Existence result for an age-structured SIS epidemic model with spatial diffusion

In this paper, we consider an age-structured SIS epidemic model with spatial diffusion. In order to make use of the previous theory of integral equations for age-structured models, we first obtain an explicit expression of the solution of a parabolic partial differential equation by using the Feynman–Kac formula in probability theory and then, we derive an integral equation and a corresponding integral operator by substituting the solution into the equation of force of infection. We prove that the spectral radius of the Fréchet derivative of the integral operator corresponds to the well-known basic reproduction number R0R0 and plays the role of a threshold value, that is, the system has a nontrivial (time-independent) endemic equilibrium if the value is greater than unity, and it has only the trivial disease-free equilibrium if the value is less than or equal to unity.