Abstract and Applied Analysis

Mathematical Analysis of a Malaria Model with Partial Immunity to Reinfection

Li-Ming Cai, Abid Ali Lashari, Il Hyo Jung, Kazeem Oare Okosun, and Young Il Seo

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A deterministic model with variable human population for the transmission dynamics of malaria disease, which allows transmission by the recovered humans, is first developed and rigorously analyzed. The model reveals the presence of the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with one or more stable endemic equilibria when the associated reproduction number is less than unity. This phenomenon may arise due to the reinfection of host individuals who recovered from the disease. The model in an asymptotical constant population is also investigated. This results in a model with mass action incidence. A complete global analysis of the model with mass action incidence is given, which reveals that the global dynamics of malaria disease with reinfection is completely determined by the associated reproduction number. Moreover, it is shown that the phenomenon of backward bifurcation can be removed by replacing the standard incidence function with a mass action incidence. Graphical representations are provided to study the effect of reinfection rate and to qualitatively support the analytical results on the transmission dynamics of malaria.

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Abstr. Appl. Anal., Volume 2013 (2013), Article ID 405258, 17 pages.

First available in Project Euclid: 18 April 2013

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Cai, Li-Ming; Lashari, Abid Ali; Jung, Il Hyo; Okosun, Kazeem Oare; Seo, Young Il. Mathematical Analysis of a Malaria Model with Partial Immunity to Reinfection. Abstr. Appl. Anal. 2013 (2013), Article ID 405258, 17 pages. doi:10.1155/2013/405258. https://projecteuclid.org/euclid.aaa/1366306768

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  • WHO, “Malaria,” http://www.who.int/malaria/world-malaria- report2010/en/index.html.
  • NIAID, “Malaria,” http://www.niaid.nih.gov/topics/malaria/ Pages/default.aspx.
  • http://www.cdc.gov/Malaria/.
  • J. L. Aron, “Mathematical modeling of immunity to malaria,” Mathematical Biosciences, vol. 90, no. 1-2, pp. 385–396, 1988.
  • N. T. J. Bailey, The Biomathematics of Malaria, Charles Griffin, London, UK, 1982.
  • G. A. Ngwa, “Modelling the dynamics of endemic malaria in growing populations,” Discrete and Continuous Dynamical Systems B, vol. 4, no. 4, pp. 1173–1202, 2004.
  • P. Hviid, “Natural acquired immunity to Plasmodium falciparum malaria in Africa,” Acta Tropica, vol. 95, pp. 265–269, 2005.
  • G. Macdonald, The Epidemiology and Control of Malaria, Oxford University Press, London, UK, 1957.
  • R. Ross, The Prevention of Malaria, Murry, London, UK, 1911.
  • R. M. Anderson and R. M. May, Infectious Diseases of Humans, Oxford University Press, London, UK, 1991.
  • Z. Ma, Y. Zhou, W. Wang, and Z. Jin, Mathematical Models and Dynamics of Infectious Diseases, China Sciences Press, Beijing, China, 2004.
  • K. Dietz, L. Molineaux, and A. Thomas, “A malaria model tested in the African Savannah,” Bulletin of the World Health Organization, vol. 50, pp. 347–357, 1974.
  • A. M. Niger and A. B. Gumel, “Mathematical analysis of the role of repeated exposure on malaria transmission dynamics,” Differential Equations and Dynamical Systems, vol. 16, no. 3, pp. 251–287, 2008.
  • H. Wan and J.-A. Cui, “A model for the transmission of malaria,” Discrete and Continuous Dynamical Systems B, vol. 11, no. 2, pp. 479–496, 2009.
  • J. Li, “A malaria model with partial immunity in humans,” Mathematical Biosciences and Engineering, vol. 5, no. 4, pp. 789–801, 2008.
  • Y. Lou and X.-Q. Zhao, “A climate-based malaria transmission model with structured vector population,” SIAM Journal on Applied Mathematics, vol. 70, no. 6, pp. 2023–2044, 2010.
  • H. Yang, H. Wei, and X. Li, “Global stability of an epidemic model for vector-borne disease,” Journal of Systems Science & Complexity, vol. 23, no. 2, pp. 279–292, 2010.
  • B. Nannyonga, J. Y. T. Mugisha, and L. S. Luboobi, “Does co-infection with malaria boost persistence of trypanosomiasis?” Nonlinear Analysis: Real World Applications, vol. 13, no. 3, pp. 1379–1390, 2012.
  • Y. Wang, Z. Jin, Z. Yang, Z.-K. Zhang, T. Zhou, and G.-Q. Sun, “Global analysis of an SIS model with an infective vector on complex networks,” Nonlinear Analysis: Real World Applications, vol. 13, no. 2, pp. 543–557, 2012.
  • L.-M. Cai and X.-Z. Li, “Global analysis of a vector-host epidemic model with nonlinear incidences,” Applied Mathematics and Computation, vol. 217, no. 7, pp. 3531–3541, 2010.
  • P. van den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Mathematical Biosciences, vol. 180, pp. 29–48, 2002.
  • W. Wang, “Backward bifurcation of an epidemic model with treatment,” Mathematical Biosciences, vol. 201, no. 1-2, pp. 58–71, 2006.
  • F. Brauer, “Backward bifurcations in simple vaccination models,” Journal of Mathematical Analysis and Applications, vol. 298, no. 2, pp. 418–431, 2004.
  • H. Wan and H. Zhu, “The backward bifurcation in compartmental models for West Nile virus,” Mathematical Biosciences, vol. 227, no. 1, pp. 20–28, 2010.
  • C. Castillo-Chavez and B. Song, “Dynamical models of tuberculosis and their applications,” Mathematical Biosciences and Engineering, vol. 1, no. 2, pp. 361–404, 2004.
  • J. K. Hale, Ordinary Differential Equations, Krierer Basel, 2nd edition, 1980.
  • J. S. Muldowney, “Compound matrices and ordinary differential equations,” The Rocky Mountain Journal of Mathematics, vol. 20, no. 4, pp. 857–872, 1990.
  • M. Y. Li and J. S. Muldowney, “Global stability for the SEIR model in epidemiology,” Mathematical Biosciences, vol. 125, no. 2, pp. 155–164, 1995.
  • H. R. Thieme, “Persistence under relaxed point-dissipativity (with application to an endemic model),” SIAM Journal on Mathematical Analysis, vol. 24, no. 2, pp. 407–435, 1993.