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

TL;DR

The paper studies a spatial SIR model with infectivity that depends on the age of infection and nonlocal transmission on an unbounded domain, with individuals fixed in space. It proves a law of large numbers for the empirical measures as the population size $N$ tends to infinity, yielding a deterministic limit described by a coupled system of integral equations for the susceptible, infectious, recovered compartments and the total infection force. The analysis hinges on a careful construction of measure-valued processes, a truncated deterministic framework on bounded domains $D_n$ to handle the unbounded setting, and rigorous well-posedness and convergence arguments. This provides a rigorous macroscopic description of spatial disease spread under nonlocal interactions and age-structured infectivity, applicable to large-scale or infinite spatial domains.

Abstract

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 $\R^d$,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 $\infty$.

Theorems & Definitions (27)

• Proposition 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.1
• Proposition 3.1
• Lemma 3.3
• proof
• Lemma 3.4