A mathematical model for the dynamics of malaria in a human host and mosquito vector with temporary immunity.

*(English)*Zbl 1117.92039Summary: We propose a model that tracks the dynamics of malaria in the human host and mosquito vector. Our model incorporates some infected humans that recover from infection and immune humans after loss of immunity to the disease to join the susceptible class again. All the new borne are susceptible to the infection and there is no vertical transmission. The stability of the system is analyzed for the existence of the disease-free and endemic equilibria points.

We established that the disease-free equilibrium point is globally asymptotically stable when the reproduction number, \(R_{0}\leq 1\) and the disease always dies out. For \(R_{0}>1\) the disease-free equilibrium becomes unstable and the endemic equilibrium is globally asymptotically stable. Thus, due to new births and immunity loss to malaria, the susceptible class will always be refilled and the disease becomes more endemic.

We established that the disease-free equilibrium point is globally asymptotically stable when the reproduction number, \(R_{0}\leq 1\) and the disease always dies out. For \(R_{0}>1\) the disease-free equilibrium becomes unstable and the endemic equilibrium is globally asymptotically stable. Thus, due to new births and immunity loss to malaria, the susceptible class will always be refilled and the disease becomes more endemic.

##### MSC:

92C60 | Medical epidemiology |

92C50 | Medical applications (general) |

34D23 | Global stability of solutions to ordinary differential equations |

34D05 | Asymptotic properties of solutions to ordinary differential equations |

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\textit{J. Tumwiine} et al., Appl. Math. Comput. 189, No. 2, 1953--1965 (2007; Zbl 1117.92039)

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