As Sridhar and Majumder argue, it is crucially important to understand the limitations of computational models and the data used to calibrate them. The model given here is intended for exploration of the relative consequences of different actions and policy interventions under different conditions.
There are two fundamentally different kinds of assumptions at play. Structural assumptions that are a consequence of the formulation of the model itself. These are eternally true in the sense that if the model is calibrated with parameters for different disease outbreaks, the structural assumptions are still there and need to be understood. Assumptions about data, meanwhile, are particular to calibration of the model to a specific outbreak. Data about a novel pathogen causing a pandemic of proportions not seen for a century are simply too scarce and inconsistent, particularly in the early days, to accurately calibrate any model without assumptions about the meaning and reliability of the data. Neither class of assumption presents insurmountable problems but they are important to recognise in order to understand what can, and cannot, be learned from a model.
We refer specifically to the SEIR-TTI ODE model with economic extensions as implemented in the PTTI software. This is fundamentally a SEIR compartmental model simulated with ordinary differential equations (ODE)s. The extensions to include testing, tracing and isolation (TTI) are based on a careful probabilistic argument that we discuss in detail in that paper. This formulation means that there are automatically several underlying assumptions, for which we provide justification here.
The outbreak, and each susceptible, exposed, infectious and removed cohort is of sufficient size that it is coherent to simulate it with ODEs. This is not the case in the very early days with dozens or small numbers of hundreds of cases where stochastic methods would be more appropriate. This is also not the case if the outbreak is nearly completely suppressed.
Justification: we are studying management strategies for large outbreaks. That is the circumstance where ODEs are applicable.
In reality, the UK is undergoing several COVID-19 outbreaks. The outbreak in Edinburgh is perhaps 10 days behind that in London. The outbreak in the North of England began declining well after London. Distancing interventions such as lock-down happened simultaneously throughout the UK but are being released at different times.
In general, a model of a single large outbreak cannot be used to reproduce the dynamics of smaller outbreaks separated in time. This is easily seen with the simplest of models (e.g. SIR) if outbreaks are widely spaced in time. Simulating the large outbreak with such a model will produce a curve for, say, infections, with one peak. Simulating two small outbreaks individually will each produce a peak at a different time. Adding the results of simulating the small outbreaks produces a curve with two peaks, different from the result of simulating the large outbreak.
This model does not represent those different, coupled outbreaks, it represents a single outbreak.
Justification: this is a simplifying assumption. In the case of COVID-19, the major outbreaks in the most populous cities are separated in time by only a couple of generations. We argue that this is close enough that, to a first approximation, the differences can be disregarded and valid insights gained by considering the ensemble as one large outbreak. Furthermore, it is possible to use the same model to study individual outbreaks.
The population is homogeneous and each individual behaves in the same way and is affected in the same way by the virus. This is manifestly untrue of the real world. It introduces the limitation that the model will not tell us how specific segments of society are affected and whether different strategies would be appropriate for different groups.
Justification: this is another simplifying assumption. The subject of this study is testing, tracing and isolation. We wish to know the effect of this on the population level and how it interacts with other interventions for a large outbreak. This is not a well-studied topic and for this reason we examine it for a simple population before introducing more complex structure. It is possible to build an analogous stratified model to account for more population structure and we leave this as future work.
The nature of our contact tracing approximation, triggered by testing, is such that it requires that testing happen on the same time-scale as disease progression, or faster.
Justification: it is not realistic to expect that testing individuals after they have recovered from the disease or died would be useful for contact tracing.
When computing costs, in some scenarios we consider the possibility of widespread testing that only identifies infectious people. This testing, for costing purposes, is considered applied only to people who are undiagnosed - identified neither by previous tests nor by contact tracing as potentially infected. The ones who are already diagnosed are considered as not being tested, since it would be redundant to do so.
Our contact tracing approximation is formulated such that contacts are traced at a rate proportional to their chance of having had at least one contact with an infectious individual. An alternative formulation could trace proportional to the number of infectious contacts: if one has had two infectious contacts, then perhaps one is twice is likely to be traced.
Justification: this is a conservative assumption. It may underestimate the effectiveness of contact tracing for outbreak suppression. For policy purposes, it is better to under-promise and over-deliver than the reverse.
The available data for the UK COVID-19 epidemic presents several challenges. The central government releases data about deaths in the whole of the UK as well as Scotland, Wales and Northern Ireland on the data.gov.uk coronavirus web site as well as the number of cases reported in England and Wales.
The availability of tests both in hospital settings and among different groups of workers and individuals has changed over time and varies by location, and the supply and provision of tests continues to be constrained. Because the reported cases depend on testing, these data are difficult to interpret reliably. Due to the lack of seroprevalence testing we also do not know the true number of cases in the population. Attempting to fit a model that produces a definite number of instantaneous or cumulative cases to such data is a fool's errand. We therefore disregard this data.
Mortality data is better but also suffers problems. Reporting of coronavirus-related deaths in hospitals is thought to be relatively consistent across the UK. The criterion for reporting of deaths outside of hospitals as coronavirus-related has changed in time and has differed by region. We nevertheless attempt to calibrate our model to this mortality data, estimating that it is not as severely defective as the case data.
Calibration of our model to mortality data involves some facts and some assumptions.
Although we suspect a slowing of activity and reduction in contact for at least a week before lockdown was ordered, we are only certain that it was ordered effective the 23rd of March. The effect of this is a reduction of the average contact rate as of that date.
Although we cannot say by how much contact increased on the 13th of May, some economic activity resumed in England on that day. Data that reflects this is, at the time of writing sparse and very recent.
