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Deterministic Seirs Epidemic Model for Modeling Vital Dynamics, Vaccinations, and Temporary Immunity

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  • Marek B. Trawicki

    (Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Drive, Madison, WI 53706, USA)

Abstract

In this paper, the author proposes a new SEIRS model that generalizes several classical deterministic epidemic models (e.g., SIR and SIS and SEIR and SEIRS) involving the relationships between the susceptible S , exposed E , infected I , and recovered R individuals for understanding the proliferation of infectious diseases. As a way to incorporate the most important features of the previous models under the assumption of homogeneous mixing (mass-action principle) of the individuals in the population N , the SEIRS model utilizes vital dynamics with unequal birth and death rates, vaccinations for newborns and non-newborns, and temporary immunity. In order to determine the equilibrium points, namely the disease-free and endemic equilibrium points, and study their local stability behaviors, the SEIRS model is rescaled with the total time-varying population and analyzed according to its epidemic condition R 0 for two cases of no epidemic ( R 0 ≤ 1) and epidemic ( R 0 > 1) using the time-series and phase portraits of the susceptible s , exposed e , infected i , and recovered r individuals. Based on the experimental results using a set of arbitrarily-defined parameters for horizontal transmission of the infectious diseases, the proportional population of the SEIRS model consisted primarily of the recovered r (0.7–0.9) individuals and susceptible s (0.0–0.1) individuals (epidemic) and recovered r (0.9) individuals with only a small proportional population for the susceptible s (0.1) individuals (no epidemic). Overall, the initial conditions for the susceptible s , exposed e , infected i , and recovered r individuals reached the corresponding equilibrium point for local stability: no epidemic (DFE X ¯ D F E ) and epidemic (EE X ¯ E E ).

Suggested Citation

  • Marek B. Trawicki, 2017. "Deterministic Seirs Epidemic Model for Modeling Vital Dynamics, Vaccinations, and Temporary Immunity," Mathematics, MDPI, vol. 5(1), pages 1-19, January.
    • Handle: RePEc:gam:jmathe:v:5:y:2017:i:1:p:7-:d:87999
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    References listed on IDEAS

    as
    1. Li, Guihua & Jin, Zhen, 2005. "Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period," Chaos, Solitons & Fractals, Elsevier, vol. 25(5), pages 1177-1184.
    2. Zhao, Zhong & Chen, Lansun & Song, Xinyu, 2008. "Impulsive vaccination of SEIR epidemic model with time delay and nonlinear incidence rate," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(3), pages 500-510.
    3. Li, Guihua & Wang, Wendi & Jin, Zhen, 2006. "Global stability of an SEIR epidemic model with constant immigration," Chaos, Solitons & Fractals, Elsevier, vol. 30(4), pages 1012-1019.
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    Cited by:

    1. Bing Li & Ziye Xiang, 2023. "Evolutionary Game of Vaccination Considering Both Epidemic and Economic Factors by Infectious Network of Complex Nodes," Mathematics, MDPI, vol. 11(12), pages 1-26, June.
    2. Svetozar Margenov & Nedyu Popivanov & Iva Ugrinova & Tsvetan Hristov, 2022. "Mathematical Modeling and Short-Term Forecasting of the COVID-19 Epidemic in Bulgaria: SEIRS Model with Vaccination," Mathematics, MDPI, vol. 10(15), pages 1-28, July.
    3. Ihtisham Ul Haq & Numan Ullah & Nigar Ali & Kottakkaran Sooppy Nisar, 2022. "A New Mathematical Model of COVID-19 with Quarantine and Vaccination," Mathematics, MDPI, vol. 11(1), pages 1-21, December.

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