Multi-patch multi-group epidemic model with varying infectivity

TL;DR

This paper develops a large population limit for a complex stochastic SIR epidemic model with spatial, group heterogeneity, and age-dependent infectivity, resulting in integral equations that capture these effects.

Contribution

It introduces a novel law of large numbers for a multi-patch, multi-group epidemic model with age-dependent infectivity, using a new proof technique based on McKean-Vlasov equations.

Findings

Derivation of Volterra-type integral equations as the population size tends to infinity.

Establishment of a law of large numbers for the multi-patch, multi-group model.

Weaker conditions on infectivity functions compared to previous models.

Abstract

This paper presents a law of large numbers result, as the size of the population tends to infinity, of SIR stochastic epidemic models, for a population distributed over $L$ distinct patches (with migrations between them) and $K$ distinct groups (possibly age groups). The limit is a set of Volterra-type integral equations, and the result shows the effects of both spatial and population heterogeneity. The novelty of the model is that the infectivity of an infected individual is infection age dependent. More precisely, to each infected individual is attached a random infection-age dependent infectivity function, such that the various random functions attached to distinct individuals are i.i.d. The proof involves a novel construction of a sequence of i.i.d. processes to invoke the law of large numbers for processes in $D$, by using the solution of a MacKean-Vlasov type Poisson-driven stochastic equation (as in the propagation of chaos theory). We also establish an identity using the Feynman-Kac formula for an adjoint backward ODE. The advantage of this approach is that it assumes much weaker conditions on the random infectivity functions than our earlier work for the homogeneous model in [20], where standard tightness criteria for convergence of stochastic processes were employed. To illustrate this new approach, we first explain the new proof under the weak assumptions for the homogeneous model, and then describe the multipatch-multigroup model and prove the law of large numbers for that model.