Individual versus cluster recoveries within a spatially structured population

Abstract

Stochastic modeling of disease dynamics has had a long tradition. Among the first epidemic models including a spatial structure in the form of local interactions is the contact process. In this article we investigate two extensions of the contact process describing the course of a single disease within a spatially structured human population distributed in social clusters. That is, each site of the d-dimensional integer lattice is occupied by a cluster of individuals; each individual can be healthy or infected. The evolution of the disease depends on three parameters, namely the outside infection rate which models the interactions between the clusters, the within infection rate which takes into account the repeated contacts between individuals in the same cluster, and the size of each social cluster. For the first model, we assume cluster recoveries, while individual recoveries are assumed for the second one. The aim is to investigate the existence of nontrivial stationary distributions for both processes depending on the value of each of the three parameters. Our results show that the probability of an epidemic strongly depends on the recovery mechanism.

A self-regulating and patch subdivided population

“…We note that this process is similar to the IRP of [2] with cluster size equal to N . Nevertheless, it has two main differences: the interaction between clusters is always active (while in the IRP it vanishes once the cluster is nonempty) and it gets more difficult to increase the number of individuals in the cluster if the cluster is crowded.…”

“…The individual recovery process (IRP) introduced in [2] can be seen as a particular case of the PRP, namely, by setting N = 1 and c = 1 {0,...,κ−1} (the patch has only one site, which can host at most κ particles).…”

“…This kind of interaction was introduced by Schinazi [11] with a cluster recovery clearing mechanism (or mass extinction), and modified by Belhadji and Lanchier with individual recoveries in [2]. Note nevertheless that this model differs from the IRP in the fact that the breeding inside the cluster becomes increasingly difficult as we approach the full carrying capacity κ.…”

“…Recall that, by [2,Theorem 5] for the IRP with parameters λ and φ and with κ maximal number of particles per vertex, there is a critical value for κ, say κ c (λ, φ), such that, for κ ≥ κ c (λ, φ), there is survival and, for κ < κ c (λ, φ), there is extinction. It is easy to prove that condition (b) of Theorem 3.4 can be relaxed, provided that we have some knowledge of the function κ c .…”

Contact Process with Destruction of Cubes and Hyperplanes: Forest Fires Versus Tornadoes

“…Even though a wide variety of stochastic models have been developed in order to understand the effect of random catastrophic events on the survival probability of a population, most of them, similarly to the generic model (1), exclude the presence of a spatial structure, i.e. the state space of these stochastic processes keeps track of a number of individuals but ignores the spatial arrangement of the population.…”