SIR Disease Model Mathematics

The basic SIR (Susceptible, Infectious, Recovered) disease model assumes 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

Thus, we now have two types of Infectious population members, those that will eventually recover at rate σ , and those that will eventually die at rate μi (of course, members in all three states still die at the background rate μ ).

Let

We modify our model to include these additional states and rates.

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 (both states combined), 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.