Global properties of a delayed HIV infection model with CTL immune response
Introduction
It is well known that the initial infection with HIV generally occurs after transfer of the body fluid from an infected individual to an uninfected one, such as sexual intercourse, contaminated needles used for intravenous drug delivery and use of blood or blood products. And the immune response plays an important role in eliminating or controlling the disease after human body is infected by virus [1], [2], [3], [4], [5], [6], [7], [21], [28]. Recently, many mathematical models have been developed to study the dynamics of human immunodeficiency virus (HIV) infection (for example, [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [22] and so on). These models have been used to explain different phenomena. Moreover, some of these models are mainly modelled on the interaction of the HIV with the CD4+ T cells. For more references and some detailed mathematical analysis on such models, we refer to the survey papers by Kirschner [19] and Perelson and Nelson [20]. Other HIV models consider the interaction process of the HIV not only with the CD4+ T cells but also with the macrophages (see, e.g. [20], [30], [31], [32], [33]). The importance of considering these HIV models is due to the observation of Perleson el al. [34].
The basic properties of the two target cells HIV models are not well studied in the literature, Elaiw [35] studied that the HIV infection model describes two co-circulation populations of target cells, potentially representing CD4+ T cells and macrophages as follows:where the state variables and parameters are described in Table 1.
To account for the time between viral entry into a target cell and the production of new virus particles, typically lasts for around 1 day for the HIV-1 infection [23], Nelson and Perlson [26] assumed that there are two types of delays that occur between the administration of drug and the observed decline in viral load: a pharmacological delay that occurs between the ingestion of drug and its appearance within cells and an intracellular delay that is between initial infection of a cell by HIV and the release of new virions. Herz et al. [24] used a discrete delay to model the intracellular delay in a HIV model and showed that the incorporation of a delay would substantially shorten the estimate for the half-life of free virus. Mittler et al. [25] argued that a γ-distribution delay would be more realistic to model the intracellular delay phenomenon and introduced such a delay into the model of Perelson et al. [29].
In order to incorporate the intracellular phase of the virus life-cycle, we assume that virus production occurs after the virus entry by the constant delays . The recruitment of virus-producing cells at time t is given by the number of the uninfected CD4+ T cells and the macrophages that were newly infected at time and are still alive at time [38], [39]. If we assume constant death rates for infected CD4+ T cells and infected macrophages but not yet produce virus particles, the probability of surviving the time period from to t is . Therefore, based on the discussion above, we obtain the following DDE model with CTL response:where the state variables and parameters are described in Table 1. All the parameters of the model (1.2) are supposed to be positive.
Section snippets
Well-posedness and equilibria of system (1.2)
Let be the Banach space of continuous functions from to R6 equipped with the sup-norm, here . It is biologically reasonable to consider the following initial conditions for (1.2):By the results in Hale and Verdugn [27], we know that there is a unique solution , of (1.2) with initial conditions (2.1). The following theorem establishes the
Global attractivity of
The global attractivity of the infected-free equilibrium by using a Lyapunov approach can be established as follows: Theorem 3.1 If , then is globally asymptotically attractive. Proof We consider a Lyapunov functionalwhere for . Therefore,
Conclusions
The global dynamics of the six-dimensional that describes the interactions of the HIV infection model with two target cells, CD4+ T-cells and macrophages and takes into account the Cytotoxic T Lymphocytes (CTL) immune response were studied in this paper. We obtain the following results by constructing Lyapunov functionals:
- (1)
if , then all solutions converge to the viral-free equilibrium ;
- (2)
if , then the immune-free equilibrium is globally asymptotically stable;
- (3)
if , then all
Acknowledgements
This work is supported by the National Natural Science Foundation of China (11171284) and program for Innovative Research Team (in Science and Technology) in University of Henan Province (2010IRTSTHN006) and Innovation Scientists and Technicians Troop Construction Projects of Henan Province (104200510011).
References (40)
- et al.
Target cell limited and immune control models of HIV infection: a comparison
J. Theor. Biol.
(1998) - et al.
A model for the treatment strategy in the chemotherapy of AIDS
Bull. Math. Biol.
(1996) - et al.
Understanding drug resistance for monotherapy treatment of HIV infection
Bull. Math. Biol.
(1997) - et al.
A model of human immunodeficiency virus infection in T-helper cell clones
J. Theor. Biol.
(1990) - et al.
Models of interaction between HIV and other pathogens
J. Theor. Biol.
(1992) - et al.
Mathematical biology of HIV infection: antigenic variation and diversity threshold
Math. Biosci.
(1991) - et al.
A delay-differential equation model of HIV infection of CD4+ T-cells
Math. Biosci.
(2000) - et al.
Influence of delayed virus production on viral dynamics in HIV-1 infected patients
Math. Biosci.
(1998) - et al.
Mathematical analysis of delay differential equation models of HIV-1 infection
Math. Biosci.
(2002) - et al.
HIV-1 infection and low steady state viral loads
Bull. Math. Biol.
(2002)
Rosenberg, HIV dynamics: modeling, data analysis, and optimal treatment protocols
J. Comput. Appl. Math.
Global properties of a class of HIV models
Nonlinear Anal. RWA
Complex dynamical behavior in the interaction between HIV and the immune system
Virus dynamics and drug therapy
Proc. Nat. Acad. Sci. USA
Analysis of immune system retrovirus equations
A model for the immune system response to HIV: AZT treatment studies
Population dynamics of immune responses to persistent viruses
Science
Global properties of a model of immune effector responses to viral infections
Adv. Complex Syst.
A delayed HIV-1 infection model with Beddington–DeAngelis functional response
Nonlinear Dyn.
Global stability of a virus dynamics model with Beddington–DeAngelis incidence and CTL immune response
Nonlinear Dyn.
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