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Mathematical Biosciences and Engineering

2011, Volume 8, Issue 4: 999-1018. doi: 10.3934/mbe.2011.8.999
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The Within-Host dynamics of malaria infection with immune response

  • 1.
    Department of Mathematics, East China University of Science and Technology, Shanghai 200237
  • 2.
    Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250
  • 3.
    Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240
  • Received: 01 December 2010 Accepted: 29 June 2018 Published: 01 August 2011

  • MSC : Primary: 34C23, 34D23; Secondary: 92C60.

  • Malaria infection is one of the most serious global health problems of our time. In this article the blood-stage dynamics of malaria in an infected host are studied by incorporating red blood cells, malaria parasitemia and immune effectors into a mathematical model with nonlinear bounded Michaelis-Menten-Monod functions describing how immune cells interact with infected red blood cells and merozoites. By a theoretical analysis of this model, we show that there exists a threshold value $R_0$, namely the basic reproduction number, for the malaria infection. The malaria-free equilibrium is global asymptotically stable if $R_0<1$. If $R_0>1$, there exist two kinds of infection equilibria: malaria infection equilibrium (without specific immune response) and positive equilibrium (with specific immune response). Conditions on the existence and stability of both infection equilibria are given. Moreover, it has been showed that the model can undergo Hopf bifurcation at the positive equilibrium and exhibit periodic oscillations. Numerical simulations are also provided to demonstrate these theoretical results.

    Citation: Yilong Li, Shigui Ruan, Dongmei Xiao. The Within-Host dynamics of malaria infection with immune response[J]. Mathematical Biosciences and Engineering, 2011, 8(4): 999-1018. doi: 10.3934/mbe.2011.8.999

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  • Malaria infection is one of the most serious global health problems of our time. In this article the blood-stage dynamics of malaria in an infected host are studied by incorporating red blood cells, malaria parasitemia and immune effectors into a mathematical model with nonlinear bounded Michaelis-Menten-Monod functions describing how immune cells interact with infected red blood cells and merozoites. By a theoretical analysis of this model, we show that there exists a threshold value $R_0$, namely the basic reproduction number, for the malaria infection. The malaria-free equilibrium is global asymptotically stable if $R_0<1$. If $R_0>1$, there exist two kinds of infection equilibria: malaria infection equilibrium (without specific immune response) and positive equilibrium (with specific immune response). Conditions on the existence and stability of both infection equilibria are given. Moreover, it has been showed that the model can undergo Hopf bifurcation at the positive equilibrium and exhibit periodic oscillations. Numerical simulations are also provided to demonstrate these theoretical results.


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