Impact of discontinuous treatments on disease dynamics in an SIRepidemic model
We consider an SIR epidemic model withdiscontinuous treatment strategies. Under some reasonableassumptions on the discontinuous treatment function, we are able todetermine the basic reproduction number $\mathcal{R}_0$, confirm thewell-posedness of the model, describe the structure of possibleequilib...
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AIMS Press
2011-11-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2012.9.97 |
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| author | Zhenyuan Guo Lihong Huang Xingfu Zou |
| author_facet | Zhenyuan Guo Lihong Huang Xingfu Zou |
| author_sort | Zhenyuan Guo |
| collection | DOAJ |
| description | We consider an SIR epidemic model withdiscontinuous treatment strategies. Under some reasonableassumptions on the discontinuous treatment function, we are able todetermine the basic reproduction number $\mathcal{R}_0$, confirm thewell-posedness of the model, describe the structure of possibleequilibria as well as establish the stability/instability of theequilibria. Most interestingly, we find that in the case that anequilibrium is asymptotically stable, the convergence to theequilibrium can actually be achieved in finite time, and wecan estimate this time in terms of the model parameters, initialsub-populations and the initial treatment strength. This suggeststhat from the view point of eliminating the disease from the hostpopulation, discontinuous treatment strategies would be superior tocontinuous ones. The methods we use to obtain the mathematicalresults are the generalized Lyapunov theory for discontinuousdifferential equations and some results on non-smooth analysis. |
| format | Article |
| id | doaj-art-47eca9534b2249af97b356c92e3785b9 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2011-11-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-47eca9534b2249af97b356c92e3785b92025-01-24T02:05:22ZengAIMS PressMathematical Biosciences and Engineering1551-00182011-11-01919711010.3934/mbe.2012.9.97Impact of discontinuous treatments on disease dynamics in an SIRepidemic modelZhenyuan Guo0Lihong Huang1Xingfu Zou2College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082We consider an SIR epidemic model withdiscontinuous treatment strategies. Under some reasonableassumptions on the discontinuous treatment function, we are able todetermine the basic reproduction number $\mathcal{R}_0$, confirm thewell-posedness of the model, describe the structure of possibleequilibria as well as establish the stability/instability of theequilibria. Most interestingly, we find that in the case that anequilibrium is asymptotically stable, the convergence to theequilibrium can actually be achieved in finite time, and wecan estimate this time in terms of the model parameters, initialsub-populations and the initial treatment strength. This suggeststhat from the view point of eliminating the disease from the hostpopulation, discontinuous treatment strategies would be superior tocontinuous ones. The methods we use to obtain the mathematicalresults are the generalized Lyapunov theory for discontinuousdifferential equations and some results on non-smooth analysis.https://www.aimspress.com/article/doi/10.3934/mbe.2012.9.97discontinuous treatmentinfectious diseaseconvergence in finite time.sir modelgeneralized lyapunov methodstability |
| spellingShingle | Zhenyuan Guo Lihong Huang Xingfu Zou Impact of discontinuous treatments on disease dynamics in an SIRepidemic model Mathematical Biosciences and Engineering discontinuous treatment infectious disease convergence in finite time. sir model generalized lyapunov method stability |
| title | Impact of discontinuous treatments on disease dynamics in an SIRepidemic model |
| title_full | Impact of discontinuous treatments on disease dynamics in an SIRepidemic model |
| title_fullStr | Impact of discontinuous treatments on disease dynamics in an SIRepidemic model |
| title_full_unstemmed | Impact of discontinuous treatments on disease dynamics in an SIRepidemic model |
| title_short | Impact of discontinuous treatments on disease dynamics in an SIRepidemic model |
| title_sort | impact of discontinuous treatments on disease dynamics in an sirepidemic model |
| topic | discontinuous treatment infectious disease convergence in finite time. sir model generalized lyapunov method stability |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2012.9.97 |
| work_keys_str_mv | AT zhenyuanguo impactofdiscontinuoustreatmentsondiseasedynamicsinansirepidemicmodel AT lihonghuang impactofdiscontinuoustreatmentsondiseasedynamicsinansirepidemicmodel AT xingfuzou impactofdiscontinuoustreatmentsondiseasedynamicsinansirepidemicmodel |