In this paper we study a model of HCV with mitotic proliferation, a saturation infection rate and a discrete intracellular delay: the delay corresponds to the time between infection of a infected target hepatocytes and production of new HCV particles. We establish the global stability of the infection-free equilibrium and existence, uniqueness, local and global stabilities of the infected equilibrium, also we establish the occurrence of a Hopf bifurcation. We will determine conditions for the permanence of model, and the length of delay to preserve stability. The unique infected equilibrium is globally-asymptotically stable for a special case, where the hepatotropic virus is non-cytopathic. We present a sensitivity analysis for the basic reproductive number. Numerical simulations are carried out to illustrate the analytical results.

There is a wide range of works that have proposed mathematical models to describe the spread of infectious diseases within human populations. Based on such models, researchers can evaluate the effect of applying different strategies for the treatment of diseases. In this article, we generalize previous models by studying an SIR epidemic model with a nonlinear incidence rate, saturated Holling type II treatment rate, and logistic growth. We compute the basic reproduction number and determine conditions for the local stability of equilibria and the existence of backward bifurcation and Hopf bifurcation. We also show that, when the disease transmission rate and treatment parameter are varied, our model undergoes a Bogdanov-Takens bifurcation of codimension 2 or 3. Simulations of the solutions and numerical continuation of equilibria are carried out to generate 2D and 3D bifurcation diagrams, as well as several related phase portraits that illustrate our results. Our work shows that incorporating these factors into epidemic models can lead to very complex dynamics.

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This paper discussed a hepatitis B virus infection with delay, spatial diffusion, and standard incidence function. The local stability of equilibrium is obtained via characteristic equations. By using comparison arguments, it is proved that if the basic reproduction number is less than unity, the infection-free equilibrium is globally asymptotically stable. If the basic reproductive number is greater than unity, by means of an iteration technique, sufficiently conditions are obtained for the global asymptotic stability of the infected steady state. Numerical simulations are carried out to illustrate our findings.

We consider a family of periodic SEIRS epidemic models with a fairly general incidence rate of the form ( ), and it is shown that the basic reproduction number determines the global dynamics of the models and it is a threshold parameter for persistence of disease. Numerical simulations are performed using a nonlinear incidence rate to estimate the basic reproduction number and illustrate our analytical findings.