This value, represented in our model as c, comes from the literature [citation] and reflects normal life. The main effect of distancing measures is to change this value.
We know from clinical studies [citation] that the incubation/latent period when infected individuals are not yet infectious is about 5 days. We likewise know that the infectious phase of the disease lasts approximately a week, after which the individuals either recover or become severely ill.
We assume that all severely ill individuals are hospitalised, and that conditions within hospitals are such as prevent onward transmission. This has consequences because the reciprocal of the parameter γ appears in the model as the duration of infectiousness and is fixed at one week.
Justification:
The model does not explicitly produce a time-series for the number of dead. Rather, it produces the number of individuals removed through recovery or death. To calibrate against mortality data it is necessary to count the dead. To do this, we need to know what fraction of infections result in death. Estimates vary and reflection in the model as the probability of infection (and consequently the overall force of the epidemic) depends strongly on this value.
Justification: scientific consensus appears to be converging on an IFR of around 0.8% for the UK, justified by the IFR in China being estimated as 0.66% and the UK population being older, as well as by a recent large scale seroprevalence survey of 70,000 people in Spain showing 5.0% infected to date (2.35m people) meaning an IFR there of 1.15% given the 27,100 Covid-19 deaths there. However we conduct sensitivity analysis on the resulting infectiousness parameter to understand how this affects the behaviour of the model.
Our strategy is not to find which model parameters are determined by the data,
but to find those that are consistent with the data and also with the broader
scientific consensus. The PTTI software includes a tool, ptti-fit, for
calibrating the model to mortality data. The fitting is conducted using the
fitting scenario that includes three interventions:
- on the 18th of February
- on the 23rd of March
- on the 13th of May
The first intervention represents a notional start to the epidemic implied by the UK government figures under our model. However, we do not believe this to be the true date as discussed below.
The number of contacts per unit time, c, was fixed at 1, and the infectiousness, β, was allowed to vary. No testing or contact tracing was included in the model for fitting purposes. Without contact tracing, the effects of β and c are indistinguishable so no information is lost by simply fixing c to 1.
To compute the difference between the model output and the mortality data we used the L2 norm or Euclidean distance, as is standard practice for nonlinear optimisation. We minimised this distance using the Nelder-Mead simplex algorithm, allowing infectiousness to vary independently in each time-period.
The result of this fitting exercise is shown below,
The vertical lines indicate the interventions and the case data begins on the 6th of March. The values for β that were discovered by the optimisation process are,
- 0.1529 (equivalent to R = 1) from the first of January to the 12th of February
- 1.59 (equivalent to R = 11) from the 12th of February until lock-down
- 0.1529 (equivalent to R = 1) during lock-down
Of these, only the third is meaningful. The first may be disregarded as it is before the notional start of the epidemic implied by the UK government mortality data. Seeding the simulation on the first of January with one infectious individual, a value implying that R = 1 will maintain this until the notional start of the epidemic. That this was found automatically through fitting is a sanity check for identifying the 12th of February as the key date.
The second figure of 1.59 (R = 11) during the initial phase of the epidemic is conspicuously high and this bears explanation. Firstly, there is a large degree of uncertainty here: a fit of comparable quality equivalent to R = 8 is also possible. Nevertheless, the early UK government data implies a much larger value than we believe to be correct. This can be clearly seen from the figure presented on a logarithmic scale. This value can be revised downwards easily by moving the notional start of the epidemic earlier and a value close to the scientific consensus value is obtained towards the end of December or the beginning of January, the specific date not being very influential for obtaining this value. The resulting curve, however, no longer matches the initial data.
The final figure of 0.1529 (R = 1) during lock-down appears to be robust against all efforts to vary the starting date and intervention timing, varying from slightly below to slightly above. Whilst the initial UK government data is suspect, the later data appears more reliable.
On this basis, then, we can infer that if it is correct that lock-down reduces contact by 70% [citation] then the initial value for the infectiousness should be 0.51, implying R0 of 3.5, at the high end of the range of scientific consensus. By adjusting the notional start of the epidemic to be in late December or early January, we obtain such an initial value without perturbing the fit or value of infectiousness during the lock-down period.
Recall that this model assumes individuals are removed through hospitalisation or death. From what we have observed of disease progression, the second phase of the illness leading to death typically lasts 2-3 weeks. If the mean is 18 days, this means that, when fitting to mortality data with this model, an adjustment of 18 days is required. Therefore, measured as the first active case, the date implied by the UK government data is not the 12th of February but the 25th of January, and by the above argument, we believe the true beginning of the epidemic in the UK to be in mid-December.
We therefore conclude that the following are consistent with the data:
- The UK government data implies an unreasonably high infectiousness and the epidemic beginning on the 25th of January.
- The COVID-19 epidemic in the UK was probably seeded at in the middle to end of December 2019.
- The reproduction number for a completely susceptible population in the absence of any interventions is about 3.3, implying an appropriate value for β in our full model (c equal to 11 without interventions) of 0.0435.
Acknowledging the uncertainty in the data, and inherent in our reasoning above, we also conducted a sensitivity analysis, exploring scenarios where infectiousness is slightly below, and slightly above this value to ascertain the effectiveness of the measures that we propose here in fitting observed deaths in June 2020.
Fitting can be verified by downloading the mortality data from the data.gov.uk coronavirus web site and running the command,
ptti-fit -y fitting.yaml --dgu coronavirus-deaths_latest.csv --mask c theta --ifr 0.011where fitting.yaml is the fitting scenario in the examples/ subdirectory
of the PTTI software distribution.