Epidemic models with varying infectivity

We introduce an epidemic model with varying infectivity and general exposed and infectious periods, where the infectivity of each individual is a random function of the elapsed time since infection, those function being independent and identically distributed for the various individuals in the population. This approach models infection-age-dependent infectivity and extends the classical SIR and SEIR models. We focus on the infectivity process (total force of infection at each time) and prove a functional law of large number (FLLN). In the deterministic limit of this FLLN, the joint evolution of the mean infectivity and of the proportion of susceptible individuals is determined by a two-dimensional deterministic integral equation. From its solutions, we then obtain expressions for the evolution of the proportions of exposed, infectious, and recovered individuals. For the early phase, we study the stochastic model directly by using an approximate (non-Markovian) branching process and show that the epidemic grows at an exponential rate on the event of nonextinction, which matches the rate of growth derived from the deterministic linearized equations. We also use these equations to derive the expression for the basic reproduction number R₀ during the early stage of an epidemic, in terms of the average individual infectivity function and the exponential rate of growth of the epidemic, and apply our results to the Covid-19 epidemic.