Contact Process under heavy-tailed renewals on finite graphs
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2019
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| _version_ | 1866913629910073344 |
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| author | Fontes, Luiz Renato Gomes, Pablo Almeida Sanchis, Remy |
| author_facet | Fontes, Luiz Renato Gomes, Pablo Almeida Sanchis, Remy |
| contents | We investigate a non-Markovian analogue of the Harris contact process in a finite connected graph G=(V,E): an individual is attached to each site x in V, and it can be infected or healthy; the infection propagates to healthy neighbors just as in the usual contact process, according to independent exponential times with a fixed rate lambda>0; however, the recovery times for an individual are given by the points of a renewal process attached to its timeline, whose waiting times have distribution mu such that mu(t,infty) = t^{-alpha}L(t), where 1/2 < alpha < 1 and L is a slowly varying function; the renewal processes are assumed to be independent for different sites. We show that, starting with a single infected individual, if |V| < 2 + (2 alpha -1)/[(1-alpha)(2-alpha)], then the infection does not survive for any lambda; and if |V| > 1/(1-alpha), then, for every lambda, the infection has positive probability to survive |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1907_00290 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Contact Process under heavy-tailed renewals on finite graphs Fontes, Luiz Renato Gomes, Pablo Almeida Sanchis, Remy Probability 60K35, 82B43 We investigate a non-Markovian analogue of the Harris contact process in a finite connected graph G=(V,E): an individual is attached to each site x in V, and it can be infected or healthy; the infection propagates to healthy neighbors just as in the usual contact process, according to independent exponential times with a fixed rate lambda>0; however, the recovery times for an individual are given by the points of a renewal process attached to its timeline, whose waiting times have distribution mu such that mu(t,infty) = t^{-alpha}L(t), where 1/2 < alpha < 1 and L is a slowly varying function; the renewal processes are assumed to be independent for different sites. We show that, starting with a single infected individual, if |V| < 2 + (2 alpha -1)/[(1-alpha)(2-alpha)], then the infection does not survive for any lambda; and if |V| > 1/(1-alpha), then, for every lambda, the infection has positive probability to survive |
| title | Contact Process under heavy-tailed renewals on finite graphs |
| topic | Probability 60K35, 82B43 |
| url | https://arxiv.org/abs/1907.00290 |