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Large population asymptotics for a multitype stochastic SIS epidemic model in randomly switched environment
- Publication Year :
- 2021
- Publisher :
- HAL CCSD, 2021.
-
Abstract
- We consider an epidemic SIS model described by a multitype birth-and-death process in a randomly switched environment. That is, the infection and cure rates of the process depend on the state of a finite Markov jump process (the environment), whose transitions also depend on the number of infectives. The total size of the population is constant and equal to some K $\in$ N * , and the number of infectives vanishes almost surely in finite time. We prove that, as K $\rightarrow$ $\infty$, the process composed of the proportions of infectives of each type X^K and the state of the environment $\Xi$^K , converges to a piecewise deterministic Markov process (PDMP) given by a system of randomly switched ODEs. The long term behaviour of this PDMP has been previously investigated by Bena{\"i}m and Strickler, and depends only on the sign 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 show that the large population asymptotics of X^K also strongly depend on the sign of $\Lambda$: if negative, then from fixed initial proportions of infectives the 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 in the situation where the origin is accessible for the linearised PDMP, the mean extinction time of X^K is logarithmically equivalent to K^p * , where p * > 0 is fully characterised. We also investigate the quasi-stationary distribution $\mu$^K of (X^K , $\Xi$^K) and show that, when $\Lambda$ < 0, weak limit points of ($\mu$^K), K>0 are supported by the extinction set, while when $\Lambda$ > 0, limit points belong to the (non empty) set of stationary distributions of the limiting PDMP which do not give mass to the extinction set.
- Subjects :
- Extinction time
Probability (math.PR)
Quasi-stationary distributions
Lyapunov exponents
Birth-and-death processes
Large population
Random switching
Epidemic models
SIS
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
Stochastic persistence
FOS: Mathematics
Quantitative Biology::Populations and Evolution
Scaling limit
Piecewise deterministic Markov processes
Mathematics - Probability
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....a3f1e62fc95c01eda2426973136052c0