Simple Dynamics of Epidemics, the Reproduction Number

In epidemiology, the basic reproduction number is the expected number of new infections from a single infection in a population where all subjects are susceptible [1]. This metric is useful because it helps determine whether or not an infectious disease can spread through a population. When , the infection will die in the long run. But if , the infection will spread in the population. Generally, the greater the value of , the harder it is to control the epidemic. This Demonstration solves a simple epidemic model and shows the conditions necessary for the outbreak of an infection to result in an epidemic.

Consider an SIR model in which we divide the population into three parts: susceptible, infected, and recovered, in the proportions , , and , so that . Suppose the birth rate is a constant equal to the death rate. The model is:

,

,

,

where is the infection rate and is the recovery rate, with the initial , , and non-negative. There is an outbreak only if , and the outbreak results in an epidemic only if ; otherwise the infection will die out [1].

The SIR equations are solved and the result presented in a plot in the - plane for various initial values of and , with initial value . You can vary the values of the infection rate and the recovery rate to follow the trajectory of the SIR system.