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A multi-city epidemic model

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  • Julien Arino
  • P. van den Driessche

Abstract

Some analytical results are given for a model that describes the propagation of a disease in a population of individuals who travel between n cities. The model is formulated as a system of 2n 2 ordinary differential equations, with terms accounting for disease transmission, recovery, birth, death, and travel between cities. The mobility component is represented as a directed graph with cities as vertices and arcs determined by outgoing (or return) travel. An explicit formula that can be used to compute the basic reproduction number, {\cal R}_0 , is obtained, and explicit bounds on {\cal R}_0 are determined in the case of homogeneous contacts between individuals within each city. Numerical simulations indicate that {\cal R}_0 is a sharp threshold, with the disease dying out if {\cal R}_0 1 .

Suggested Citation

  • Julien Arino & P. van den Driessche, 2003. "A multi-city epidemic model," Mathematical Population Studies, Taylor & Francis Journals, vol. 10(3), pages 175-193.
    • Handle: RePEc:taf:mpopst:v:10:y:2003:i:3:p:175-193
    DOI: 10.1080/08898480306720
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    Cited by:

    1. Sheryl Le Chang & Mahendra Piraveenan & Mikhail Prokopenko, 2019. "The Effects of Imitation Dynamics on Vaccination Behaviours in SIR-Network Model," IJERPH, MDPI, vol. 16(14), pages 1-31, July.
    2. Pablo D. Fajgelbaum & Amit Khandelwal & Wookun Kim & Cristiano Mantovani & Edouard Schaal, 2021. "Optimal Lockdown in a Commuting Network," American Economic Review: Insights, American Economic Association, vol. 3(4), pages 503-522, December.
    3. Mahajan, Shveta & Kumar, Deepak & Verma, Atul Kumar & Sharma, Natasha, 2023. "Dynamic analysis of modified SEIR epidemic model with time delay in geographical networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 629(C).
    4. Westerink-Duijzer, L.E. & van Jaarsveld, W.L. & Wallinga, J. & Dekker, R., 2015. "Dose-optimal vaccine allocation over multiple populations," Econometric Institute Research Papers EI2015-29, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    5. Tri Nguyen-Huu & Pierre Auger & Ali Moussaoui, 2023. "On Incidence-Dependent Management Strategies against an SEIRS Epidemic: Extinction of the Epidemic Using Allee Effect," Mathematics, MDPI, vol. 11(13), pages 1-25, June.
    6. Chang, Sheryl L. & Piraveenan, Mahendra & Prokopenko, Mikhail, 2020. "Impact of network assortativity on epidemic and vaccination behaviour," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    7. Yi Xie & Ziheng Zhang & Yan Wu & Shuang Li & Liuyong Pang & Yong Li, 2024. "Time-Delay Dynamic Model and Cost-Effectiveness Analysis of Major Emergent Infectious Diseases with Transportation-Related Infection and Entry-Exit Screening," Mathematics, MDPI, vol. 12(13), pages 1-25, July.
    8. Liu, Shasha & Yamamoto, Toshiyuki, 2022. "Role of stay-at-home requests and travel restrictions in preventing the spread of COVID-19 in Japan," Transportation Research Part A: Policy and Practice, Elsevier, vol. 159(C), pages 1-16.
    9. Wanduku, Divine, 2017. "Complete global analysis of a two-scale network SIRS epidemic dynamic model with distributed delay and random perturbations," Applied Mathematics and Computation, Elsevier, vol. 294(C), pages 49-76.
    10. Wang, Jianrong & Liu, Maoxing & Li, Youwen, 2013. "Analysis of epidemic models with demographics in metapopulation networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(7), pages 1621-1630.
    11. Jiang, Jiehui & Ma, Jie & Chen, Xiaojing, 2024. "Multi-regional collaborative mechanisms in emergency resource reserve and pre-dispatch design," International Journal of Production Economics, Elsevier, vol. 270(C).
    12. Constanza Fosco, 2012. "Spatial Difusion and Commuting Flows," Documentos de Trabajo en Economia y Ciencia Regional 30, Universidad Catolica del Norte, Chile, Department of Economics, revised Sep 2012.
    13. Jiang, Jiehui & Ma, Jie, 2023. "Dynamic analysis of pandemic cross-regional transmission considering quarantine strategies in the context of limited medical resources," Applied Mathematics and Computation, Elsevier, vol. 450(C).
    14. Liu, Junli & Zhou, Yicang, 2009. "Global stability of an SIRS epidemic model with transport-related infection," Chaos, Solitons & Fractals, Elsevier, vol. 40(1), pages 145-158.
    15. Elisa F. Long & Eike Nohdurft & Stefan Spinler, 2018. "Spatial Resource Allocation for Emerging Epidemics: A Comparison of Greedy, Myopic, and Dynamic Policies," Manufacturing & Service Operations Management, INFORMS, vol. 20(2), pages 181-198, May.
    16. Jana, Soovoojeet & Haldar, Palash & Kar, T.K., 2016. "Optimal control and stability analysis of an epidemic model with population dispersal," Chaos, Solitons & Fractals, Elsevier, vol. 83(C), pages 67-81.
    17. Sharma, Natasha & Gupta, Arvind Kumar, 2017. "Impact of time delay on the dynamics of SEIR epidemic model using cellular automata," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 471(C), pages 114-125.
    18. Saket Saurabh & Ayush Trivedi & Nithilaksh P. Lokesh & Bhagyashree Gaikwad, 2020. "Sustaining the economy under partial lockdown: A pandemic centric approach," Papers 2005.08273, arXiv.org.
    19. Yin, Qian & Wang, Zhishuang & Xia, Chengyi & Dehmer, Matthias & Emmert-Streib, Frank & Jin, Zhen, 2020. "A novel epidemic model considering demographics and intercity commuting on complex dynamical networks," Applied Mathematics and Computation, Elsevier, vol. 386(C).

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