*(English)*Zbl 1107.92047

Summary: We present an ordinary differential equation mathematical model for the spread of malaria in human and mosquito populations. Susceptible humans can be infected when they are bitten by an infectious mosquito. They then progress through the exposed, infectious, and recovered classes, before reentering the susceptible class. Susceptible mosquitoes can become infected when they bite infectious or recovered humans, and once infected they move through the exposed and infectious classes. Both species follow a logistic population model, with humans having immigration and disease-induced death.

We define a reproductive number, ${R}_{0}$, for the number of secondary cases that one infected individual will cause through the duration of the infectious period. We find that the disease-free equilibrium is locally asymptotically stable when ${R}_{0}<1$ and unstable when ${R}_{0}>1$. We prove the existence of at least one endemic equilibrium point for all ${R}_{0}>1$. In the absence of disease-induced death, we prove that the transcritical bifurcation at ${R}_{0}=1$ is supercritical (forward). Numerical simulations show that for larger values of the disease-induced death rate, a subcritical (backward) bifurcation is possible at ${R}_{0}=1$.

##### MSC:

92D30 | Epidemiology |

37N25 | Dynamical systems in biology |

34D05 | Asymptotic stability of ODE |

92C60 | Medical epidemiology |