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Transmission dynamics and optimal control of measles epidemics

Author

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  • Pang, Liuyong
  • Ruan, Shigui
  • Liu, Sanhong
  • Zhao, Zhong
  • Zhang, Xinan

Abstract

Based on the mechanism and characteristics of measles transmission, we propose a susceptible-exposed-infectious-recovered (SEIR) measles epidemic model with vaccination and investigate the effect of vaccination in controlling the spread of measles. We obtain two critical threshold values, μc1 and μc2, of the vaccine coverage ratio. Measles will be extinct when the vaccination ratio μ>μc1, endemic when μc2<μ<μc1, and outbreak periodically when μ<μc2. In addition, we apply the optimal control theory to obtain an optimal vaccination strategy μ∗(t) and give some numerical simulations for those theoretical findings. Finally, we use our model to simulate the data of measles cases in the U.S. from 1951 to 1962 and design a control strategy.

Suggested Citation

  • Pang, Liuyong & Ruan, Shigui & Liu, Sanhong & Zhao, Zhong & Zhang, Xinan, 2015. "Transmission dynamics and optimal control of measles epidemics," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 131-147.
    • Handle: RePEc:eee:apmaco:v:256:y:2015:i:c:p:131-147
    DOI: 10.1016/j.amc.2014.12.096
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    1. Matthew J. Ferrari & Rebecca F. Grais & Nita Bharti & Andrew J. K. Conlan & Ottar N. Bjørnstad & Lara J. Wolfson & Philippe J. Guerin & Ali Djibo & Bryan T. Grenfell, 2008. "The dynamics of measles in sub-Saharan Africa," Nature, Nature, vol. 451(7179), pages 679-684, February.
    2. Xia, Cheng-yi & Wang, Zhen & Sanz, Joaquin & Meloni, Sandro & Moreno, Yamir, 2013. "Effects of delayed recovery and nonuniform transmission on the spreading of diseases in complex networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(7), pages 1577-1585.
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    2. Nudee, K. & Chinviriyasit, S. & Chinviriyasit, W., 2019. "The effect of backward bifurcation in controlling measles transmission by vaccination," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 400-412.
    3. Li, Can & Guo, Zun-Guang & Zhang, Zhi-Yu, 2017. "Transmission dynamics of a brucellosis model: Basic reproduction number and global analysis," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 161-172.
    4. Kimberly M. Thompson, 2016. "Evolution and Use of Dynamic Transmission Models for Measles and Rubella Risk and Policy Analysis," Risk Analysis, John Wiley & Sons, vol. 36(7), pages 1383-1403, July.
    5. Juliet Nakakawa & Joseph Y. T. Mugisha & Michael W. Shaw & William Tinzaara & Eldad Karamura, 2017. "Banana Xanthomonas Wilt Infection: The Role of Debudding and Roguing as Control Options within a Mixed Cultivar Plantation," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2017, pages 1-13, December.
    6. Xu, Rui & Wang, Zhili & Zhang, Fengqin, 2015. "Global stability and Hopf bifurcations of an SEIR epidemiological model with logistic growth and time delay," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 332-342.
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    8. Abdelaziz, Mahmoud A.M. & Ismail, Ahmad Izani & Abdullah, Farah A. & Mohd, Mohd Hafiz, 2020. "Codimension one and two bifurcations of a discrete-time fractional-order SEIR measles epidemic model with constant vaccination," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    9. Mingdong Lyu & Kuofu Liu & Randolph W. Hall, 2024. "Spatial Interaction Analysis of Infectious Disease Import and Export between Regions," IJERPH, MDPI, vol. 21(5), pages 1-19, May.
    10. Ali Khaleel Dhaiban & Baydaa Khalaf Jabbar, 2021. "An optimal control model of COVID-19 pandemic: a comparative study of five countries," OPSEARCH, Springer;Operational Research Society of India, vol. 58(4), pages 790-809, December.
    11. Farman, Muhammad & Xu, Changjin & Shehzad, Aamir & Akgul, Ali, 2024. "Modeling and dynamics of measles via fractional differential operator of singular and non-singular kernels," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 221(C), pages 461-488.
    12. Fu, Xinjie & Wang, JinRong, 2022. "Dynamic stability and optimal control of SISqIqRS epidemic network," Chaos, Solitons & Fractals, Elsevier, vol. 163(C).
    13. Jose Diamantino Hernández Guillén & Ángel Martín del Rey & Roberto Casado Vara, 2020. "On the Optimal Control of a Malware Propagation Model," Mathematics, MDPI, vol. 8(9), pages 1-16, September.
    14. Yujiang Liu & Shujing Gao & Di Chen & Bing Liu, 2024. "Modeling the Transmission Dynamics and Optimal Control Strategy for Huanglongbing," Mathematics, MDPI, vol. 12(17), pages 1-23, August.
    15. Qureshi, Sania & Jan, Rashid, 2021. "Modeling of measles epidemic with optimized fractional order under Caputo differential operator," Chaos, Solitons & Fractals, Elsevier, vol. 145(C).
    16. Berhe, Hailay Weldegiorgis & Makinde, Oluwole Daniel & Theuri, David Mwangi, 2019. "Co-dynamics of measles and dysentery diarrhea diseases with optimal control and cost-effectiveness analysis," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 903-921.

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