English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider an epidemic SIS model described by a multitype birth-and-deathprocess in a randomly switched environment. That is, the infection and curerates of the process depend on the state of a finite Markov jump process (theenvironment), whose transitions also depend on the number of infectives. Thetotal size of the population is constant and equal to some K $\in$ N * , andthe number of infectives vanishes almost surely in finite time. We prove that,as K $\rightarrow$ $\infty$, the process composed of the proportions ofinfectives of each type X^K and the state of the environment $\Xi$^K ,converges to a piecewise deterministic Markov process (PDMP) given by a systemof randomly switched ODEs. The long term behaviour of this PDMP has beenpreviously investigated by Bena{\"i}m and Strickler, and depends only on thesign of the top Lyapunov exponent $\Lambda$ of the linearised PDMP at 0: if$\Lambda$ < 0, the proportion of infectives in each group converges to zero,while if $\Lambda$ > 0, the disease becomes endemic. In this paper, we showthat the large population asymptotics of X^K also strongly depend on the signof $\Lambda$: if negative, then from fixed initial proportions of infectivesthe disease disappears in a time of order at most log(K), while if positive,the typical extinction time grows at least as a power of K. We prove that inthe situation where the origin is accessible for the linearised PDMP, the meanextinction time of X^K is logarithmically equivalent to K^p * , where p * > 0is fully characterised. We also investigate the quasi-stationary distribution$\mu$^K of (X^K , $\Xi$^K) and show that, when $\Lambda$ < 0, weak limit pointsof ($\mu$^K), K>0 are supported by the extinction set, while when $\Lambda$ >0, limit points belong to the (non empty) set of stationary distributions ofthe limiting PDMP which do not give mass to the extinction set.

Large population asymptotics for a multitype stochastic SIS epidemic model in randomly switched environment