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

Abstract

This paper is focused on SIS epidemic dynamics (also known as the contact process) on stationary Poisson point processes of the Euclidean plane, when the infection rate of a susceptible point is proportional to the number of infected points in a ball around it. Two models are discussed, the first with a static point process, and the second where points are subject to some random motion. For both models, we use conservation equations for moment measures to analyze the stationary point processes of infected and susceptible points. A heuristic factorization of the third moment measure is then proposed to derive simple polynomial equations allowing one to derive closed form approximations for the fraction of infected nodes and 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. According to this phase diagram, the survival of the epidemic is not always an increasing function of 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 nodes when the epidemic survives. The phase diagram is also partly substantiated by the simulation of the mean survival time of the epidemic on large tori. The phase diagram accurately predicts the parameter regions where the mean survival time increases or decreases with the motion rate.