Epidemic threshold conditions for seasonally forced SEIR models
In this paper we derive threshold conditions for eradication of diseases that can be described by seasonally forced susceptible-exposed-infectious-recovered (SEIR) models or their variants. For autonomous models, the basic reproduction number $\mathcal{R}_0 < 1$ is usually both necessary and suff...
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AIMS Press
2005-10-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2006.3.161 |
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| author | Junling Ma Zhien Ma |
| author_facet | Junling Ma Zhien Ma |
| author_sort | Junling Ma |
| collection | DOAJ |
| description | In this paper we derive threshold conditions for eradication of diseases that can be described by seasonally forced susceptible-exposed-infectious-recovered (SEIR) models or their variants. For autonomous models, the basic reproduction number $\mathcal{R}_0 < 1$ is usually both necessary and sufficient for the extinction of diseases. For seasonally forced models, $\mathcal{R}_0$ is a function of time $t$. We find that for models without recruitment of susceptible individuals (via births or loss of immunity), max$_t{\mathcal{R}_0(t)} < 1$ is required to prevent outbreaks no matter when and how the disease is introduced. For models with recruitment, if the latent period can be neglected, the disease goes extinct if and only if the basic reproduction number $\bar{\mathcal{R}}$ of the time-average systems (the autonomous systems obtained by replacing the time-varying parameters with their long-term time averages) is less than 1. Otherwise, $\bar{\mathcal{R}} < 1$ is sufficient but not necessary for extinction. Thus, reducing $\bar{\mathcal{R}}$ of the average system to less than 1 is sufficient to prevent or curtail the spread of an endemic disease. |
| format | Article |
| id | doaj-art-626b705912224579a9b88b85e9eb4c34 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2005-10-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-626b705912224579a9b88b85e9eb4c342025-01-24T01:51:11ZengAIMS PressMathematical Biosciences and Engineering1551-00182005-10-013116117210.3934/mbe.2006.3.161Epidemic threshold conditions for seasonally forced SEIR modelsJunling Ma0Zhien Ma1Department of Mathematics and Statistics, McMaster University, Hamilton, ON Canada L8S 4K1Department of Applied Mathematics, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049In this paper we derive threshold conditions for eradication of diseases that can be described by seasonally forced susceptible-exposed-infectious-recovered (SEIR) models or their variants. For autonomous models, the basic reproduction number $\mathcal{R}_0 < 1$ is usually both necessary and sufficient for the extinction of diseases. For seasonally forced models, $\mathcal{R}_0$ is a function of time $t$. We find that for models without recruitment of susceptible individuals (via births or loss of immunity), max$_t{\mathcal{R}_0(t)} < 1$ is required to prevent outbreaks no matter when and how the disease is introduced. For models with recruitment, if the latent period can be neglected, the disease goes extinct if and only if the basic reproduction number $\bar{\mathcal{R}}$ of the time-average systems (the autonomous systems obtained by replacing the time-varying parameters with their long-term time averages) is less than 1. Otherwise, $\bar{\mathcal{R}} < 1$ is sufficient but not necessary for extinction. Thus, reducing $\bar{\mathcal{R}}$ of the average system to less than 1 is sufficient to prevent or curtail the spread of an endemic disease.https://www.aimspress.com/article/doi/10.3934/mbe.2006.3.161seasonal forcingbasic reproduction number.epidemic models |
| spellingShingle | Junling Ma Zhien Ma Epidemic threshold conditions for seasonally forced SEIR models Mathematical Biosciences and Engineering seasonal forcing basic reproduction number. epidemic models |
| title | Epidemic threshold conditions for seasonally forced SEIR models |
| title_full | Epidemic threshold conditions for seasonally forced SEIR models |
| title_fullStr | Epidemic threshold conditions for seasonally forced SEIR models |
| title_full_unstemmed | Epidemic threshold conditions for seasonally forced SEIR models |
| title_short | Epidemic threshold conditions for seasonally forced SEIR models |
| title_sort | epidemic threshold conditions for seasonally forced seir models |
| topic | seasonal forcing basic reproduction number. epidemic models |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2006.3.161 |
| work_keys_str_mv | AT junlingma epidemicthresholdconditionsforseasonallyforcedseirmodels AT zhienma epidemicthresholdconditionsforseasonallyforcedseirmodels |