}\\\\\n\\\\[3pt]\n\\text{total deaths} &= \\frac{[\\text{observed deaths}] \\times [\\text{true infections}]}{[\\text{observed cases}]}\\\\\n\\\\[3pt]\nD_{jt} &= \\frac{ [\\sigma\\rho\\mu_j I_{jt}] \\times [I_{jt}] }{ [\\sigma\\rho I_{jt}] }\n\\end{aligned}\n\\end{equation}\\]\nTable 4.9: Table 4.10: CFR Values Beta Shape Parameters AFRO Countries\n\nFigure 4.27: Case Fatality Rate (CFR) Total Cases Country AFRO Region 2014 2024. Panel : Case Fatality Ratio (CFR) 95% confidence intervals. Panel B: total number cholera cases. AFRO Region highlighted black, countries less 3/0.2 = 150 total reported cases assigned mean CFR AFRO.\n\nFigure 4.28: Beta distributions overall Case Fatality Rate (CFR) 2014 2024. Examples show overall CFR AFRO region (2%) black, Congo highest CFR (7%) red, South Sudan lowest CFR (0.1%) blue.\n","code":""},{"path":"model-description.html","id":"demographics-1","chapter":"4 Model description","heading":"4.7 Demographics","text":"model includes basic demographic change using reported birth death rates \\(j\\) countries, \\(b_j\\) \\(d_j\\) respectively. rates static defined United Nations Department Economic Social Affairs Population Division World Population Prospects 2024. Values \\(b_j\\) \\(d_j\\) derived crude rates converted birth rate per day death rate per day (shown Table 4.11).\nTable 4.11: Table 4.12: Demographic AFRO countries 2023. Data include: total population January 1, 2023, daily birth rate, daily death rate. Values calculate crude birth death rates UN World Population Prospects 2024.\n","code":""},{"path":"model-description.html","id":"the-reproductive-number","chapter":"4 Model description","heading":"4.8 The reproductive number","text":"reproductive number common metric epidemic growth represents average number secondary cases generated primary case specific time epidemic. track \\(R\\) changes time estimating instantaneous reproductive number \\(R_t\\) described Cori et al 2013. track \\(R_t\\) across metapopulations model give \\(R_{jt}\\) using following formula:\\[\\begin{equation}\nR_{jt} = \\frac{I_{jt}}{\\sum_{\\Delta t=1}^{t} g(\\Delta t) I_{j,t-\\Delta t}}\n\\tag{4.17}\n\\end{equation}\\]\\(I_{jt}\\) number new infections destination \\(j\\) time \\(t\\), \n\\(g(\\Delta t)\\) represents probability value generation time distribution cholera. accomplished using weighed sum denominator highly influenced generation time distribution.","code":""},{"path":"model-description.html","id":"the-generation-time-distribution","chapter":"4 Model description","heading":"4.8.1 The generation time distribution","text":"generation time distribution gives time individual infected infect subsequent individuals. parameterized quantity using Gamma distribution mean 5 days:\\[\\begin{equation}\ng(\\cdot) \\sim \\text{Gamma}(0.5, 0.1).\n\\tag{4.18}\n\\end{equation}\\], shape=0.5, rate=0.1, mean given shape/rate. Previous studies use mean 5 days (Kahn et al 2020 Azman 2016), however mean 3, 5, 7, 10 days may admissible (Azman 2012).\nFigure 4.29: generation time\n\nTable 4.13: Table 4.14: Generation Time Weeks\n","code":""},{"path":"model-description.html","id":"initial-conditions","chapter":"4 Model description","heading":"4.9 Initial conditions","text":"Since first version model begin Jan 2023 (take advantage available weekly data), initial conditions surrounding population immunity must estimated. set initial conditions, use historical data find total number reported cases location previous X years, multiply \\(1/\\sigma\\) estimate total infections symptomatic cases reported, adjust based waning immunity. also sum total number vaccinations past X years adjust vaccine efficacy \\(\\phi\\) waning immunity vaccination \\(\\omega\\).total number infected? reported cases… back symptomatic asymptomatictotal number infected? reported cases… back symptomatic asymptomaticTotal number immune due natural infections past X yearsTotal number immune due natural infections past X yearstotal number immune due past vaccinations X yearstotal number immune due past vaccinations X yearsUse deconvolution based immune decay estimated vaccine section","code":""},{"path":"model-description.html","id":"model-calibration","chapter":"4 Model description","heading":"4.10 Model calibration","text":"model calibrated using Latin hypercube sampling hyper-parameters model likelihoods fit incidence deaths.model calibrated using Latin hypercube sampling hyper-parameters model likelihoods fit incidence deaths.important challenge flexibly fitting data often missing available aggregated forms.important challenge flexibly fitting data often missing available aggregated forms.