SIR(S) Disease Model Mathematics

The basic SIR (Susceptible, Infectious, Removed) and SIRS (Susceptible, Infectious, Recovered, Susceptible) disease models assume a uniform population at a single location and that the population members are well "mixed", meaning that they are equally likely to meet and infect each other. This model, for a normalized population, is defined by the three equations below:

Where:

Following basically the same derivation as outlined for the SI model, these become:

Let

We modify our model to include this additional rate.

Spatial Adaptation

Let Sl = s Pl be the number of Susceptible population members at location l. Similarly, let Il = i Pl be the number of population members at location l that are Infectious, and let r Pl be the Recovered population. For readability, we drop the l subscript and substitute.

Substituting

Continuing with i = I/Pl , we have: Letting β* = βl/Pl = β (dl/(APDPl)) gives: TSF

Neighboring Infectious Populations

Specific statistics on the total number of births, deaths and deaths due to the disease can be computed by adding the appropriate terms of the equations above.