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Dynamics and optimal control of a non-linear epidemic model with relapse and cure

Author

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  • Lahrouz, A.
  • El Mahjour, H.
  • Settati, A.
  • Bernoussi, A.

Abstract

In this work, we introduce the basic reproduction number R0 for a general epidemic model with graded cure, relapse and nonlinear incidence rate in a non-constant population size. We established that the disease free-equilibrium state Ef is globally asymptotically exponentially stable if R0<1 and globally asymptotically stable if R0=1. If R0>1, we proved that the system model has at least one endemic state Ee. Then, by means of an appropriate Lyapunov function, we showed that Ee is unique and globally asymptotically stable under some acceptable biological conditions. On the other hand, we use two types of control to reduce the number of infectious individuals. The optimality system is formulated and solved numerically using a Gauss–Seidel-like implicit finite-difference method.

Suggested Citation

  • Lahrouz, A. & El Mahjour, H. & Settati, A. & Bernoussi, A., 2018. "Dynamics and optimal control of a non-linear epidemic model with relapse and cure," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 496(C), pages 299-317.
    • Handle: RePEc:eee:phsmap:v:496:y:2018:i:c:p:299-317
    DOI: 10.1016/j.physa.2018.01.007
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    References listed on IDEAS

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    1. Helen J Wearing & Pejman Rohani & Matt J Keeling, 2005. "Appropriate Models for the Management of Infectious Diseases," PLOS Medicine, Public Library of Science, vol. 2(7), pages 1-1, July.
    2. Jin, Yu & Wang, Wendi & Xiao, Shiwu, 2007. "An SIRS model with a nonlinear incidence rate," Chaos, Solitons & Fractals, Elsevier, vol. 34(5), pages 1482-1497.
    3. C. P. Farrington & M. N. Kanaan & N. J. Gay, 2001. "Estimation of the basic reproduction number for infectious diseases from age‐stratified serological survey data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 50(3), pages 251-292.
    4. Xu, Rui & Ma, Zhien, 2009. "Stability of a delayed SIRS epidemic model with a nonlinear incidence rate," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2319-2325.
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    Cited by:

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    2. Ghosh, M. & Olaniyi, S. & Obabiyi, O.S., 2020. "Mathematical analysis of reinfection and relapse in malaria dynamics," Applied Mathematics and Computation, Elsevier, vol. 373(C).
    3. Kar, T.K. & Nandi, Swapan Kumar & Jana, Soovoojeet & Mandal, Manotosh, 2019. "Stability and bifurcation analysis of an epidemic model with the effect of media," Chaos, Solitons & Fractals, Elsevier, vol. 120(C), pages 188-199.
    4. Jang, Junyoung & Kwon, Hee-Dae & Lee, Jeehyun, 2020. "Optimal control problem of an SIR reaction–diffusion model with inequality constraints," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 171(C), pages 136-151.
    5. El Fatini, Mohamed & El Khalifi, Mohamed & Gerlach, Richard & Laaribi, Aziz & Taki, Regragui, 2019. "Stationary distribution and threshold dynamics of a stochastic SIRS model with a general incidence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 534(C).

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