Spatial SIR epidemic model with varying infectivity without movement of individuals: Law of Large Numbers
Armand Kanga, Etienne Pardoux
TL;DR
The paper studies spatially structured SIR dynamics with variable infectivity in a fixed population distributed over a compact domain $D\subset\mathbb{R}^d$ and without individual movement. It defines empirical measures for susceptible, infected, recovered individuals and the total infection pressure, and proves a law of large numbers as $N\to\infty$, yielding a deterministic limit described by a coupled system of integral-differential equations with a nonlinear kernel $M(x,y)=\dfrac{K(x,y)}{[\int_D K(z,y)\overline{\mu}(dz)]^{\gamma}}$. The analysis develops a McKean-Vlasov reformulation of the limiting dynamics and proves existence and uniqueness of the limit, while addressing potential degeneracy in the normalization through a robust variant using $\Phi(u)$. The results provide a rigorous bridge from stochastic spatial epidemics with time-varying infectivity to tractable deterministic descriptions, enabling study of spatial heterogeneity, contact kernels, and non-exponential infectious durations in large populations.
Abstract
In this work, we use a new approach to study the spread of an infectious disease. Indeed, we study a SIR epidemic model with variable infectivity, where the individuals are distributed over a compact subset $D$ of $\R^d$. We define empirical measures which describe the evolution of the state (susceptible, infectious, recovered) of the individuals in the various locations, and the total force of infection in the population. In our model, the individuals do not move. We establish a law of large numbers for these measures, as the population size tends to infinity.