["model"] ["bug"] Epi question regarding rate of new infection in SIR model

[wyusuf068]

Summary

I was looking for clarification as to why you have stated in the documentation that the rate of new infection in the SIR model is g=βS−γ as opposed to g=βS.

Additional details

In the SIR model, the number of infected patients is calculated as follows:

It+1 = It + βStIt - γIt in which It is the starting number of infected patients, βStIt is the number of susceptible patients who become infected (i.e. new infections), and γIt is the number of infected patients who become recovered (i.e. new recoveries). To me this means that the rate of new infections is mediated by βS and that γ mediates the rate of new recoveries.

Suggested fix

Using g=βS to estimate rate of new infections, which may have implications in estimating doubling time.

[DougManuel]

An issue is γ is inconsistently defined and mislabelled in parts of the documentation. γ is 1/mean infectious period. In the documentation, please provide reference and rationale for "γ: the CDC is recommending 14 days of self-quarantine, we'll use γ=1/14)". γ is not the maximum period of infectivity or period of illness or symptoms.

[DougManuel]

This issue of γ results in a large impact on projected estimates for larger estimates of social distancing (>60%). As social distancing increases, doubling time increases and β becomes more influenced by γ.

β=(g+γ), therefore β → y, when g → 0 (when double-time [Td] increases). Therefore, the concern will not be seen when validating the model on past observed hospitalizations.

Related, the more commonly-used period of infectivity is 7 days. See issue #492. Changing the period of infectivity does not have a large influence on the β when doubling time is short, but there is a large effect when doubling time decreases.

[jlubken]

#533 modifies both the default doubling_time and infectious_days.

[DougManuel]

The math is correct and the suggested fix doesn't need to be incorporated. This issue can be closed.

The default doubling_time and infectious_days fix is a separate issue has been discussed in issue #492 and elsewhere. The new defaults are an improvement but people would suggest further modifications, as discussed.