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

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 L 1 -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 R 0 . 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.