Abstract
The role of incidence rate is very important in the study of epidemiological models. In this article, the analysis of an epidemic problem for the transmission dynamic of HBV with saturated incidence rate is presented, which is more generalize than bilinear incidence rate. After formulating the new mathematical model, the threshold quantity reproduction number R_0 is investigated by using the well known approach i.e. next generation matrix and investigate the possible equilibriums such as disease free and endemic equilibria. Then for the local and global behavior of the proposed problem, the local asymptotic stability analysis as well as global asymptotic stability analysis are proved. To prove the global asymptotic stability at disease free equilibrium, the classic Lyapunov function theory is considered. Similarly to show global asymptotic stability at endemic equilibria, the geometrical approach is used, which is the generalization of Lyapunov theory. Finally, numeric of the proposed problem are carried out to show the feasibility of the obtained results and the role of saturated incidence rate.
Highlights
Mathematical modeling is a power full tool to study the dynamic of different diseases in the real world phenomena (Zaman et al 2008, 2009; Mann and Roberts 2011; Zhao et al 2000)
From the evidence it is clear that Hepatitis B virus infection is responsible for about 80 % of the primary liver cancer
We investigate the possible equilibriums i.e. disease free and endemic equilibria and show the local asymptotic stability as well as global asymptotic stability at both equilibriums
Summary
Mathematical modeling is a power full tool to study the dynamic of different diseases in the real world phenomena (Zaman et al 2008, 2009; Mann and Roberts 2011; Zhao et al 2000). We prove the local stability of our model at illness free and endemic equilibrium, the global asymptotic stability. Local stability analysis For the local dynamic of the proposed model at disease free and endemic equilibrium points, we state and prove the following results. Theorem 2 If R0 > 1, the model (1) is locally asymptotically stable at endemic equilibrium point F1 and if R0 < 1, it is the unstable. Regarding the global stability of the proposed model at disease free and endemic equilibrium points, we have the following results. Theorem 3 If R0 < 1, the model (1) is globally asymptotically stable at disease free equilibrium point F0 and unstable otherwise. Proof To show the global stability at disease free equilibrium point F0, we use Lypanavo function theory, so consider the following Lypanavo function, such that. Proof Let J2 and J3 be the Jacobian matrix and second additive compound matrix containing only the first three equation of the model (1), we have
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
