Periodic Solutions of a Periodic Nonlinear Delay Differential Equation

Abstract

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 $\omega > 0$. 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 $0 \leqq y( t ) \leqq 1$. It is shown that, if b is nonconstant, a positive periodic solution exists provided that c is less than a certain threshold value $c_T $. If c is greater or equal to $c_T $ 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 $c_T $. The main methods of analysis that are used are fixed point theorems for operators on cones and Lyapunov functions for delay-differential equations.