Abstract: 

In this talk we will discuss several stochastic epidemic models recently developed to account for general infectious durations, infection-age dependent infectivity and/or progress loss of immunity/varying susceptibility, extending the standard epidemic models. We construct individual based stochastic models, and prove scaling limits for the associated epidemic dynamics in large populations. Each individual is associated with a random function/process that represents the infection-age dependent infectivity force to exert on other individuals. We extend this formulation to associate each individual with a random function that represents the loss of immunity/varying susceptibility. A typical infectivity function first increases and then decreases from the epoch of becoming infected to the time of recovery, while a typical immunity/susceptibility function gradually increases from the time of recovery to the time of losing immunity and becoming susceptible. The scaling limits are deterministic and stochastic Volterra integral equations. We present some new PDEs models arising from the scaling limits. We also discuss some spatial models with varying infectivity and the associated scaling limits and PDE models. (This talk is based on recent joint work with Etienne Pardoux, Raphael Forien, and Arsene Brice Zosta Ngoufack.)