Global Stability of the Viral Dynamics With Crowley-Martin Functional Response

Abstract

Abstract. It is well known that the mathematical models provide very important information for the research of human immunodeciency virus type. However, the infection rate of almost all mathematical models is linear. The linearity shows the simple interaction between the T-cells and the viral particles. In this paper, a differential equation model of HIV infection of CD4 + T-cells with Crowley-Martin function response is studied. We prove that if the basic reproduction number R 0 < 1, the HIV infection is cleared from the T-cell population and the disease dies out; if R 0 > 1, the HIV infection persists in the host. We find that the chronic disease steady state is globally asymptotically stable if R 0 > 1. Numerical simulations are presented to illustrate the results.

Global dynamics for a class of age-infection HIV models with nonlinear infection rate

“…Then, from Theorems 2.1 and 2.2, we obtain the following corollary: Furthermore, we set the maximum age for the viral production as a † = 10 and δ(a) = 0.4 1 + sin (a − 5) π 10 , p(a) = 300 1 + sin (a − 5) π 10 , 0 ≤ a ≤ 10, so that each of the averages is equal to 0.4 and 300, respectively, which were used in [40]. Let β = 0.4 as in [40] and observe the dynamical behavior of solutions when α varies.…”

“…It is well known that nonlinear incidence rates are frequently used to describe the viral infection process based on experiments data and reasonable assumptions [23] and it is important to account for a number of nonlinear features of the biological phenomena involved, which is influenced by the availability of susceptible cells and by the force of infection of viral cells. For example, Holling type II functional response [12], saturation infection rate [20,38], Beddington-DeAngelis functional response [10], Crowley-Martin functional response [39,40] and general nonlinear incidence c(x)f (v) [6], where c(x) denotes the contact rate function at concentration of the target cells x and f (v) denotes the force of infection by virus at concentration v. Motivated by the works mentioned above (see, [6,10,13,[20][21][22]24,38]), in this paper, we develop the model (1.2) with nonlinear incidence rate and investigate the global stability of its equilibrium.…”

Dynamics of a Class of HIV Infection Models with Cure of Infected Cells in Eclipse Stage

“…Furthermore, when a 3 ¼ 0, the Hattaf's response is simplified to Beddington-DeAnglis functional response which was introduced by Beddington (1975) and DeAngelis et al (1975) and was used in Huang et al (2009Huang et al ( , 2011. Also, when a 3 ¼ a 1 a 2 , the Hattaf's response is simplified to the Crowley-Martin functional response which was introduced by Crowley and Martin (1989) and was used by Zhou and Cui (2011). Recently, the more generalized incidence function f(T, V) is used in Adnani et al (2013), Lotfi et al (2014), Hattaf and Yousfi (2014).…”

Dynamics of a Fractional Order HIV Infection Model with Specific Functional Response and Cure Rate

“…Further, we obtain the model of Liu et al [11] when 3 = 0. It is very important to note that when = 1, system (1) becomes a model with an ordinary derivative which is the generalization of the ODE models presented in [12][13][14][15].…”