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On a Discrete SEIR Epidemic Model with Exposed Infectivity, Feedback Vaccination and Partial Delayed Re-Susceptibility

Author

Listed:
  • Manuel De la Sen

    (Institute of Research and Development of Processes, IIDP, University of the Basque Country, 48080 Biscay, Spain)

  • Santiago Alonso-Quesada

    (Institute of Research and Development of Processes, IIDP, University of the Basque Country, 48080 Biscay, Spain)

  • Asier Ibeas

    (Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona, UAB, 08193 Barcelona, Spain)

Abstract

A new discrete Susceptible-Exposed-Infectious-Recovered (SEIR) epidemic model is proposed, and its properties of non-negativity and (both local and global) asymptotic stability of the solution sequence vector on the first orthant of the state-space are discussed. The calculation of the disease-free and the endemic equilibrium points is also performed. The model has the following main characteristics: (a) the exposed subpopulation is infective, as it is the infectious one, but their respective transmission rates may be distinct; (b) a feedback vaccination control law on the Susceptible is incorporated; and (c) the model is subject to delayed partial re-susceptibility in the sense that a partial immunity loss in the recovered individuals happens after a certain delay. In this way, a portion of formerly recovered individuals along a range of previous samples is incorporated again to the susceptible subpopulation. The rate of loss of partial immunity of the considered range of previous samples may be, in general, distinct for the various samples. It is found that the endemic equilibrium point is not reachable in the transmission rate range of values, which makes the disease-free one to be globally asymptotically stable. The critical transmission rate which confers to only one of the equilibrium points the property of being asymptotically stable (respectively below or beyond its value) is linked to the unity basic reproduction number and makes both equilibrium points to be coincident. In parallel, the endemic equilibrium point is reachable and globally asymptotically stable in the range for which the disease-free equilibrium point is unstable. It is also discussed the relevance of both the vaccination effort and the re-susceptibility level in the modification of the disease-free equilibrium point compared to its reached component values in their absence. The influences of the limit control gain and equilibrium re-susceptibility level in the reached endemic state are also explicitly made viewable for their interpretation from the endemic equilibrium components. Some simulation examples are tested and discussed by using disease parameterizations of COVID-19.

Suggested Citation

  • Manuel De la Sen & Santiago Alonso-Quesada & Asier Ibeas, 2021. "On a Discrete SEIR Epidemic Model with Exposed Infectivity, Feedback Vaccination and Partial Delayed Re-Susceptibility," Mathematics, MDPI, vol. 9(5), pages 1-32, March.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:5:p:520-:d:509125
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    References listed on IDEAS

    as
    1. Cooper, Ian & Mondal, Argha & Antonopoulos, Chris G., 2020. "A SIR model assumption for the spread of COVID-19 in different communities," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    2. Abbasi, Zohreh & Zamani, Iman & Mehra, Amir Hossein Amiri & Shafieirad, Mohsen & Ibeas, Asier, 2020. "Optimal Control Design of Impulsive SQEIAR Epidemic Models with Application to COVID-19," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    3. Wang, Xinhe & Lu, Junwei & Wang, Zhen & Li, Yuxia, 2020. "Dynamics of discrete epidemic models on heterogeneous networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 539(C).
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