Effect of Control Measure on the Transmission of Leishmaniasis
In this paper we develop a mathematical model to describe the transmission of the disease Leishmaniasis between three populations, a human host population, a reservoir host population (dogs) and vector population (sand flies). The dynamical equations of the model are analyzed using the standard method. The conditions for the stability of the model are determined. It was found that there are two equilibrium points, disease free equilibrium and endemic equilibrium. The basic reproductive number that represents the epidemic indicator is obtained from the spectral radius of the next generation matrix. It is seen that if the basic reproductive number is less than one, the disease free equilibrium is local asymptotically stable, meaning that the disease will died out but if the basic reproductive number is greater than one, the endemic equilibrium will be local asymptotically stable, meaning that the disease will persist in the community. The numerical simulations are presented to illustrate the results. In addition, we show that vector control by the use of insecticide is the best method for controlling the disease.