Spatial SIR epidemic model with varying infectivity in an unbounded domain: Law of Large Numbers

We consider a spatial SIR epidemic model where the infectivity of infected individuals depends upon their age of infection, and infections are non local. The domain is an unbounded subset of ${\mathbb{R}}^d$,and the individuals do not move. We extend our earlier result in \cite{AK-EP}, where the domain was bounded, and prove a law of large numbers as the size of the population tends to $\mathrm{\infty }$.