  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.
