Dynamics of a malaria infection model with time delay

Qian Ding; Jian Liu; Zhiming Guo; Qian Ding; Jian Liu; Zhiming Guo

doi:10.3934/mbe.2019246

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 4885-4907. doi: 10.3934/mbe.2019246

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Dynamics of a malaria infection model with time delay

School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, 510006, P.R. China

Received: 30 January 2019 Accepted: 09 May 2019 Published: 29 May 2019

In this paper, a new mathematical model (a system of delay differential equation) is proposed to describe dynamical behaviors of malaria in an infected host with red blood cells (RBCs), infected red blood cells (iRBCs) and immune factors. The basic reproduction number $\Re_0$ of the malaria infection is derived. If $\Re_{0}\leq1$, the uninfected equilibrium $E_0$ is globally asymptotically stable. If $\Re_{0}>1$, there exists two kinds of infection equilibria. The conditions of these equilibria with respect to the existence, stability and uniform persistence are given. Furthermore, fluctuations occur when the model undergoes Hopf bifurcation, and periodic solution appears near the positive equilibrium. The direction and stability of Hopf bifurcation are also obtained by applying the center manifold method and the normal form theory. Numerical simulations are provided to demonstrate the theoretical results.
- malaria,
- delay,
- stability,
- persistence,
- Hopf bifurcation
Citation: Qian Ding, Jian Liu, Zhiming Guo. Dynamics of a malaria infection model with time delay[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 4885-4907. doi: 10.3934/mbe.2019246

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Abstract

In this paper, a new mathematical model (a system of delay differential equation) is proposed to describe dynamical behaviors of malaria in an infected host with red blood cells (RBCs), infected red blood cells (iRBCs) and immune factors. The basic reproduction number $\Re_0$ of the malaria infection is derived. If $\Re_{0}\leq1$, the uninfected equilibrium $E_0$ is globally asymptotically stable. If $\Re_{0}>1$, there exists two kinds of infection equilibria. The conditions of these equilibria with respect to the existence, stability and uniform persistence are given. Furthermore, fluctuations occur when the model undergoes Hopf bifurcation, and periodic solution appears near the positive equilibrium. The direction and stability of Hopf bifurcation are also obtained by applying the center manifold method and the normal form theory. Numerical simulations are provided to demonstrate the theoretical results.

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