Stability and Hopf Bifurcations of nonlinear delay malaria epidemic model

The objective of this paper is to systematically study the boundedness, persistence and stability of the nonlinear malaria epidemic model with latent periods. First, we consider the simplified model with the approximation f(t−η)≃f(t)−ηf′(t)f(t−η)≃f(t)−ηf′(t), when ηη is small enough so that the function ff does not vary too rapidly over the time interval [t−η,tt−η,t], and study the stability of the trivial and the positive equilibrium points. Second, when the latent periods are equal (and not small enough), we will investigate the stability of the positive equilibrium point and prove the existence of Hopf Bifurcations and discuss the stability independent of the delays. Third, in the case when the latent periods are different, we will employ the Lyapunov functional method to establish some sufficient conditions for the local asymptotic stability of the positive equilibrium point.