Burundi, a country in East Africa with a temperate climate, has experienced in recent years a worrying growth of the Malaria epidemic. In this paper, a deterministic model of the transmission dynamics of malaria paras...Burundi, a country in East Africa with a temperate climate, has experienced in recent years a worrying growth of the Malaria epidemic. In this paper, a deterministic model of the transmission dynamics of malaria parasite in mosquito and human populations was formulated. The mathematical model was developed based on the SEIR model. An epidemiological threshold, <em>R</em><sub>0</sub>, called the basic reproduction number was calculated. The disease-free equilibrium point was locally asymptotically stable if <em>R</em><sub>0</sub> < 1 and unstable if <em>R</em><sub>0</sub> > 1. Using a Lyapunov function, we proved that this disease-free equilibrium point was globally asymptotically stable whenever the basic reproduction number is less than unity. The existence and uniqueness of endemic equilibrium were examined. With the Lyapunov function, we proved also that the endemic equilibrium is globally asymptotically stable if <em>R</em><sub>0</sub> > 1. Finally, the system of equations was solved numerically according to Burundi’s data on malaria. The result from our model shows that, in order to reduce the spread of Malaria in Burundi, the number of mosquito bites on human per unit of time (<em>σ</em>), the vector population of mosquitoes (<em>N<sub>v</sub></em>), the probability of being infected for a human bitten by an infectious mosquito per unit of time (<em>b</em>) and the probability of being infected for a mosquito per unit of time (<em>c</em>) must be reduced by applying optimal control measures.展开更多
文摘Burundi, a country in East Africa with a temperate climate, has experienced in recent years a worrying growth of the Malaria epidemic. In this paper, a deterministic model of the transmission dynamics of malaria parasite in mosquito and human populations was formulated. The mathematical model was developed based on the SEIR model. An epidemiological threshold, <em>R</em><sub>0</sub>, called the basic reproduction number was calculated. The disease-free equilibrium point was locally asymptotically stable if <em>R</em><sub>0</sub> < 1 and unstable if <em>R</em><sub>0</sub> > 1. Using a Lyapunov function, we proved that this disease-free equilibrium point was globally asymptotically stable whenever the basic reproduction number is less than unity. The existence and uniqueness of endemic equilibrium were examined. With the Lyapunov function, we proved also that the endemic equilibrium is globally asymptotically stable if <em>R</em><sub>0</sub> > 1. Finally, the system of equations was solved numerically according to Burundi’s data on malaria. The result from our model shows that, in order to reduce the spread of Malaria in Burundi, the number of mosquito bites on human per unit of time (<em>σ</em>), the vector population of mosquitoes (<em>N<sub>v</sub></em>), the probability of being infected for a human bitten by an infectious mosquito per unit of time (<em>b</em>) and the probability of being infected for a mosquito per unit of time (<em>c</em>) must be reduced by applying optimal control measures.
