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Existence of Traveling Waves of a Diffusive Susceptible–Infected–Symptomatic–Recovered Epidemic Model with Temporal Delay

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Listed:
  • Julio C. Miranda

    (Departamento de Matematicas y Estadistica, Universidad de Cordoba, Monteria 230002, Colombia)

  • Abraham J. Arenas

    (Departamento de Matematicas y Estadistica, Universidad de Cordoba, Monteria 230002, Colombia)

  • Gilberto González-Parra

    (Department of Mathematics, New Mexico Tech, Socorro, NM 87801, USA
    Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera, s/n, 46022 Valencia, Spain)

  • Luis Miguel Villada

    (GIMNAP-Departamento de Matemáticas, Universidad del Bío-Bío, Casilla 5-C, Concepción 4051381, Chile
    CI 2 MA, Universidad de Concepción, Casilla 160-C, Concepción 4030000, Chile)

Abstract

The aim of this article is to investigate the existence of traveling waves of a diffusive model that represents the transmission of a virus in a determined population composed of the following populations: susceptible ( S ) , infected ( I ) , asymptomatic ( A ) , and recovered ( R ) . An analytical study is performed, where the existence of solutions of traveling waves in a bounded domain is demonstrated. We use the upper and lower coupled solutions method to achieve this aim. The existence and local asymptotic stability of the endemic ( E e ) and disease-free ( E 0 ) equilibrium states are also determined. The constructed model includes a discrete-time delay that is related to the incubation stage of a virus. We find the crucial basic reproduction number R 0 , which determines the local stability of the steady states. We perform numerical simulations of the model in order to provide additional support to the theoretical results and observe the traveling waves. The model can be used to study the dynamics of SARS-CoV-2 and other viruses where the disease evolution has a similar behavior.

Suggested Citation

  • Julio C. Miranda & Abraham J. Arenas & Gilberto González-Parra & Luis Miguel Villada, 2024. "Existence of Traveling Waves of a Diffusive Susceptible–Infected–Symptomatic–Recovered Epidemic Model with Temporal Delay," Mathematics, MDPI, vol. 12(5), pages 1-36, February.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:5:p:710-:d:1347835
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    References listed on IDEAS

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    1. Benito Chen-Charpentier, 2023. "Delays and Exposed Populations in Infection Models," Mathematics, MDPI, vol. 11(8), pages 1-22, April.
    2. Li, Jing & Sun, Gui-Quan & Jin, Zhen, 2014. "Pattern formation of an epidemic model with time delay," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 403(C), pages 100-109.
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