Stochastic epidemic models with a backward bifurcation

Two new stochastic epidemic models, a continuous-time Markov chainmodel and a stochastic differential equation model, are formulated. These arebased on a deterministic model that includes vaccination and is applicableto pertussis. For some parameter values, the deterministic model exhibitsa backwa...

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Main Authors: Linda J. S. Allen, P. van den Driessche
Format: Article
Language:English
Published: AIMS Press 2006-04-01
Series:Mathematical Biosciences and Engineering
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Online Access:https://www.aimspress.com/article/doi/10.3934/mbe.2006.3.445
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author Linda J. S. Allen
P. van den Driessche
author_facet Linda J. S. Allen
P. van den Driessche
author_sort Linda J. S. Allen
collection DOAJ
description Two new stochastic epidemic models, a continuous-time Markov chainmodel and a stochastic differential equation model, are formulated. These arebased on a deterministic model that includes vaccination and is applicableto pertussis. For some parameter values, the deterministic model exhibitsa backward bifurcation if the vaccine is imperfect. Thusa region of bistability exists in a subset of parameter space.The dynamics of the stochastic epidemic models are investigated in this regionof bistability, and compared with those of the deterministic model. In this region the probabilitydistribution associated with the infective population exhibits bimodality withone mode at the disease-free equilibrium and the other at the larger endemicequilibrium. For population sizes $N\geq 1000$, the deterministic and stochastic models agree,butfor small population sizes the stochastic models indicate that the backward bifurcation may have little effect on the disease dynamics.
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spelling doaj-art-695874da4eb041f6b4135aee9c47c6122025-01-24T01:51:49ZengAIMS PressMathematical Biosciences and Engineering1551-00182006-04-013344545810.3934/mbe.2006.3.445Stochastic epidemic models with a backward bifurcationLinda J. S. Allen0P. van den Driessche1Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042Two new stochastic epidemic models, a continuous-time Markov chainmodel and a stochastic differential equation model, are formulated. These arebased on a deterministic model that includes vaccination and is applicableto pertussis. For some parameter values, the deterministic model exhibitsa backward bifurcation if the vaccine is imperfect. Thusa region of bistability exists in a subset of parameter space.The dynamics of the stochastic epidemic models are investigated in this regionof bistability, and compared with those of the deterministic model. In this region the probabilitydistribution associated with the infective population exhibits bimodality withone mode at the disease-free equilibrium and the other at the larger endemicequilibrium. For population sizes $N\geq 1000$, the deterministic and stochastic models agree,butfor small population sizes the stochastic models indicate that the backward bifurcation may have little effect on the disease dynamics.https://www.aimspress.com/article/doi/10.3934/mbe.2006.3.445stochastic differential equations.bistabilityvaccinationcontinuous-time markov chainepidemic model
spellingShingle Linda J. S. Allen
P. van den Driessche
Stochastic epidemic models with a backward bifurcation
Mathematical Biosciences and Engineering
stochastic differential equations.
bistability
vaccination
continuous-time markov chain
epidemic model
title Stochastic epidemic models with a backward bifurcation
title_full Stochastic epidemic models with a backward bifurcation
title_fullStr Stochastic epidemic models with a backward bifurcation
title_full_unstemmed Stochastic epidemic models with a backward bifurcation
title_short Stochastic epidemic models with a backward bifurcation
title_sort stochastic epidemic models with a backward bifurcation
topic stochastic differential equations.
bistability
vaccination
continuous-time markov chain
epidemic model
url https://www.aimspress.com/article/doi/10.3934/mbe.2006.3.445
work_keys_str_mv AT lindajsallen stochasticepidemicmodelswithabackwardbifurcation
AT pvandendriessche stochasticepidemicmodelswithabackwardbifurcation