Global stability and persistence of HIV models with switching parameters and pulse control

Abstract

This paper studies some HIV (Human Immunodeficiency Virus) models with switching parameters and pulse control. By taking into account of the effects of reverse transcriptase inhibitor (RTI) drugs, protease inhibitor (PI) drugs and the variable transmission rate, we propose a new HIV model with switching parameters and derive a general threshold value that measures the persistence of the disease. Furthermore, pulse vaccination is applied to the HIV model and some novel threshold conditions are established to ensure the existence and stability of the periodic infection-free solution. In contrast to the standard HIV models, it is shown that our proposed models are more practical and useful. Moreover, pulse vaccination strategies are proven to be more effective in determining whether or not the disease is eradicated. Numerical simulations are carried out to illustrate the effectiveness of the obtained results, and future research directions are suggested.

Introduction

The HIV epidemic with its high rate of global spread has attracted the attention of numerous medical specialists as well as theoretical researchers. The virus attacks many types of cells, particularly the $\mathrm{CD}{4}^{+}$ T cells, which are essential in the human immune system  [30], [31], [37]. When human immune functions are damaged, a human body may incur incurable infections or tumors, some of which may be fatal  [44]. The virus has caused millions of deaths, and millions of people are currently living with HIV  [29]. In June 2001, governments and civil society representatives attending a special session of the General Assembly of the United Nations on AIDS (Acquired Immunodeficiency Syndrome) made a commitment to ensure that resources for the global response to HIV/AIDS are substantial, sustained, and geared towards achieving results  [30]. However, there are few effective vaccines to prevent HIV infection and few treatments to cure AIDS  [17], [27], [32].

In recent years, mathematical models have been used to predict the viral dynamics of HIV, which plays an important role in analyzing various drug therapy strategies  [22]. The main mathematical models for HIV include Ordinary Differential Equations (ODEs) and Delay Differential Equations (DDEs). For example, Alan and Patrick  [3] used ODEs to demonstrate many quantitative features of the interaction between HIV-1, the virus that causes AIDS, and the cells that are infected by the virus. Patrick and Alan  [26] provided some general results regarding the stability of non-linear delay differential equations to model an HIV-1 infection. In  [24], Narendra and Alan studied HIV dynamics with multiple infections of target cells. Similarly, Wodarz and Levy  [40] discussed the effect of different models of viral spread on the dynamics of HIV infected cells. Global asymptotic stability and persistence of the disease are key issues in the study of epidemic models. In fact, these issues may be considered by the basic reproduction number $R_0$, which is obtained as the spectral radius of the next generation operator  [14]. Some global stability and persistence results are obtained  [5], [6], [9], [15], [38], [39], [41], [42]. More recently, the control of the HIV virus has also been studied in  [1], [2], [10], [12], [25], [28], [33], [36], [43] and references therein. In these works, the basic reproduction number is utilized to examine whether or not infectious disease will be eradicated  [28], and to understand the local and global stabilities of the disease-free equilibrium and the endemic equilibrium  [5], the existence of periodic solutions  [1], and the persistence and extinction of the disease  [43]. However, to the best of our knowledge, HIV models with switching parameters and pulse control have not been studied in the existing literatures, and few results on the global stability and persistence of the disease have been derived.

In the present paper, we assume that certain time-varying parameters (such as the RTI effect $\gamma$, the PI effect $\eta$, and the transmission coefficient $\beta$) are switching parameters of the HIV model. This assumption is physically reasonable since the effects of RTI drugs and PI drugs can gradually decrease before the next dose is taken; thus, these parameters can be regarded as time-varying. Traditionally, the transmission coefficient, which is defined as the average number of transmissions between uninfected cells and infective virus particles per unit time, is constant in time. However, the spread of an infectious disease is influenced by a number of factors, such as the behavior of the human population and the environment in which it spread. Note that a body’s temperature variation can make host behavior change so that biological and environmental parameters naturally vary in time. When the drug effect fades away with time, the virus becomes more active, which will cause the transmission coefficient to decrease. Thus, a more realistic approach is to assume that the transmission coefficient is a time-varying parameter.

Motivated by the above discussion, we develop a new HIV model called a switched HIV model. A switched system is a distinct type of hybrid system which is composed of several subsystems and a switching law that designates the switching between subsystems  [16], [18], [23]. Switched systems have been applied in a variety of areas, such as engineering control systems, neural networks, ecosystems, and biological systems. Liu and Stechlinski  [19] assumed that the control rate was time-varying, and then modeled the spread of an infectious disease as a switched system. Liu and Stechlinski  [20] discussed an SIR model with time-varying parameters and switched nonlinear incidence rate. Hence, the HIV model with switching parameters is modeled in the present study.

Pulse vaccination is another important issue in the study of epidemic models, which is used to prevent and control infectious diseases such as smallpox and measles (see  [8], [7], [13], [21], [34], [35]). Mathematical modeling is often devoted to the design and assessment of vaccination strategies  [11]. Liu et al.  [21] established two SVIR epidemic models to describe a continuous vaccination strategy and a pulse vaccination strategy. Shulgin et al.  [34] noted that a measles pulse vaccination strategy could be distinguished from conventional strategies in leading to disease eradication at relatively low values of vaccination. The dynamics of general HIV models with impulse vaccination has been studied in  [8], [13], [34], [35]. However, pulse vaccination has not been incorporated into switched HIV models in the existing literature, which motivates the present study.

Motivated by the above discussion, this paper focuses on the global stability and persistence of HIV models with switching parameters and pulse vaccination. The rest of this paper is organized as follows: In Section  2, the HIV model with switching parameters is introduced and new threshold conditions are obtained to show that the disease persists. In Section  3, pulse treatment is applied to the switched HIV model, and global asymptotic stability of the periodic infection-free solution is obtained. Furthermore, sufficient conditions on the persistence of the disease are derived, which could imply that the treatment has failed under some switching laws. Numerical examples are given to verify the analytical results in Section  4. Some discussions and conclusions are presented in Section  5.