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Global Analysis and Optimal Control of a Periodic Visceral Leishmaniasis Model

Author

Listed:
  • Ibrahim M. ELmojtaba

    (Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, Muscat 123, Oman)

  • Santanu Biswas

    (Agricultural and Ecological Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700108, India)

  • Joydev Chattopadhyay

    (Agricultural and Ecological Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700108, India)

Abstract

In this paper, we propose and analyze a mathematical model for the dynamics of visceral leishmaniasis with seasonality. Our results show that the disease-free equilibrium is globally asymptotically stable under certain conditions when R 0 , the basic reproduction number, is less than unity. When R 0 > 1 and under some conditions, then our system has a unique positive ω -periodic solution that is globally asymptotically stable. Applying two controls, vaccination and treatment, to our model forces the system to be non-periodic, and all fractions of infected populations settle on a very low level.

Suggested Citation

  • Ibrahim M. ELmojtaba & Santanu Biswas & Joydev Chattopadhyay, 2017. "Global Analysis and Optimal Control of a Periodic Visceral Leishmaniasis Model," Mathematics, MDPI, vol. 5(4), pages 1-18, December.
    • Handle: RePEc:gam:jmathe:v:5:y:2017:i:4:p:80-:d:122924
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    References listed on IDEAS

    as
    1. Coutinho, F.A. B. & Burattini, M.N. & Lopez, L.F. & Massad, E., 2005. "An approximate threshold condition for non-autonomous system: An application to a vector-borne infection," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 70(3), pages 149-158.
    2. Luis Fernando Chaves & Mercedes Pascual, 2006. "Climate Cycles and Forecasts of Cutaneous Leishmaniasis, a Nonstationary Vector-Borne Disease," PLOS Medicine, Public Library of Science, vol. 3(8), pages 1-9, August.
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