Chapter 2.5, Problem 6P

Epidemics. The use of mathematical methods to study the spread of contagious diseases goes back at least to some work by Daniel Bernoulli in $1760$ on smallpox. In more recent years, many mathematical models have been proposed and studied for many different diseases. Problems $6$ through $8$ deal with a few of the simpler models and the conclusions that can be drawn from them. Similar models have also been used to describe the spread of rumors and of consumer products.

Suppose that a given population can be divided into two parts: those who have a given disease and can infect others, and those who do not have it but are susceptible. Let $x$ be the proportion of susceptible individuals and $y$ the proportion of infectious individuals; then $x+y=1$. Assume that the disease spreads by contact between sick and well members of the population and that the rate of spread $dy/dt$ is proportional to the number of such contacts. Further, assume that members of both groups move about freely among each other, so the number of contacts is proportional to the product of $x$ and $y$. Since $x=1-y$, we obtain the initial value problem

$\begin{array}{ccc}dy/dt=\alpha y\left(1-y\right),& y\left(0\right)=y_0& (i)\end{array}$

where $\alpha$ is a positive proportionality factor, and $y_0$ is the initial proportion of infectious individuals.

Find the equilibrium points for the differential equation $\left(i\right)$ and determine whether each is asymptotically stable, semi stable, or unstable.

Solve the initial value problem $\left(i\right)$ and verify that the conclusions you reached in part (a) are correct. Show that $y\left(t\right)\to 1$ as $t\to \infty$, which means that ultimately the disease spreads through the entire population.