[Fig: different spatial temporal scales available data]","code":""},{"path":"model-description.html","id":"caveats","chapter":"4 Model description","heading":"4.11 Caveats","text":"Simplest model start. Easier initial spatial structure minimum additional compartments calibrate available data (vaccination, cases, deaths).Country level aggregations. First generation data 2023/24…Assumes vaccinating susceptible individuals.climate, summarizing whole country.","code":""},{"path":"model-description.html","id":"table-of-parameters","chapter":"4 Model description","heading":"4.12 Table of parameters","text":"Table 4.15: Descriptions model parameters along prior distributions sources applicable.","code":""},{"path":"model-description.html","id":"references","chapter":"4 Model description","heading":"4.13 References","text":"","code":""},{"path":"scenarios.html","id":"scenarios","chapter":"5 Scenarios","heading":"5 Scenarios","text":"key aim MOSAIC model provide near-term forecasts cholera transmission Sub-Saharan Africa (SSA) using current data available. However, MOSAIC just forecasting tool; dynamic model designed explore various scenarios influence critical factors vaccination, environmental conditions, Water, Sanitation, Hygiene (WASH) interventions.","code":""},{"path":"scenarios.html","id":"vaccination","chapter":"5 Scenarios","heading":"5.1 Vaccination","text":"","code":""},{"path":"scenarios.html","id":"spatial-and-temporal-strategies","chapter":"5 Scenarios","heading":"5.1.1 Spatial and Temporal Strategies","text":"Understanding spatial temporal distribution cholera vaccination efforts crucial effective outbreak control. Key resources include:Stockpile Status: availability oral cholera vaccine emergency stockpiles can tracked UNICEF’s Emergency Stockpile Availability.OCV Dashboard: dashboard (link) provides insights deployment oral cholera vaccines (OCV) across different regions.","code":""},{"path":"scenarios.html","id":"reactive-vaccination","chapter":"5 Scenarios","heading":"5.1.2 Reactive Vaccination","text":"timing logistics reactive vaccination campaigns critical controlling ongoing outbreaks. Relevant resources include:Recommended Timing: Guidelines recommendations timing reactive OCV campaigns available (link).Requests Delay Time Distributions: Information vaccine request processes distribution delays vaccine deployment can accessed GTFCC OCV Dashboard (link).","code":""},{"path":"scenarios.html","id":"impacts-of-climate-change","chapter":"5 Scenarios","heading":"5.2 Impacts of Climate Change","text":"","code":""},{"path":"scenarios.html","id":"severe-weather-events","chapter":"5 Scenarios","heading":"5.2.1 Severe Weather Events","text":"Projections climate shocks, including frequency severity cyclones floods, essential modeling future impacts climate change cholera transmission. Key references include:Chen Chavas 2020: study cyclone season dynamics climate change scenarios (link).Sparks Toumi 2024: Research projected flood frequencies due climate change (link).Switzer et al. 2023: analysis climate shock impacts cholera outbreaks (link).","code":""},{"path":"scenarios.html","id":"long-term-trends","chapter":"5 Scenarios","heading":"5.2.2 Long-Term Trends","text":"Long-term trends weather variables various climate change scenarios can explored using following resource:Weather Variables Climate Change: OpenMeteo Climate API provides access projected weather data different climate change scenarios (link).","code":""},{"path":"usage.html","id":"usage","chapter":"6 Usage","heading":"6 Usage","text":"open-source code used run MOSAIC currently development presented future.","code":""},{"path":"news.html","id":"news","chapter":"7 News","heading":"7 News","text":"","code":""},{"path":"news.html","id":"past-versions-of-mosaic","chapter":"7 News","heading":"7.1 Past versions of MOSAIC","text":"Table 7.1: Current future planned model versions brief descriptions.","code":""},{"path":"references-1.html","id":"references-1","chapter":"8 References","heading":"8 References","text":"","code":""}] diff --git a/docs/usage.html b/docs/usage.html index 2c49fbd..9b187e2 100644 --- a/docs/usage.html +++ b/docs/usage.html @@ -79,7 +79,7 @@

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