Chapter 2.5, Problem 7P

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.

Some diseases (such as typhoid fever) are spread largely by carriers, individuals who can transmit the disease but who exhibit no overt symptoms. Let $x$ and $y$, respectively, denote the proportion of susceptible and carriers in the population. Suppose that carriers are identified and removed from the population at a rate $\beta$, so

$\begin{array}{cc}dy/dt=-\beta y& \left(i\right)\end{array}$

Suppose also that the disease spreads at a rate proportional to the product of $x$ and $y$; thus

$\begin{array}{cc}dx/dt=-\alpha xy& \left(ii\right)\end{array}$

Determine y at any time t by solving Eq. $\left(i\right)$ subject to the initial condition $y\left(0\right)=y_0$.

Use the result of part (a) to find $x$ at any time $t$ by solving Eq. $\left(ii\right)$ subject to the initial condition $x\left(0\right)=x_0$.

Find the proportion of the population that escapes the epidemic by finding the limiting value of $x$ as $t\to \infty$.