Global stability in a viral infection model with lytic and nonlytic immune responses

Abstract

This paper investigates the global stability of a viral infection model with lyric and nonlytic immune responses. If the basic reproductive ratio of the virus is less than or equal to one, by the LaSalle's invariance principle and center manifold theorem, the disease-free steady state is globally asymptotically stable. If the basic reproductive ratio of the virus is greater than one, then the virus persists in the host and the disease steady state is locally asymptotically stable. Furthermore, by the method of Lyapunov function, the global stability of the disease steady state is established. At the same time, if we neglect the efficacy of the lyric component, using a geometrical approach, we obtain a different type of conditions for the global stability of the disease steady state.

The importance of immune responses in a model of hepatitis B virus

“…Nowak and Bangham [11] presented models which incorporate the effect of the cell-mediated immunity into a basic model for HIV and HBV infection. Bartholdy et al [12] and Wodarz et al [13] constructed a mathematical model considering both lytic and nonlytic immune responses, and Wang et al [14] studied mathematically its global dynamics. Wang and Liu [15] proposed a model to study the effect of immune impairment via mathematical analysis and numerical simulations of the model.…”

“…Furthermore, many researches have shown that although protection against infection is a multifactorial phenomenon depending on both the innate and adaptive immune mechanisms, CTL cells play a vital role in the control of human HBV infection via both cytolytic and noncytolytic mechanisms [6,[12][13][14][17][18][19]. Cytolytic effector mechanisms kill infected cells, whereas noncytolytic effector mechanisms "cure" the infected cells [13].…”

Dynamics of a HIV-1 Infection model with cell-mediated immune response and intracellular delay

“…Indeed, it is believed that CTLs are the main host immune factor that determines virus load (e.g., [1]). Therefore, the dynamics of HIV-1 infection with CTLs response has recently drawn much attention of researchers in the related areas (see, e.g., [1,15,21,7] and the references therein), and is also the main concern of this research. Letting ( ) be the concentration of CTLs, model (1) can be modified to…”

“…The function ( , , ) describes the rate of immune response activated by the infected cells. Arnaout et al [1] and Wang et al [21] assumed that the production of CTLs depends only on the concentration of infected cells and chose the linear dependent former ( , , ) = ( ). In [15], Nowak and Bangham assumed that the production of CTLs is also dependent upon the concentration of CTL cells, and used ( , , ) = ( ) ( ), leading to the following concrete model:…”

Stability of pathogen dynamics models with viral and cellular infections and immune impairment

“…[18,24,25,27,32,33,37,38,48]). The function φ(y, x) has been chosen in different forms such as: (i) constant, φ(y, x) = c 1 [31]; (ii) linear, φ(y, x) = ρy [1,23,44]; (iii) nonlinear φ(y, x) = c 2 yx (see e.g [27,30]). In these works, most pathogen infection models assume that the presence of the antigen can stimulate immunity and neglect the CTL immune impairment.…”