We analyze the bifurcation behavior of an SIS model with 2-periodic constant demography. A traditional non-periodic SIS model does not result in period doubling bifurcations; however, when adding periodicity the model undergoes a period doubling route to chaos. We numerically detect the first period doubling bifurcation as a function of one and multiple parameters. We simplify the original transcendental equation by using the Taylor approximation of the transmission rate. The total population is globally attracted to a 2-cycle, so we must use the 2-fold composition of the infected class equation to encompass both population values.
By using the Period Doubling Bifurcation Theorem, we are able to analytically find parameter values that give rise to these bifurcations. The two major conditions in this theorem reduce to two cubic equations in I, the infected class, and the 5 model parameters. Using conditions imposed on the parameters in our model together with two other reasonable conditions on the parameters, we establish that each of the cubic equations has one real root. Equating these real roots gives an equation in terms of our parameters that, when satisfied, results in a period doubling bifurcation.