Modeling diseases with latency and relapse
A general mathematical model for a disease with an exposed(latent) period and relapse is proposed. Such a model isappropriate for tuberculosis, including bovine tuberculosis incattle and wildlife, and for herpes. For this model with ageneral probability of remaining in the exposed class, the bas...
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| Format: | Article |
| Language: | English |
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AIMS Press
2007-01-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2007.4.205 |
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| author | P. van den Driessche Lin Wang Xingfu Zou |
| author_facet | P. van den Driessche Lin Wang Xingfu Zou |
| author_sort | P. van den Driessche |
| collection | DOAJ |
| description | A general mathematical model for a disease with an exposed(latent) period and relapse is proposed. Such a model isappropriate for tuberculosis, including bovine tuberculosis incattle and wildlife, and for herpes. For this model with ageneral probability of remaining in the exposed class, the basicreproduction number $\R_0$ is identified and its thresholdproperty is discussed. In particular, the disease-free equilibriumis proved to be globally asymptotically stable if $\R_01$, and the disease is shown to be uniformly persistentwith the infective population size either approaching oroscillating about the endemic level. Numerical simulations (forparameters appropriate for bovine tuberculosis in cattle) with$\mathcal{R}_0>1$ indicate that solutions tend to this endemicstate. |
| format | Article |
| id | doaj-art-76dfd16adbc24ce7853b565c28272489 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2007-01-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-76dfd16adbc24ce7853b565c282724892025-01-24T01:53:27ZengAIMS PressMathematical Biosciences and Engineering1551-00182007-01-014220521910.3934/mbe.2007.4.205Modeling diseases with latency and relapseP. van den Driessche0Lin Wang1Xingfu Zou2Department of Mathematics and Statistics, University of Victoria, Victoria B.C., Canada V8W 3P4Department of Mathematics and Statistics, University of Victoria, Victoria B.C., Canada V8W 3P4Department of Mathematics and Statistics, University of Victoria, Victoria B.C., Canada V8W 3P4A general mathematical model for a disease with an exposed(latent) period and relapse is proposed. Such a model isappropriate for tuberculosis, including bovine tuberculosis incattle and wildlife, and for herpes. For this model with ageneral probability of remaining in the exposed class, the basicreproduction number $\R_0$ is identified and its thresholdproperty is discussed. In particular, the disease-free equilibriumis proved to be globally asymptotically stable if $\R_01$, and the disease is shown to be uniformly persistentwith the infective population size either approaching oroscillating about the endemic level. Numerical simulations (forparameters appropriate for bovine tuberculosis in cattle) with$\mathcal{R}_0>1$ indicate that solutions tend to this endemicstate.https://www.aimspress.com/article/doi/10.3934/mbe.2007.4.205bovine tuberculosisdelay differential equationuniform persistence.endemic equilibriumglobal asymptotic stabilitydisease-free equilibriumtuberculosis |
| spellingShingle | P. van den Driessche Lin Wang Xingfu Zou Modeling diseases with latency and relapse Mathematical Biosciences and Engineering bovine tuberculosis delay differential equation uniform persistence. endemic equilibrium global asymptotic stability disease-free equilibrium tuberculosis |
| title | Modeling diseases with latency and relapse |
| title_full | Modeling diseases with latency and relapse |
| title_fullStr | Modeling diseases with latency and relapse |
| title_full_unstemmed | Modeling diseases with latency and relapse |
| title_short | Modeling diseases with latency and relapse |
| title_sort | modeling diseases with latency and relapse |
| topic | bovine tuberculosis delay differential equation uniform persistence. endemic equilibrium global asymptotic stability disease-free equilibrium tuberculosis |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2007.4.205 |
| work_keys_str_mv | AT pvandendriessche modelingdiseaseswithlatencyandrelapse AT linwang modelingdiseaseswithlatencyandrelapse AT xingfuzou modelingdiseaseswithlatencyandrelapse |