Periodic solutions to a discrete model for the spread of infectious disease

The difference equation x(l)=Σ_s=l-T^l-1 F(s, x(s)) is used to model the spread of infectious disease. Here, x(l) represents the proportion of the population infected at time l, F(l,x(l)) denotes the proportion of the population newly infected between times l and (l+1), and T is the length of time an individual remains Infectious. Criteria will be established for the existence of a nontrivial and nonnegative periodic solution for the difference equation. The results are easy to implement numerically, and only require basic information of the Contact rate q(l)=lim_x→0 (F(l,x)/x). An algorithm and some illustrative examples will be given.