Multi-patch multi-group epidemic model with varying infectivity

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