Question

A cattle rancher discovers a viral disease spreading among his 4000 cattle. Suppose the rate at which the disease spreads is proportional to the product of the number of cattle that have the disease and the number that do not. Write a differential equation that describes this scenario.

Answer 1

The rate of spread of the disease is:
\[
\frac{dP}{dt}
\]We also know that the population is \(P\) with the initial population being \(P_0\). So, our product now becomes:
\[
(P_0-P)(P)
\]We know that the rate is proportional to this product, hence, we receive:
\[
\frac{dP}{dt}\propto(P_0-P)(P)
\]

Answer 2

\(P\) denotes the population with the disease.