IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i5p710-d1347835.html
   My bibliography  Save this article

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

Author

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/5/710/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/5/710/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Halet Ismail & Amar Debbouche & Soundararajan Hariharan & Lingeshwaran Shangerganesh & Stanislava V. Kashtanova, 2024. "Stability and Optimality Criteria for an SVIR Epidemic Model with Numerical Simulation," Mathematics, MDPI, vol. 12(20), pages 1-29, October.
    2. Kerr, Gilbert & González-Parra, Gilberto & Sherman, Michele, 2022. "A new method based on the Laplace transform and Fourier series for solving linear neutral delay differential equations," Applied Mathematics and Computation, Elsevier, vol. 420(C).
    3. Wang, Yi & Cao, Jinde & Sun, Gui-Quan & Li, Jing, 2014. "Effect of time delay on pattern dynamics in a spatial epidemic model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 412(C), pages 137-148.
    4. Andrei D. Polyanin & Vsevolod G. Sorokin, 2023. "Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays," Mathematics, MDPI, vol. 11(3), pages 1-25, January.
    5. Kerr, Gilbert & González-Parra, Gilberto, 2022. "Accuracy of the Laplace transform method for linear neutral delay differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 197(C), pages 308-326.
    6. Vsevolod G. Sorokin & Andrei V. Vyazmin, 2022. "Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration," Mathematics, MDPI, vol. 10(11), pages 1-39, May.
    7. Andrei D. Polyanin & Vsevolod G. Sorokin, 2023. "Exact Solutions of Reaction–Diffusion PDEs with Anisotropic Time Delay," Mathematics, MDPI, vol. 11(14), pages 1-19, July.
    8. Zheng, Qianqian & Shen, Jianwei & Pandey, Vikas & Guan, Linan & Guo, Yantao, 2023. "Turing instability in a network-organized epidemic model with delay," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    9. Zhang, Zizhen & Kundu, Soumen & Tripathi, Jai Prakash & Bugalia, Sarita, 2020. "Stability and Hopf bifurcation analysis of an SVEIR epidemic model with vaccination and multiple time delays," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    10. Liu, Pan-Ping, 2015. "Periodic solutions in an epidemic model with diffusion and delay," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 275-291.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:12:y:2024:i:5:p:710-:d:1347835. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.