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# Dynamics of Seasonal Epidemics

To study how the probability of getting a disease varies with the seasons of the year, this Demonstration analyzes a susceptibility-infection-recovery (SIR) model with a periodically varying infection rate. The model is a set of nonlinear differential equations with periodically varying parameters.

### DETAILS

The periodically forced SIR model is:
,
,
.
Here , , and are the fractions of susceptible, infective, and recovered individuals in the population; is the birth and death rate; is the seasonally dependent infection rate; and is the recovery rate. We take to be periodic; , where is the length of the infection season and is a constant.
The reproduction number for this system is ; if the integral is greater than one, the disease will not disappear and may undergo interesting and complex phenomena of nonlinear parametric resonance [1].
You can vary the time window of observation, the length of the infection season, the fraction initially infected, and the rates of initial infection, recovery, and death to follow the trajectory of the SIR system.
Reference
[1] Wikipedia. "Compartmental Models in Epidemiology." (Jul 2, 2012) en.wikipedia.org/wiki/Compartmental_models_in _epidemiology.

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