6.2.7: Epidemics

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Another interesting area of application of differential equation is in predicting the spread of disease. Typically, one has a population of susceptible people or animals. Several infected individuals are introduced into the population and one is interested in how the infection spreads and if the number of infected people drastically increases or dies off. Such models are typically nonlinear and we will look at what is called the SIR model in the next chapter. In this section we will model a simple linear model.

Let us break the population into three classes. First, we let \(S(t)\) represent the healthy people, who are susceptible to infection. Let \(I(t)\) be the number of infected people. Of these infected people, some will die from the infection and others could recover. We will consider the case that initially there is one infected person and the rest, say \(N\), are healthy. Can we predict how many deaths have occurred by time \(t\) ?

We model this problem using the compartmental analysis we had seen for mixing problems. The total rate of change of any population would be due to those entering the group less those leaving the group. For example, the number of healthy people decreases due infection and can increase when some of the infected group recovers. Let’s assume that a) the rate of infection is proportional to the number of healthy people, \(a S\), and \(b\) ) the number who recover is proportional to the number of infected people, \(r I\). Thus, the rate of change of healthy people is found as

\[\dfrac{d S}{d t}=-a S+r I \text {. } \nonumber \]

Let the number of deaths be \(D(t)\). Then, the death rate could be taken to be proportional to the number of infected people. So,

\[\dfrac{d D}{d t}=d I \nonumber \]

Finally, the rate of change of infected people is due to healthy people getting infected and the infected people who either recover or die. Using the corresponding terms in the other equations, we can write the rate of change of infected people as

\[\dfrac{d I}{d t}=a S-r I-d I.\nonumber \]

This linear system of differential equations can be written in matrix form.

\[\dfrac{d}{d t}\left(\begin{array}{c} S \\ I \\ D \end{array}\right)=\left(\begin{array}{ccc} -a & r & 0 \\ a & -d-r & 0 \\ 0 & d & 0 \end{array}\right)\left(\begin{array}{c} S \\ I \\ D \end{array}\right) \nonumber \]

The reader can find the solutions of this system and determine if this is a realistic model.