An SEIR epidemic model with constant latency time and infectious period
We present a two delays SEIR epidemic model with a saturation incidencerate. One delay is the time taken by the infected individuals to becomeinfectious (i.e. capable to infect a susceptible individual), the seconddelay is the time taken by an infectious individual to be removed from theinfection. B...
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AIMS Press
2011-07-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2011.8.931 |
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| author | Edoardo Beretta Dimitri Breda |
| author_facet | Edoardo Beretta Dimitri Breda |
| author_sort | Edoardo Beretta |
| collection | DOAJ |
| description | We present a two delays SEIR epidemic model with a saturation incidencerate. One delay is the time taken by the infected individuals to becomeinfectious (i.e. capable to infect a susceptible individual), the seconddelay is the time taken by an infectious individual to be removed from theinfection. By iterative schemes and the comparison principle, we provideglobal attractivity results for both the equilibria, i.e. the disease-freeequilibrium $\mathbf{E}_{0}$ and the positive equilibrium $\mathbf{E}_{+}$,which exists iff the basic reproduction number $\mathcal{R}_{0}$ is largerthan one. If $\mathcal{R}_{0}>1$ we also provide a permanence result for themodel solutions. Finally we prove that the two delays are harmless in thesense that, by the analysis of the characteristic equations, which result tobe polynomial trascendental equations with polynomial coefficients dependentupon both delays, we confirm all the standard properties of an epidemicmodel: $\mathbf{E}_{0}$ is locally asymptotically stable for $\mathcal{R}%_{0}1$, while if $\mathcal{R}_{0}>1$then $\mathbf{E}_{+}$ is always asymptotically stable. |
| format | Article |
| id | doaj-art-eca3bf2319a04b74824742d28397a3b1 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2011-07-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-eca3bf2319a04b74824742d28397a3b12025-01-24T02:02:16ZengAIMS PressMathematical Biosciences and Engineering1551-00182011-07-018493195210.3934/mbe.2011.8.931An SEIR epidemic model with constant latency time and infectious periodEdoardo Beretta0Dimitri Breda1CIMAB, University of Milano, via C. Saldini 50, I20133 MilanoCIMAB, University of Milano, via C. Saldini 50, I20133 MilanoWe present a two delays SEIR epidemic model with a saturation incidencerate. One delay is the time taken by the infected individuals to becomeinfectious (i.e. capable to infect a susceptible individual), the seconddelay is the time taken by an infectious individual to be removed from theinfection. By iterative schemes and the comparison principle, we provideglobal attractivity results for both the equilibria, i.e. the disease-freeequilibrium $\mathbf{E}_{0}$ and the positive equilibrium $\mathbf{E}_{+}$,which exists iff the basic reproduction number $\mathcal{R}_{0}$ is largerthan one. If $\mathcal{R}_{0}>1$ we also provide a permanence result for themodel solutions. Finally we prove that the two delays are harmless in thesense that, by the analysis of the characteristic equations, which result tobe polynomial trascendental equations with polynomial coefficients dependentupon both delays, we confirm all the standard properties of an epidemicmodel: $\mathbf{E}_{0}$ is locally asymptotically stable for $\mathcal{R}%_{0}1$, while if $\mathcal{R}_{0}>1$then $\mathbf{E}_{+}$ is always asymptotically stable.https://www.aimspress.com/article/doi/10.3934/mbe.2011.8.931nonlinear incidence ratetime delayslocal stability analysis.permanenceglobal attractivitydelay differential equationsepidemic model |
| spellingShingle | Edoardo Beretta Dimitri Breda An SEIR epidemic model with constant latency time and infectious period Mathematical Biosciences and Engineering nonlinear incidence rate time delays local stability analysis. permanence global attractivity delay differential equations epidemic model |
| title | An SEIR epidemic model with constant latency time and infectious period |
| title_full | An SEIR epidemic model with constant latency time and infectious period |
| title_fullStr | An SEIR epidemic model with constant latency time and infectious period |
| title_full_unstemmed | An SEIR epidemic model with constant latency time and infectious period |
| title_short | An SEIR epidemic model with constant latency time and infectious period |
| title_sort | seir epidemic model with constant latency time and infectious period |
| topic | nonlinear incidence rate time delays local stability analysis. permanence global attractivity delay differential equations epidemic model |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2011.8.931 |
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