Periodic Solutions of a Periodic Nonlinear Delay Differential Equation

In this paper the scalar delay-differential equation y' (t) = b (t) y (t - T) 1 - y (t) - cy (t), is studied, where c and T are positive constants and b is a positive periodic function of minimal period > 0. This equation models the proportion of infectious persons with a communicable disease carried by a vector; hence, interest is centered on the solutions which obey 0 y (t) 1. It is shown that, if b is nonconstant, a positive periodic solution exists provided that c is less than a certain threshold value cT. If c is greater or equal to cT no positive periodic solution exists. The stability of the positive periodic as well as of the zero solution are discussed and bounds are obtained for the critical value cT. The main methods of analysis that are used are fixed point theorems for operators on cones and Lyapunov functions for delay-differential equations.