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Abstract
In this paper, a virus dynamics model with Crowley-Martin functional response of the infec- tion rate is investigated. By analyzing the corresponding characteristic equations, the local stability of an infection-free equilibrium point and infection equilibrium point are discussed. By constructing suitable Lyapunov functions and using LaSalles invariance principle, the global stability also are established, it is proved that if the basic reproductive number, R0, is less than or equal to one, the infection-free equilib- rium point is globally asymptotically stable, if R0, is more than one, the infection equilibrium point is globally asymptotically stable.
Highlights
Mathematical models have been used to model the dynamic of viral infections, such as human immunodeficiency virus type I (HIV-I), hepatitis C virus (HCV), hepatitis B virus (HBV), and human T-cell lymphotropic virus I (HTLV-I) (Perelson et al, 1993, 1996; Bonhoeffer et al, 1997; Perelson and Nelson, 1999; Nowak and May, 2000; Nowak et al, 1996; Korobeinikov, 2004 )
= kI − μV, t > 0, where susceptible cells are produced at rate λ, die at rate dT and become infected at rate βT V ; infected cells are produced at rate βT V and die at rate pI; free viruses are produced from infected cells at rate kI
We will study the global dynamics of (1.2) by constructing a suitable Lyapunov function and using LaSalle’s invariance principle rather than by using the theory of competitive systems, as has been done in Zhou and Cui (2010). This will enable us to obtain the global asymptotic stability of the infection equilibrium point under weaker hypotheses than those used in Zhou and Cui (2010) and by a simpler method
Summary
Mathematical models have been used to model the dynamic of viral infections, such as human immunodeficiency virus type I (HIV-I), hepatitis C virus (HCV), hepatitis B virus (HBV), and human T-cell lymphotropic virus I (HTLV-I) (Perelson et al, 1993, 1996; Bonhoeffer et al, 1997; Perelson and Nelson, 1999; Nowak and May, 2000; Nowak et al, 1996; Korobeinikov, 2004 ). We will study the global dynamics of (1.2) by constructing a suitable Lyapunov function and using LaSalle’s invariance principle rather than by using the theory of competitive systems, as has been done in Zhou and Cui (2010) This will enable us to obtain the global asymptotic stability of the infection equilibrium point under weaker hypotheses than those used in Zhou and Cui (2010) and by a simpler method. Notice that R0 > R∗, Theorem 2.2.(iii) becomes: (iii)′ If R0 > 1 holds, E∗ is locally asymptotically stable. To use the center manifold theory, as described in Castillo-Chavez and Song (2004) (Theorem 4.1) To apply this method, the following simplification and change of variables are made first. The infection equilibrium point E∗ is locally asymptotically stable for R0 near 1. Note that the result in Theorem 2.3 holds for R0 > 1 but close to 1
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