IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i2p326-d1322266.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/2/326/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/12/2/326/
    Download Restriction: no
    ---><---

    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.
    3. Wang, Zhishuang & Guo, Quantong & Sun, Shiwen & Xia, Chengyi, 2019. "The impact of awareness diffusion on SIR-like epidemics in multiplex networks," Applied Mathematics and Computation, Elsevier, vol. 349(C), pages 134-147.
    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.
    7. Gabriel Sepulveda & Abraham J. Arenas & Gilberto González-Parra, 2023. "Mathematical Modeling of COVID-19 Dynamics under Two Vaccination Doses and Delay Effects," Mathematics, MDPI, vol. 11(2), pages 1-30, January.
    8. Kuilin Wu & Kai Zhou, 2019. "Traveling Waves in a Nonlocal Dispersal SIR Model with Standard Incidence Rate and Nonlocal Delayed Transmission," Mathematics, MDPI, vol. 7(7), pages 1-22, July.
    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. Zhu, Linhe & Liu, Wenshan & Zhang, Zhengdi, 2021. "Interplay between epidemic and information spreading on multiplex networks," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 188(C), pages 268-279.
    2. Huang, He & Chen, Yahong & Ma, Yefeng, 2021. "Modeling the competitive diffusions of rumor and knowledge and the impacts on epidemic spreading," Applied Mathematics and Computation, Elsevier, vol. 388(C).
    3. Wang, Jianpeng & Wang, Kai & Zheng, Tingting & Zhou, Pan & Teng, Zhidong, 2024. "Qualitative analysis on a reaction–diffusion SIS epidemic model with nonlinear incidence and Dirichlet boundary," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).
    4. Wang, Haiying & Moore, Jack Murdoch & Wang, Jun & Small, Michael, 2021. "The distinct roles of initial transmission and retransmission in the persistence of knowledge in complex networks," Applied Mathematics and Computation, Elsevier, vol. 392(C).
    5. Naz, Sidra & Raja, Muhammad Asif Zahoor & Kausar, Aneela & Zameer, Aneela & Mehmood, Ammara & Shoaib, Muhammad, 2022. "Dynamics of nonlinear cantilever piezoelectric–mechanical system: An intelligent computational approach," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 196(C), pages 88-113.
    6. An, Xuming & Ding, Li & Hu, Ping, 2020. "Information propagation with individual attention-decay effect on activity-driven networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 556(C).
    7. Wu, Jiang & Zuo, Renxian & He, Chaocheng & Xiong, Hang & Zhao, Kang & Hu, Zhongyi, 2022. "The effect of information literacy heterogeneity on epidemic spreading in information and epidemic coupled multiplex networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 596(C).
    8. Wang, Mengyao & Pan, Qiuhui & He, Mingfeng, 2020. "The effect of individual attitude on cooperation in social dilemma," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 555(C).
    9. Reinhard Schlickeiser & Martin Kröger, 2024. "Mathematics of Epidemics: On the General Solution of SIRVD, SIRV, SIRD, and SIR Compartment Models," Mathematics, MDPI, vol. 12(7), pages 1-45, March.
    10. Chaoyu Zheng & Benhong Peng & Xin Sheng & Anxia Wan, 2021. "Haze risk: information diffusion based on cellular automata," Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards, Springer;International Society for the Prevention and Mitigation of Natural Hazards, vol. 107(3), pages 2605-2623, July.
    11. Cheng, Le & Li, Xianghua & Han, Zhen & Luo, Tengyun & Ma, Lianbo & Zhu, Peican, 2022. "Path-based multi-sources localization in multiplex networks," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
    12. Huang, He & Chen, Yahong & Yan, Zhijun, 2021. "Impacts of social distancing on the spread of infectious diseases with asymptomatic infection: A mathematical model," Applied Mathematics and Computation, Elsevier, vol. 398(C).
    13. Abboubakar, Hamadjam & Kombou, Lausaire Kemayou & Koko, Adamou Dang & Fouda, Henri Paul Ekobena & Kumar, Anoop, 2021. "Projections and fractional dynamics of the typhoid fever: A case study of Mbandjock in the Centre Region of Cameroon," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    14. Li, Xiaopeng & Sun, Shiwen & Xia, Chengyi, 2019. "Reputation-based adaptive adjustment of link weight among individuals promotes the cooperation in spatial social dilemmas," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 810-820.
    15. Xu, Yuan-Hao & Wang, Hao-Jie & Lu, Zhong-Wen & Hu, Mao-Bin, 2023. "Impact of awareness dissemination on epidemic reaction–diffusion in multiplex networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 621(C).
    16. Juang, Jonq & Liang, Yu-Hao, 2024. "Epidemic models in well-mixed multiplex networks with distributed time delay," Applied Mathematics and Computation, Elsevier, vol. 474(C).
    17. Huang, Jiechen & Wang, Juan & Xia, Chengyi, 2020. "Role of vaccine efficacy in the vaccination behavior under myopic update rule on complex networks," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    18. Mendonça, J.P. & Brum, Arthur A. & Lyra, M.L. & Lira, Sérgio A., 2024. "Evolutionary game dynamics and the phase portrait diversity in a pandemic scenario," Applied Mathematics and Computation, Elsevier, vol. 475(C).
    19. Yuan, Yiran & Li, Ning, 2022. "Optimal control and cost-effectiveness analysis for a COVID-19 model with individual protection awareness," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 603(C).
    20. Defterli, Ozlem, 2021. "Comparative analysis of fractional order dengue model with temperature effect via singular and non-singular operators," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).

    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:2:p:326-:d:1322266. 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.