Abstract
In this work, we extend the mathematical model of leptospirosis disease by taking into account the exposed individuals, the related death rate and the transmission coefficients between susceptible human and infected vector. Initially, we present the local asymptotical stability of both the disease-free and endemic equilibrium. We use the Lyapunov function theory with some sufficient conditions. This shows the global stability of both the disease-free and endemic equilibrium. Further, we present the bifurcation of the model and exhibit that the local asymptotical stability of the disease-free and endemic equilibrium co-exists with the threshold quantity. Finally, we discuss the numerical results.