Journal of Applied Mathematics

Volume 2018, Article ID 8276317, 11 pages

https://doi.org/10.1155/2018/8276317

## Modelling In Vivo HIV Dynamics under Combined Antiretroviral Treatment

^{1}Department of Mathematics, Laikipia University, P.O. Box 1100-20300, Nyahururu, Kenya^{2}Department of Mathematics, Masinde Muliro University of Science and Technology, P.O. Box 190-50100, Kakamega, Kenya

Correspondence should be addressed to B. Mobisa; ek.ca.aipikial@asibomb

Received 19 September 2018; Revised 19 November 2018; Accepted 26 November 2018; Published 10 December 2018

Academic Editor: Qingdu Li

Copyright © 2018 B. Mobisa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper a within host mathematical model for Human Immunodeficiency Virus (HIV) transmission incorporating treatment is formulated. The model takes into account the efficacy of combined antiretroviral treatment on viral growth and T cell population in the human blood. The existence of an infection free and positive endemic equilibrium is established. The basic reproduction number is derived using the method of next generation matrix. We perform local and global stability analysis of the equilibria points and show that if , then the infection free equilibrium is globally asymptotically stable and theoretically the virus is cleared and the disease dies out and if , then the endemic equilibrium is globally asymptotically stable implying that the virus persists within the host. Numerical simulations are carried out to investigate the effect of treatment on the within host infection dynamics.

#### 1. Introduction

Human Immunodeficiency Virus (HIV) remains a major threat to human life for the last three and half decades. HIV infection in humans causes Acquired Immunodeficiency Virus (AIDS), a disease that has ravaged human population all over the world. Since its discovery in the early 1980s, there has been tremendous research work on how to contain or eradicate the disease. Mathematical modelling of viral infections has led to greater understanding of virus dynamics and helped in predicting and controlling the spread of viral diseases such as HIV, Hepatitis B Virus (HBV), Hepatitis C Virus (HCV), and Dengue Fever. One of the early models of HIV infection known as the basic model was used by Nowak and May [1] and by Perelson and Nelson (1999) and was successful in numerically reproducing the dynamics of the early stages of HIV and its target CD4+ cells following an infection event. Recent studies have focused on HIV viral and cellular infections incorporating dynamics such as intracellular delays, latent infection and viral mutation, and spatial heterogeneity [2–5]. For instance, [6] investigated the global stability of within host virus models with cell-to-cell viral transmission and obtained a complete analytic description of equilibria. A four-dimensional system of delayed differential equations, where the production and removal rates of the virus and cells are given by general nonlinear functions, was proposed by [4]. Their model investigated the dynamical behaviour of virus target and cell target incidences incorporating humoral immune response. They established three key equilibrium results, an infection free equilibrium, a chronic free equilibrium with inactive humoral immune response, and chronic infection equilibrium with active humoral immune response. With dynamics governed by two bifurcation parameters basic reproduction numbers and the humoral immunity numbers and using Lyapunov functionals and Lasalle’s invariance principle, the authors proved the global stability of the equilibria.

The inclusion of treatment, at within and between host levels in mathematical modelling, has gained considerable attention in recent years. For instance, at between host levels, epidemiological models with saturated treatment function have been proposed by [7, 8]. Research on within host models that incorporate treatment has been carried out over the years, with early models highlighting the effects of AZT on viral replication [9]. Among the key findings, viral decline is drug dependent. A study of combined drug therapy of HIV infection was conducted by [10]; the mathematical model developed was used to simulate chemotherapy treatment of HIV infection. The simulations were based on clinical data of treatment with combinations of antiviral drugs involving reverse transcriptase inhibitors (RTI) and protease inhibitors (PI) and focused on the timing of treatment. The findings revealed that the success of treatment is based on longer survival times equated to the CD4+ T cells. Global dynamics of delay distributed HIV infection models with differential drug efficacy in cocirculating target cells was investigated [3]. Recent work by [11] sought to mathematically analyze the potential of Prophylaxis treatment in preventing and slowing the spread of HIV/AIDS in the population. In this study early use of Prophylaxis drug was shown to slow the rate of HIV transmission.

Whereas extensive research on HIV transmission dynamics has been carried out, mathematical modelling of HIV with combined treatment still remains an area of active research among mathematicians and biologists.

In this paper, we propose a within host HIV infection model with a logistic incidence rate that explicitly incorporates the two levels of antiretroviral treatment, namely, the reverse transcriptase inhibitors (RTIs) which prevent the reverse transcription of viral RNA into DNA. In this way the RTIs serve to reduce the rate of infection of activated cells. The other category is protease inhibitors (PIs) which prevent HIV-1 protease from clearing the HIV polyprotein into functional units, thereby causing infected cells to produce immature virus particles that are not capable of infecting additional cells; hence PIs decrease the number of newly infectious virus (virions) that are produced [12]. A mathematical analysis of the effects of treatment on the within host infection dynamics is carried out.

