Chapter 4.2, Problem 104E

Differential equations can be used to model disease epidemics. In the next set of problems, we examine the change of size of two sub-populations of people living in a city: individuals who are infected and individuals who are susceptible to infection. S represents the size of the susceptible population, and I represents the size of the infected population. We assume that if a susceptible person interacts with an infected person, there is a probability c that the susceptible person will become infected. Each infected person recovers from the infection at a rate r and becomes susceptible again. We consider the case of influenza, where we assume that no one dies from the disease, so we assume that the total population size of the two sub-populations is a constant number, N. The differential equations that model these population sizes are

S' = rI − cSI and

I' =cSI − rI.

Here c represents the contact rate and r is the recovery rate.

104. Show that, by our assumption that the total population size is constant (S +I = N), you can reduce the system to a single differential equation in I: I' = c(N-I)I-rI.