Endemic threshold results for an age-structured SIS epidemic model with periodic parameters

Abstract

The main contribution of this paper is to obtain a threshold value for the existence and uniqueness of a nontrivial endemic periodic solution of an age-structured SIS epidemic model with periodic parameters. Under the assumption of the weak ergodicity of a non-autonomous Lotka–McKendrick system, we formulate a normalized system for an infected population as an initial boundary value problem of a partial differential equation. The existence problem for endemic periodic solutions is reduced to a fixed point problem of a nonlinear integral operator acting on a Banach space of locally integrable periodic L1-valued functions. We prove that the spectral radius of the Fréchet derivative of the integral operator at zero plays the role of a threshold for the existence and uniqueness of a nontrivial fixed point of the operator corresponding to a nontrivial periodic solution of the original differential equation in a weak sense. If the Malthusian parameter of the host population is equal to zero, our threshold value is equal to the well-known epidemiological threshold value, the basic reproduction number R0. However, if it is not the case, then two threshold values are different from each other and we have to pay attention on their actual biological implications.