#### 2. Model Description and Formulation

A mathematical model of within host HIV infection dynamics is considered. The model is composed of three interacting variables, namely, uninfected CD4+ T cells , actively infected cells , and free virus particles . The uninfected CD4+ T cells are produced at rate and die naturally at the rate . The total number of T cells in the body remains bounded; thus the growth of T cells is governed by the logistic proliferation term which limits T cell growth as the cell population approaches the limit . The uninfected CD4+ T cells become infected by free virus and actively infected cells according to the simple mass infection terms and , respectively. This generates actively infected cells, , which die naturally at the rate . The infected cells produce free viruses at the rate and are cleared from circulation at rate per virus. This viral decline is a function of the efficiency of the combined treatment of Reverse Transcriptase Inhibitor (RTI) and Protease Inhibitor (PI), which are represented by the parameters and , respectively. From the description and definitions made, the infection dynamics are summarized by the following system of ODEs:

#### 3. Analysis of the Model

Since the model describes cell and virus populations dynamics, all the model variables are nonnegative for . In the absence of the virus, the T cell population has a steady state value ; hence the initial conditions for the model (1) are , , and . It can be shown that with positive initial data the solutions of model (1) will remain positive and bounded in the feasible region , .

##### 3.1. Basic Reproduction Number

The basic reproduction number is defined as the average number of secondary infections produced by one infectious virion and one infected cell over the course of their infectious period in uninfected CD4+ T cell population. We compute for model (1) using the next generation matrix method as used in [13, 14]. Model (1) has two infected compartments and . Let be the rate of appearance of new infections in compartment and as the transfer of individuals out of compartment for the two compartments, respectively, and are given in partitioned form as follows:andThe Jacobian of and evaluated at the Infection Free Equilibrium yieldswhere is nonnegative and is nonsingular. The basic reproduction number is thus given by , where is the spectral radius of the matrix . Hence

##### 3.2. Local Stability Analysis of the Infection Free Equilibrium

We investigate the local stability properties of the infection free equilibrium by approximating the nonlinear system of the differential equations (1) with the linear system at the infection free equilibrium .

Theorem 1. *The infection free equilibrium is locally asymptotically stable if and only if .*

*Proof. *Evaluating the Jacobian of model (1) at , we obtainand clearlyis one of the eigenvalues of the matrix in (7), which is negative because for a population that is growing in numbers; the rate of production (birth rate) is greater than the death rate, that is . The nature of the remaining roots of (7) can be determined from the reduced matrix:Using Routh-Hurwitz stability criteria, matrix in (9) will have negative real roots if and only if the and ; thusandand using (6), (11) reduces to From (10) and (12), and if and only if . Thus is locally asymptotically stable whenever and unstable otherwise.

This means that if a small number of free virus particles enter the blood stream, each virus will infect on average less than one uninfected cell in its entire period of infectivity whenever . Theoretically this shows that the virus is cleared from the body if .

##### 3.3. Global Stability Analysis of the Infection Free Equilibrium

In this section we study the global stability of the infection free equilibrium of model (1) using the theorem by Castillo-Chavez et al. [13]. We rewrite model (1) in the formwhere denotes the number of uninfected cells and denotes the number of actively infected cells and free virus particles, respectively. The infection free equilibrium (IFE) is now denoted byThe conditions and below must be met in order to guarantee global asymptotic stability:(i)() For is Globally Asymptotically Stable (GAS)(ii)() , for

where is an M-matrix (the off-diagonal elements of are nonnegative) and is the region where the model makes biological sense. If system (13) satisfies conditions and , then the following theorem holds.

Theorem 2. *The fixed point is Globally Asymptotically Stable equilibrium of (13) provided that and that assumptions () and () are satisfied.*

*Proof. *Let , , , and whereFrom (16) this implies that . Therefore, is globally asymptotically stable when .

This means that, at any perturbation of the equilibrium point by the introduction of free virus particles, the model solutions will always converge to the IFE, whenever .

##### 3.4. Existence of the Endemic Equilibrium (EE)

Theorem 3. *A positive endemic equilibrium exists provided .*

*Proof. *The endemic equilibrium satisfiesFrom (19) we haveSubstituting (20) in (18) we getSubstituting and in (17) we obtainThe endemic equilibrium (EE) is given as Clearly if and only if .

##### 3.5. Local Stability Analysis of the Endemic Equilibrium

Theorem 4. *The endemic equilibrium is locally asymptotically stable whenever .*

*Proof. *The Jacobian matrix of model (1) at is as follows:The characteristic equation of (24) is in the formwhere The number of possible negative real roots of (25) depends on the signs of , , and . This can be established by applying Descartes Rule of Signs as used in [11].According to this rule the number of negative real zeros of is either equal to the number of sign changes of or less by an even number, as shown in Table 1.

From Table 1 the maximum number of sign changes in is 2; hence the characteristic polynomial (27) has two negative roots. Thushas negative real roots. Hence for and if cases 1 to 8 are satisfied then the endemic equilibrium is locally asymptotically stable.