Outbreaks

This chapter returns to the problem of the SIR model, framing an example problem in a small town. There is a general discussion of potential factors that might go into a model of infectious disease spread. Next, the idea of differential equations is introduced as a generalization of the inverse problem in the previous chapter. For the SIR model, the population is divided into three groups: the susceptibles (S), those currently infected (I), and the recovered or resistant (R). We find that the rate of change of each of these three quantities depends on the others; this idea is formalized with the three differential equations of the SIR model. We then discuss that while there are no analytic solutions to these equations, we can approximately solve them using spreadsheets; there are free software tools that give very good approximations. We first focus on the two parameters of the SIR model: the infectivity parameter, r , and the recovery rate, a . We define R 0 : the basic reproductive number of a disease that determines whether its introduction into a susceptible population will spark an outbreak. Then the model is expanded to include births and deaths, which can give rise to waves of infection. Finally, vaccination is added to the model, allowing us to discover herd immunity : the projection of unvaccinated individuals when the vaccination rate is high enough to prevent further disease outbreaks.