An SEIRS epidemic model with stochastic transmission
Abstract For an SEIRS epidemic model with stochastic perturbations on transmission from the susceptible class to the latent and infectious classes, we prove the existence of global positive solutions. For sufficiently small values of the perturbation parameter, we prove the almost surely exponential...
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Online Access: | http://link.springer.com/article/10.1186/s13662-017-1166-6 |
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doaj-74bfe106a99248daa0db79dd2abccbe92020-11-24T23:12:20ZengSpringerOpenAdvances in Difference Equations1687-18472017-04-012017111610.1186/s13662-017-1166-6An SEIRS epidemic model with stochastic transmissionPeter J Witbooi0Department of Mathematics and Applied Mathematics, University of the Western CapeAbstract For an SEIRS epidemic model with stochastic perturbations on transmission from the susceptible class to the latent and infectious classes, we prove the existence of global positive solutions. For sufficiently small values of the perturbation parameter, we prove the almost surely exponential stability of the disease-free equilibrium whenever a certain invariant R σ $\mathcal{R}_{\sigma}$ is below unity. Here R σ < R $\mathcal{R}_{\sigma}< \mathcal{R}$ , the latter being the basic reproduction number of the underlying deterministic model. Biologically, the main result has the following significance for a disease model that has an incubation phase of the pathogen: A small stochastic perturbation on the transmission rate from susceptible to infectious via the latent phase will enhance the stability of the disease-free state if both components of the perturbation are non-trivial; otherwise the stability will not be disturbed. Simulations illustrate the main stability theorem.http://link.springer.com/article/10.1186/s13662-017-1166-6SEIRS modelstochastic transmissionalmost sure exponential stability |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Peter J Witbooi |
spellingShingle |
Peter J Witbooi An SEIRS epidemic model with stochastic transmission Advances in Difference Equations SEIRS model stochastic transmission almost sure exponential stability |
author_facet |
Peter J Witbooi |
author_sort |
Peter J Witbooi |
title |
An SEIRS epidemic model with stochastic transmission |
title_short |
An SEIRS epidemic model with stochastic transmission |
title_full |
An SEIRS epidemic model with stochastic transmission |
title_fullStr |
An SEIRS epidemic model with stochastic transmission |
title_full_unstemmed |
An SEIRS epidemic model with stochastic transmission |
title_sort |
seirs epidemic model with stochastic transmission |
publisher |
SpringerOpen |
series |
Advances in Difference Equations |
issn |
1687-1847 |
publishDate |
2017-04-01 |
description |
Abstract For an SEIRS epidemic model with stochastic perturbations on transmission from the susceptible class to the latent and infectious classes, we prove the existence of global positive solutions. For sufficiently small values of the perturbation parameter, we prove the almost surely exponential stability of the disease-free equilibrium whenever a certain invariant R σ $\mathcal{R}_{\sigma}$ is below unity. Here R σ < R $\mathcal{R}_{\sigma}< \mathcal{R}$ , the latter being the basic reproduction number of the underlying deterministic model. Biologically, the main result has the following significance for a disease model that has an incubation phase of the pathogen: A small stochastic perturbation on the transmission rate from susceptible to infectious via the latent phase will enhance the stability of the disease-free state if both components of the perturbation are non-trivial; otherwise the stability will not be disturbed. Simulations illustrate the main stability theorem. |
topic |
SEIRS model stochastic transmission almost sure exponential stability |
url |
http://link.springer.com/article/10.1186/s13662-017-1166-6 |
work_keys_str_mv |
AT peterjwitbooi anseirsepidemicmodelwithstochastictransmission AT peterjwitbooi seirsepidemicmodelwithstochastictransmission |
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1725601269000699904 |