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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AIMS Press
2006-04-01
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| 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. |
| format | Article |
| id | doaj-art-695874da4eb041f6b4135aee9c47c612 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2006-04-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| 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 |