[MS01] Delay differential equations in mathematical biology and life sciences

[MS01-05] Periodic solution for a class of distributed delay differential equations

*Nakata Yukihiko¹ (1. Aoyama Gakuin University (Japan))

Keywords:

periodic solution,distributed delay differential equation,epidemic model,Hamiltonian system

Motivated by the periodic oscillations observed in an SIRS epidemic model, we investigate the existence of periodic solutions for a differential equation with distributed delay. We show that, for a class of distributed delay differential equations, a symmetric period-two solution can be obtained as a periodic solution of a Hamiltonian system of ordinary differential equations, thus examples that admit periodic solutions expressed in terms of Jacobi elliptic functions can be given. This work extends the result of Kaplan and Yorke (J. Math. Anal. Appl., 1974) for a discrete delay differential equation with an odd nonlinear function. We also discuss cases where the nonlinear function is not necessarily an odd function.