A computational framework for evaluating the role of mobility on the propagation of epidemics on point processes

TL;DR

This paper develops a mathematical framework to analyze how random movement of individuals affects the spread and persistence of epidemics modeled by point processes, providing analytical approximations and phase diagrams validated by simulations.

Contribution

It introduces a novel model combining epidemic dynamics with random mobility on Poisson point processes, deriving polynomial equations and phase diagrams for epidemic survival.

Findings

Polynomial equations accurately predict infected fraction in steady state.

Phase diagram delineates parameter regions for epidemic survival and extinction.

Increased motion rate does not always lead to epidemic extinction.

Abstract

This paper is focused on SIS (Susceptible-Infected-Susceptible) epidemic dynamics (also known as the contact process) on populations modelled by homogeneous Poisson point processes of the Euclidean plane, where the infection rate of a susceptible individual is proportional to the number of infected individuals in a disc around it. The main focus of the paper is a model where points are also subject to some random motion. Conservation equations for moment measures are leveraged to analyze the stationary regime of the point processes of infected and susceptible individuals. A heuristic factorization of the third moment measure is then proposed to obtain simple polynomial equations allowing one to derive closed form approximations for the fraction of infected individuals in the steady state. These polynomial equations also lead to a phase diagram which tentatively delineates the regions of the space of parameters (population density, infection radius, infection and recovery rate, and motion rate) where the epidemic survives and those where there is extinction. A key take-away from this phase diagram is that the extinction of the epidemic is not always aided by a decrease in the motion rate. These results are substantiated by simulations on large two dimensional tori. These simulations show that the polynomial equations accurately predict the fraction of infected individuals when the epidemic survives. The simulations also show that the proposed phase diagram accurately predicts the parameter regions where the mean survival time of the epidemic increases (resp. decreases) with motion rate.