Abstract
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, R0, called the basic reproduction number was calculated. The disease-free equilibrium point was locally asymptotically stable if R0 R0 > 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 R0 > 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 (σ), the vector population of mosquitoes (Nv), the probability of being infected for a human bitten by an infectious mosquito per unit of time (b) and the probability of being infected for a mosquito per unit of time (c) must be reduced by applying optimal control measures.
Highlights
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
With the Lyapunov function, we proved that the endemic equilibrium is globally asymptotically stable if R0 > 1
It is estimated that nearly forty percent of the population live in areas where malaria is endemic as reported in World Health Organization (WHO) fact sheet (2009)
Summary
It seems important to us, in this context, to make a scientific contribution in the quest for a good understanding of the transmission dynamics of the latter in order to optimize interventions for effective control and eradication of the disease. According to data from the National Health Information System 2017, malaria is the leading cause of morbidity and mortality in Burundi with an incidence rate of 815‰. It constitutes 45.4% of the reasons for general consultations recorded in health facilities in 2017 with a rate of 50.5% in children under 5 years old.
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