Stability of the Epidemiological Model SIR with Loss of Immunity
DOI:
https://doi.org/10.5433/1679-0375.2023.v44.47860Keywords:
SIR model, ordinary differential equations, fixed points, stabilityAbstract
This study approaches the analysis of the stability of the epidemiological model SIR with loss of immunity. This is a model given by a system of ordinary differential equations. Initially, we present the model and its interpretation. Then we define the constants and elements that compose the model, so we present the results obtained using the qualitative theory of ordinary differential equations, especially the theory of planar systems related to the dynamics of fixed points. Finally, we show that the system representing the SIR model is globally stable and they have two types of dynamic that {depend on model constants}, and their meaning for epidemiology.
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References
Arrowsmith, D. K., & Place, C. M. (1982). Ordinary differential equations: A qualitative approach with applications. Chapman and Hall.
Báez-Sánchez, A. D., & Bobko, N. (2020). On equilibria stability in an epidemiological sir model with recovery-dependent infection rate. TEMA, 21, 409–424. DOI: https://doi.org/10.5540/tema.2020.021.03.409
Chowell, G., Miller, M., & Viboud, C. (2008). Seasonal Influenza in the United States, France, and Australia: Transmission and Prospects for Control. Epidemiology & Infection, 136(6), 852–864. DOI: https://doi.org/10.1017/S0950268807009144
Daley, D. J., & Gani, J. (2001). Epidemic Modelling: An Introduction. Cambridge University Press.
Doering, C. I., & Lopes, A. O. (2016). Equações Diferenciais Ordinárias. Editora do IMPA.
Frost, S., Walsh, A., & Thompson, J. (2018). Epirecipes: A Cookbook of Epidemiological Models. http: //epirecip.es/epicookbook/chapters/simple
Keeling, M. J., & Rohani, P. (2008). Modeling Infectious Diseases in Humans and Animals. Princeton University Press. DOI: https://doi.org/10.1515/9781400841035
Kermack, W. O., & McKendrick, A. G. (1927). Contributions to the Mathematical Theory of Epidemics, Part I. Proceedings of the Royal Society of London, 115, 700–721. DOI: https://doi.org/10.1098/rspa.1927.0118
López-Flores, M. M., Marchesin, D., Matos, V., & Schecter, S. (2021). Equações Diferenciais e Modelos Epidemiológicos. Editora do IMPA.
Nesteruk, I. (2021). The real COVID-19 pandemic dynamics in Qatar in 2021: simulations, predictions and verifications of the SIR model. Semina: Ciências Exatas e Tecnológicas, 42(1Supl), 55– 62. DOI: https://doi.org/10.5433/1679-0375.2021v42n1Suplp55
Pachi, C. G. F. (2006). Modelo Matemático para o Estudo da Propagação de Informações por Campanhas Educativas e Rumores [Dissertation, Universidade de São Paulo]. Universidade de São Paulo.
Perko, L. (2013). Differential Equations and Dynamical Systems (Vol. 7). Springer Science & Business Media.
Ross, R. (1911). The Prevention of Malaria. John Murray.
Shil, P. (2016). Mathematical Modeling of Viral Epidemics: A Review. Biomedical Research Journal, 3(2), 195–215. DOI: https://doi.org/10.4103/2349-3666.240612
Sotomayor, J. (2011). Equações Diferenciais Ordinárias. Editora Livraria da Física.
World Health Organization. (2003). Consensus Document on the Epidemiology of Severe Acute Respiratory Syndrome (SARS). World Health Organization. https://apps.who.int/iris/handle/10665/70863
Zaparolli, D. (2020). O Desafio de Calcular o R. Pesquisa FAPESP, 293, 46–47. https://revistapesquisa. fapesp.br/o-desafio-de-calcular-o-r/.
Zill, D. G. (2003). Equações Diferenciais com Aplicações em Modelagem. Cengage Learning Editores.
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Copyright (c) 2023 Adeval Lino Ferreira, Kalel Bispo Gimenez Araujo
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