SPDE for stochastic SIR epidemic models with infection-age dependent infectivity

We study the stochastic SIR epidemic model with infection-age dependent infectivity for which a measure-valued process is used to describe the ages of infection for each individual. We establish a functional law of large numbers (FLLN) and a functional central limit theorem (FCLT) for the properly scaled measure-valued processes together with the other epidemic processes to describe the evolution dynamics. In the FLLN, assuming that the hazard rate function of the infection periods is bounded and the ages at time 0 of the infections of the initially infected individuals are bounded, we obtain a PDE limit for the LLN-scaled measure-valued process, for which we characterize its solution explicitly. The PDE is linear with a boundary condition given by the unique solution to a set of Volterra-type nonlinear integral equations. In the FCLT, we obtain an SPDE for the CLT-scaled measure-valued process, driven by two independent white noises coming from the infection and recovery processes. The SPDE is also linear and coupled with the solution to a system of stochastic Volterra-type linear integral equations driven by three independent Gaussian noises, one from the random infection functions in addition to the two white noises mentioned above. The solution to the SPDE can be also explicitly characterized, given this auxiliary process. The uniqueness of the SPDE solution is established under stronger assumptions (density and its derivative being locally bounded) on the distribution function of an infectious duration.