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;
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 .
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.