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It is well known that the mathematical biology and dynamical systems give very important information for the study and research of viral infection models such as HIV, HBV, HCV, Ebola and Influenza. This paper deals with the global dynamics of generalized virus model with logistic growth rate for target cells, general incidence rate and cellular immunity. The results will be obtained by using Lyapunov's second method and LaSalle's invariance principle. We prove the global stability of the rest points of the system by the value of basic reproduction number (R 0) and the immune response reproduction number (R CTL). We have found that if R 0 < 1 , then the infection-free equilibrium is globally asymptotically stable. For R 0 > 1 and R CTL < 1 , under certain conditions on incidence rate function, immune-free equilibrium is globally asymptotically stable. Finally, we prove that if R 0 > 1 and R CTL > 1 , then under certain conditions on incidence rate function the endemic equilibrium is globally asymptotically stable. Since the logistic growth rate for target cells and general incidence rate have been included in this manuscript, our obtained results are the generalization of those in the previous literatures. Moreover, the results have been obtained with weaker assumptions in comparison with the previous ones. Numerical simulations are presented to support and illustrate our analytical results. [ABSTRACT FROM AUTHOR]

In this paper, a fractional-order model for HIV with drug-resistance and CTL immune response is established. Two cases (subsystem with drug-sensitive and subsystem with drug-resistant) are considered. For both subsystems: firstly, the existence and uniqueness of the positive solution is proved; secondly, the sufficient conditions for the stability of the disease-free equilibrium are obtained; finally, some numerical simulations are performed to verify the theoretical results. After that, the main system is analyzed in a similar way. Calculation indicates that the basic reproduction number of the main system is the maximum value of the basic reproduction numbers of the corresponding subsystems. In addition, through numerical simulation, we know that the drug efficacy plays an important role in the treatment of HIV. • A fractional-order model for HIV with drug-resistance and CTL immune response is constructed. • The basic reproduction number of the main model is the maximum value of the corresponding two subsystems. • Through numerical simulations, the relationship of basic reproduction number with respect to fractional order was found. [ABSTRACT FROM AUTHOR]

In this paper, a cytokine-enhanced viral infection model with time delays and CTL immune response is proposed and analyzed. Multiple time delays, namely intracellular delay τ 1 , virus replication delay τ 2 and immune response delay τ 3 are included. Firstly, two key parameters with strong biological significance, namely the virus reproductive number R 0 and the CTL immune reproductive number R 1 are derived. And then the stability of equilibria of the model is investigated by constructing suitable Lyapunov functionals and using LaSalle's invariance principle. We prove that if R 0 < 1 , then the infection-free equilibrium E 0 is globally asymptotically stable for any τ 1 , τ 2 , τ 3 ≥ 0 ; if R 0 > 1 , R 1 < 1 , then the immunity-inactivated equilibrium E 1 is globally asymptotically stable for any τ 1 , τ 2 , τ 3 ≥ 0 ; and if R 1 > 1 , then the immunity-activated equilibrium E ∗ is globally asymptotically stable for any τ 1 , τ 2 ≥ 0 and τ 3 = 0. For τ 1 , τ 2 ≥ 0 , τ 3 > 0 , theoretical analysis and numerical simulations show that with the increase of τ 3 , the equilibrium E ∗ loses its stability and the system generates Hopf bifurcation, which means the system produces periodic oscillation under certain conditions. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model describing the dynamics of hepatitis C virus infection is proposed. Distributed delay, saturation incidence and CTL immune response in the liver and extrahepatic tissue are taken into account in the model. The model is rigorously analyzed, and a detailed global stability analysis is performed. A novel method is used to analyze the existence of the equilibria. By constructing suitable Lyapunov functionals and using LaSalle-type theorem for delay differential equations, sufficient condition for the global asymptotic stability of the equilibrium point is obtained. The results show that when R 0 < 1 , there is no virus and immune responses existed in the end; when R 1 < 1 < R 0 , the virus survives, but immune response gradually disappears; when R 0 > 1 and R 1 > 1 , the system will be globally stable with both virus and immune responses. Numerical analysis confirms the theorems and suggests that time delay play a positive role in virus infection and virus production process in order to eliminate infected liver and blood cells and virus. Furthermore, the simulation also explains the personalized difference in clinical test value. • Distributed delay, saturation incidence and two target cells in the liver and extrahepatic tissue are taken into account in HCV model. • A novel method is used to analyze the existence of the equilibria. • By constructing suitable Lyapunov functionals, a sufficient condition for the global asymptotic stability of the equilibrium point is obtained. [ABSTRACT FROM AUTHOR]

• A CTL-inclusive HIV-1 infection model with both cell-to-cell and virus-to-cell transmissions and three time delays is designed. • The complete asymptotic behavior of the model is established, using Lyapunov-LaSalle techniques. • An in-depth bifurcation analysis is performed, where the local stability and Hopf bifurcation of the endemic equilibrium in the general case with all delays being positive are established. • An increase of the immune delay can stabilize and/or destabilize the endemic equilibrium. In this paper, we assess the viral dynamics of a HIV-1 mathematical model that incorporates virus-to-cell and cell-to-cell transmissions, CTL response immune and three delays describing intracellular delays and immune response delay. The model includes a constant production rate of CTLs export from thymus. Our focus is on the effect of three delays on the infection dynamics. The basic reproduction number, R 0 , which depends on the intracellular delay, is shown to determine stability conditions of the model steady states. Employing a suitable Lyapunov function, we show that the infection-free equilibrium is globally asymptotically stable including the three time delays when the basic reproduction number is a smaller amount than one. Additionally, for the special case where there is no immune delay in the model, the global stability of the unique endemic equilibrium is also established (using suitable Lyapunov functions) whenever R 0 is larger than one. Furthermore, we study the local and hopf bifurcation of the endemic equilibrium in general case with all delays being positive. When the three delays are positive, we determine some conditions for stability switches of the endemic equilibrium by using the immune delay as a bifurcation parameter. Numerical simulations indicate that an increase of the immune delay can stabilize and/or destabilize the endemic equilibrium. [ABSTRACT FROM AUTHOR]