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On the Analytical Solution of the SIRV-Model for the Temporal Evolution of Epidemics for General Time-Dependent Recovery, Infection and Vaccination Rates

Author

Listed:
  • Martin Kröger

    (Magnetism and Interface Physics & Computational Polymer Physics, Department of Materials, ETH Zurich, Leopold-Ruzicka-Weg 4, CH-8093 Zurich, Switzerland)

  • Reinhard Schlickeiser

    (Institut für Theoretische Physik, Lehrstuhl IV: Weltraum- und Astrophysik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
    Institut für Theoretische Physik und Astrophysik, Christian-Albrechts-Universität zu Kiel, Leibnizstr. 15, D-24118 Kiel, Germany)

Abstract

The susceptible–infected–recovered/removed–vaccinated (SIRV) epidemic model is an important generalization of the SIR epidemic model, as it accounts quantitatively for the effects of vaccination campaigns on the temporal evolution of epidemic outbreaks. Additional to the time-dependent infection ( a ( t ) ) and recovery ( μ ( t ) ) rates, regulating the transitions between the compartments S → I and I → R , respectively, the time-dependent vaccination rate v ( t ) accounts for the transition between the compartments S → V of susceptible to vaccinated fractions. An accurate analytical approximation is derived for arbitrary and different temporal dependencies of the rates, which is valid for all times after the start of the epidemics for which the cumulative fraction of new infections J ( t ) ≪ 1 . As vaccination campaigns automatically reduce the rate of new infections by transferring persons from susceptible to vaccinated, the limit J ( t ) ≪ 1 is even better fulfilled than in the SIR-epidemic model. The comparison of the analytical approximation for the temporal dependence of the rate of new infections J ˚ ( t ) = a ( t ) S ( t ) I ( t ) , the corresponding cumulative fraction J ( t ) , and V ( t ) , respectively, with the exact numerical solution of the SIRV-equations for different illustrative examples proves the accuracy of our approach. The considered illustrative examples include the cases of stationary ratios with a delayed start of vaccinations, and an oscillating ratio of recovery to infection rate with a delayed vaccination at constant rate. The proposed analytical approximation is self-regulating as the final analytical expression for the cumulative fraction J ∞ after infinite time allows us to check the validity of the original assumption J ( t ) ≤ J ∞ ≪ 1 .

Suggested Citation

  • Martin Kröger & Reinhard Schlickeiser, 2024. "On the Analytical Solution of the SIRV-Model for the Temporal Evolution of Epidemics for General Time-Dependent Recovery, Infection and Vaccination Rates," Mathematics, MDPI, vol. 12(2), pages 1-19, January.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:2:p:326-:d:1322266
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    References listed on IDEAS

    as
    1. Marco Gribaudo & Mauro Iacono & Daniele Manini, 2021. "COVID-19 Spatial Diffusion: A Markovian Agent-Based Model," Mathematics, MDPI, vol. 9(5), pages 1-12, February.
    2. Wang, Jinliang & Zhang, Ran & Kuniya, Toshikazu, 2021. "A reaction–diffusion Susceptible–Vaccinated–Infected–Recovered model in a spatially heterogeneous environment with Dirichlet boundary condition," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 848-865.
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    4. Abdulrahman B. Albidah, 2023. "A Proposed Analytical and Numerical Treatment for the Nonlinear SIR Model via a Hybrid Approach," Mathematics, MDPI, vol. 11(12), pages 1-15, June.
    5. Ameen, I. & Baleanu, Dumitru & Ali, Hegagi Mohamed, 2020. "An efficient algorithm for solving the fractional optimal control of SIRV epidemic model with a combination of vaccination and treatment," Chaos, Solitons & Fractals, Elsevier, vol. 137(C).
    6. Xiaoni Li & Xining Li & Qimin Zhang, 2020. "Time to extinction and stationary distribution of a stochastic susceptible-infected-recovered-susceptible model with vaccination under Markov switching," Mathematical Population Studies, Taylor & Francis Journals, vol. 27(4), pages 259-274, October.
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