Global stability for epidemicmodel with constant latency and infectious periods
In recent years many delay epidemiological models have been proposedto study at which stage of the epidemics the delays can destabilizethe disease free equilibrium, or the endemic equilibrium, givingrise to stability switches. One of these models is the SEIR modelwith constant latency time and infec...
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AIMS Press
2012-02-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2012.9.297 |
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| author | Gang Huang Edoardo Beretta Yasuhiro Takeuchi |
| author_facet | Gang Huang Edoardo Beretta Yasuhiro Takeuchi |
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| description | In recent years many delay epidemiological models have been proposedto study at which stage of the epidemics the delays can destabilizethe disease free equilibrium, or the endemic equilibrium, givingrise to stability switches. One of these models is the SEIR modelwith constant latency time and infectious periods [2],for which the authors have proved that the two delays are harmlessin inducing stability switches. However, it is left open the problemof the global asymptotic stability of the endemic equilibriumwhenever it exists. Even the Lyapunov functions approach, recentlyproposed by Huang and Takeuchi to study many delay epidemiologicalmodels, fails to work on this model. In this paper, an age-infectionmodel is presented for the delay SEIR epidemic model, such that theproperties of global asymptotic stability of the equilibria of theage-infection model imply the same properties for the originaldelay-differential epidemic model. By introducing suitable Lyapunovfunctions to study the global stability of the disease freeequilibrium(when $\mathcal{R}_0\leq 1$) and of the endemic equilibria (whenever $\mathcal{R}_0>1$) of the age-infection model, we can infer thecorrespondingglobal properties for the equilibria of the delay SEIR model in [2], thus proving that the endemic equilibrium in[2] is globally asymptotically stable whenever itexists.   Furthermore, we also present a review of the SIR, SEIR epidemicmodels, with and without delays, appeared in literature, that can beseen as particular cases of the approach presented in the paper. |
| format | Article |
| id | doaj-art-734d33ec94a84054943ef16eaeec7593 |
| institution | Kabale University |
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| language | English |
| publishDate | 2012-02-01 |
| publisher | AIMS Press |
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| spelling | doaj-art-734d33ec94a84054943ef16eaeec75932025-01-24T02:05:29ZengAIMS PressMathematical Biosciences and Engineering1551-00182012-02-019229731210.3934/mbe.2012.9.297Global stability for epidemicmodel with constant latency and infectious periodsGang Huang0Edoardo Beretta1Yasuhiro Takeuchi2School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074In recent years many delay epidemiological models have been proposedto study at which stage of the epidemics the delays can destabilizethe disease free equilibrium, or the endemic equilibrium, givingrise to stability switches. One of these models is the SEIR modelwith constant latency time and infectious periods [2],for which the authors have proved that the two delays are harmlessin inducing stability switches. However, it is left open the problemof the global asymptotic stability of the endemic equilibriumwhenever it exists. Even the Lyapunov functions approach, recentlyproposed by Huang and Takeuchi to study many delay epidemiologicalmodels, fails to work on this model. In this paper, an age-infectionmodel is presented for the delay SEIR epidemic model, such that theproperties of global asymptotic stability of the equilibria of theage-infection model imply the same properties for the originaldelay-differential epidemic model. By introducing suitable Lyapunovfunctions to study the global stability of the disease freeequilibrium(when $\mathcal{R}_0\leq 1$) and of the endemic equilibria (whenever $\mathcal{R}_0>1$) of the age-infection model, we can infer thecorrespondingglobal properties for the equilibria of the delay SEIR model in [2], thus proving that the endemic equilibrium in[2] is globally asymptotically stable whenever itexists.   Furthermore, we also present a review of the SIR, SEIR epidemicmodels, with and without delays, appeared in literature, that can beseen as particular cases of the approach presented in the paper.https://www.aimspress.com/article/doi/10.3934/mbe.2012.9.297epidemic modelinfectious periodglobal stabilityage-structure. |
| spellingShingle | Gang Huang Edoardo Beretta Yasuhiro Takeuchi Global stability for epidemicmodel with constant latency and infectious periods Mathematical Biosciences and Engineering epidemic model infectious period global stability age-structure. |
| title | Global stability for epidemicmodel with constant latency and infectious periods |
| title_full | Global stability for epidemicmodel with constant latency and infectious periods |
| title_fullStr | Global stability for epidemicmodel with constant latency and infectious periods |
| title_full_unstemmed | Global stability for epidemicmodel with constant latency and infectious periods |
| title_short | Global stability for epidemicmodel with constant latency and infectious periods |
| title_sort | global stability for epidemicmodel with constant latency and infectious periods |
| topic | epidemic model infectious period global stability age-structure. |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2012.9.297 |
| work_keys_str_mv | AT ganghuang globalstabilityforepidemicmodelwithconstantlatencyandinfectiousperiods AT edoardoberetta globalstabilityforepidemicmodelwithconstantlatencyandinfectiousperiods AT yasuhirotakeuchi globalstabilityforepidemicmodelwithconstantlatencyandinfectiousperiods |