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Stability and Optimality Criteria for an SVIR Epidemic Model with Numerical Simulation

Author

Listed:
  • Halet Ismail

    (Department of Applied Sciences, National Institute of Technology Goa, Cuncolim 403703, India)

  • Amar Debbouche

    (Department of Mathematics, Guelma University, Guelma 24000, Algeria)

  • Soundararajan Hariharan

    (Department of Applied Sciences, National Institute of Technology Goa, Cuncolim 403703, India
    Department of Mathematics, School of Engineering, Dayananda Sagar University, Bengaluru 562112, India)

  • Lingeshwaran Shangerganesh

    (Department of Applied Sciences, National Institute of Technology Goa, Cuncolim 403703, India)

  • Stanislava V. Kashtanova

    (World-Class Research Center for Advanced Digital Technologies, Peter the Great St. Petersburg Polytechnic University, 195251 St. Petersburg, Russia)

Abstract

The mathematical modeling of infectious diseases plays a vital role in understanding and predicting disease transmission, as underscored by recent global outbreaks; to delve deep into the dynamic of infectious disease considering latent period presciently is inevitable as it bridges the gap between realistic nature and mathematical modeling. This study extended the classical Susceptible–Infected–Recovered (SIR) model by incorporating vaccination strategies during incubation. We introduced multiple time delays to an account incubation period to capture realistic disease dynamics better. The model is formulated as a system of delay differential equations that describe the transmission dynamics of diseases such as polio or COVID-19, or diseases for which vaccination exists. Critical aspects of the study include proving the positivity of the model’s solutions, calculating the basic reproduction number ( R 0 ) using next-generation matrix theory, and identifying disease-free and endemic equilibrium points. The local stability of these equilibria is then analyzed using the Routh–Hurwitz criterion. Due to the complexity introduced by the delay components, we examine the stability by studying the roots of a fourth-degree exponential polynomial. The effects of educational campaigns and vaccination efficacy are also investigated as control measures. Furthermore, an optimization problem is formulated, based on Pontryagin’s maximum principle, to minimize the number of infections and associated intervention costs. Numerical simulations of the delay differential equations are conducted, and a modified Runge–Kutta method with delays is used to solve the optimal control problem. Finally, we present a few simulation results to illustrate the analytical findings.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:20:p:3231-:d:1499380
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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. Kuniya, Toshikazu & Muroya, Yoshiaki, 2015. "Global stability of a multi-group SIS epidemic model with varying total population size," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 785-798.
    3. Khan, Muhammad Altaf & Khan, Yasir & Islam, Saeed, 2018. "Complex dynamics of an SEIR epidemic model with saturated incidence rate and treatment," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 493(C), pages 210-227.
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