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Eric Foxall

Publications and source records attributed to Eric Foxall.

24 records · Page 2Linked to original sources
arXiv2015

The SEIS model, or, the contact process with a latent stage

The susceptible-exposed-infectious-susceptible (SEIS) model is well-known in mathematical epidemiology as a model of infection in which there is a latent period between the moment of infection and the onset of infectiousness. The compartment model is well studied, but the corresponding particle system has so far received no attention. For the particle system model in one spatial dimension, we give upper and lower bounds on the critical values, prove convergence of critical values in the limit of small and large latent time, and identify a limiting process to which the SEIS model converges in the limit of large latent time.

math.PR↗
arXiv2015

Critical Behaviour of the Partner Model

We consider a stochastic model of infection spread incorporating monogamous partnership dynamics. In previous work a basic reproduction number $R_0$ is defined with the property that if $R_0<1$ the infection dies out within $O(\log N)$ units of time, while if $R_0>1$ the infection survives for at least $e^{γN}$ units of time, for some $γ>0$. Here we consider the critical case $R_0=1$ and show that the infection dies out within $O(\sqrt{N})$ units of time, and moreover that this estimate is sharp.

math.PR↗
arXiv2015

Survival and extinction results for a patch model with sexual reproduction

This article is concerned with a version of the contact process with sexual reproduction on a graph with two levels of interactions modeling metapopulations. The population is spatially distributed into patches and offspring are produced in each patch at a rate proportional to the number of pairs of individuals in the patch (sexual reproduction) rather than simply the number of individuals as in the basic contact process. Offspring produced at a given patch either stay in their parents' patch or are sent to a nearby patch with some fixed probabilities. As the patch size tends to infinity, we identify a mean-field limit consisting of an infinite set of coupled differential equations. For the mean-field equations, we find explicit conditions for survival and extinction that we call expansion and retreat. Using duality techniques to compare the stochastic model to its mean-field limit, we find that expansion and retreat are also precisely the conditions needed to ensure survival and extinction of the stochastic model when the patch size is large. In addition, we study the dependence of survival on the dispersal range. We find that, with probability close to one and for a certain set of parameters, the metapopulation survives in the presence of nearest neighbor interactions while it dies out in the presence of long range interactions, suggesting that the best strategy for the population to spread in space is to use intermediate dispersal ranges.

math.PR↗
arXiv2014

Duality and Complete Convergence for Multi-type Additive Growth Models

We consider a class of multi-type particle systems having similar structure to the contact process and show that additivity is equivalent to the existence of a dual process, extending a result of Harris. We give two additional characterizations of these systems, in spacetime as percolation models, and biologically as population models in which the interactions are due to crowding. We prove a necessary and sufficient condition for the model to preserve positive correlations. We then show that complete convergence on $\mathbb{Z}^d$ holds for a large subclass of models including the two-stage contact process and a household model, and give examples.

math.PR↗
arXiv2014

New Results for the Two-Stage Contact Process

Here we continue the work started by Steve Krone on the two-stage contact process. We give a simplified proof of the duality relation, and answer most of the open questions posed in that paper. We also fill in the details of an incomplete proof.

math.PR↗