Stability analysis of epidemiological models incorporating heterogeneous infectivity
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Stability analysis of epidemiological models incorporating heterogeneous infectivity Anna Lígia Oenning Soares1
· Rodney Carlos Bassanezi2
Received: 2 December 2019 / Revised: 30 June 2020 / Accepted: 7 August 2020 / Published online: 14 August 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper we analyze general deterministic epidemiological models described by autonomous ordinary differential equations taking into account heterogeneity related to the infectivity and vital dynamics, in which the flow into the compartment of the susceptible individuals is given by a generic function. Our goal is to provide a new tool that facilitates the qualitative analysis of equilibrium points, which represent the disease free population, generalizing the result presented by Leite et al. (Math Med Biol J IMA 17:15–31, 2000) , and population extinction. The epidemiological models exposed are the type SEIRS (Susceptible-Exposed-Infectious-Recovered-Susceptible) and SEIR (Susceptible-ExposedInfectious-Recovered) with vaccination. Moreover, we computed the basic reproduction number from the models by van den Driessche and Watmough (Math Biosci 180:29–48, 2002) and correlate this threshold parameter with the stability of the equilibrium point representing the disease free population. Keywords Stability · Epidemiological models · Heterogeneous infectivity Mathematics Subject Classification 92B05
1 Introduction Many epidemiological models consider that the population is compartmentalized into groups and the infectious individuals have the same probability of meeting any susceptible individual. Furthermore, it is assumed that there is homogeneity in both susceptibility and infectivity, and there is no differentiation related to age, behaviour, spatial position and/or stage of disease. (Anderson 1991; Kermack and McKendrick 1991; Capasso 1993; Soares 2010).
Communicated by Marcos Eduardo Valle.
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Anna Lígia Oenning Soares [email protected]
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Departamento de Matemática, Universidade Federal de Mato Grosso, Av. Fernando Corrêa da Costa 2367, Cuiabá 78060-900, Brazil
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Departamento de Matemática Aplicada, Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda 651, Campinas 13083-859, Brazil
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A. L. O. Soares, R. C. Bassanezi
The assumptions above aim to simplify the understanding of an epidemic. Aron (1989) claims that simple models head to general conclusions of a qualitative analysis and more detailed models are effective for quantitative analysis. Moreover, according to Sattenspiel and Simon (1988) the heterogeneity in the population may interfere with the spread of diseases. Shaman et al. (2020) granted a model of the transmission dynamics of COVID-19 which takes into account the heterogeneity in infectivity, showing that the rapid spread of the virus can be elucidated by the contagion of infectious individuals who do not present severe symptoms. This work was of great relevance to combat COVID-19, also showing the importance of popu
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