mathematical analysis of visceral leishmaniasis model

In this work, we consider a mathematical model describing the dynamics of visceral leishmaniasis in a population of dogs D. First, we consider the case of constant total population D, this is the case where birth and death rates are equal, in this case transcritical bifurcation occurs when the basic reproduction number ℛ0 is equal to one, and global stability is shown by the mean of suitable Lyapunov functions. After that, we consider the case where the birth and death rates are different, if the birth rate is great than death rate the total dog population increases exponentially, while the infectious dogs I dies out if the basic reproduction number is less than one, if it is great than one then D goes to infinity. We also prove that the total population D will extinct for birth rate less than death rate. Finally we give numerical simulations.