  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.
