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# 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.

### DETAILS

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.
Reference
[1] Wikipedia. "Compartmental Models in Epidemiology." (Dec 14, 2012) en.wikipedia.org/wiki/Compartmental_models_in _epidemiology.

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