# Modelling In Vivo HIV Dynamics under Combined Antiretroviral Treatment

Authors: B. Mobisa; G. O. Lawi; J. K. Nthiiri
Journal: Journal of Applied Mathematics
 (2018)

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## 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 numberR0 is derived using the method of next generation matrix. We perform local and global stability analysis of the equilibria points and show that if R0<1, then the infection free equilibrium is globally asymptotically stable and theoretically the virus is cleared and the disease dies out and if R0>1, 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.


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## Body

## 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 R~0c and the humoral immunity numbers R~1G 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 activatedCD4+T 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 cellsT(t), actively infected cells T⋆(t), and free virus particles V(t). The uninfected CD4+ T cells are produced at rate r and die naturally at the rate μ. The total number of T cells in the body remains bounded; thus the growth of T cells is governed by the logistic proliferation term 1-T/Tmax which limits T cell growth as the cell population approaches the limit Tmax. The uninfected CD4+ T cells become infected by free virus and actively infected cells according to the simple mass infection terms βVT and αTT⋆, respectively. This generates actively infected cells, T⋆, which die naturally at the rate κ. The infected cells produce free viruses V at the rate ωκT⋆ and are cleared from circulation at rate c 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 ρ and ϑ, respectively. From the description and definitions made, the infection dynamics are summarized by the following system of ODEs:(1)dTdt=rT1-TTmax-1-ρβVT-1-ϑαTT⋆-μTdT⋆dt=1-ρβVT+1-ϑαTT⋆-κT⋆dVdt=ωκT⋆-cV


## 3. Analysis of the Model

Since the model describes cell and virus populations dynamics, all the model variables are nonnegative fort≥0. In the absence of the virus, the T cell population has a steady state value T0; hence the initial conditions for the model (1) are T(0)=T0>0, T⋆(0)=T0⋆≥0, and V(0)=V0>0. It can be shown that with positive initial data the solutions of model (1) will remain positive and bounded in the feasible region Ω=T,T⋆,V∈R+3:T+T⋆+V≤Tmax1-μ/r, ∀t.


### 3.1. Basic Reproduction Number

The basic reproduction numberR0 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 R0 for model (1) using the next generation matrix method as used in [13, 14]. Model (1) has two infected compartments T⋆ and V. Let fi be the rate of appearance of new infections in compartment i and νi as the transfer of individuals out of compartment i for the two compartments, respectively, and are given in partitioned form as follows:(2)f=1-ρβVT+α1-ϑTT⋆0and(3)ν=κT⋆-ωκT⋆+cVThe Jacobian of f and ν evaluated at the Infection Free Equilibrium E0=(Tmax,0,0) yields(4)F=α1-ϑT0β1-ρT000(5)V=κ0-ωκcwhere F is nonnegative and V is nonsingular. The basic reproduction number is thus given by R0=ρ(FV-1), where ρ(FV-1) is the spectral radius of the matrix FV-1. Hence(6)R0=Tmax1-μr1-ϑακ+Tmax1-μrωβc1-ρ


### 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 E0=(Tmax(1-μ/r),0,0).Theorem 1.
The infection free equilibriumE0 is locally asymptotically stable if and only if R0<1.Proof.
Evaluating the Jacobian of model (1) at E0, we obtain(7)JE0=μ-r-αTmax1-μr1-ϑ-βTmax1-μr1-ρ0αTmax1-μr1-ϑ-κβTmax1-μr1-ρ0ωκ-cand clearly(8)λ1=μ-ris 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 r>μ. The nature of the remaining roots of (7) can be determined from the reduced matrix:(9)A=αTmax1-μr1-ϑ-κβTmax1-μr1-ρωκ-cUsing Routh-Hurwitz stability criteria, matrix A in (9) will have negative real roots if and only if the tr⁡A<0 and det⁡A>0; thus(10)tr⁡A=R0-1-Tmax1-μr1-ρβωc-ckand(11)det⁡A=-ακTmax1-μr1-ϑ+βωcTmax1-μr1-ρ+1and using (6), (11) reduces to(12)det⁡A=1-R0 From (10) and (12), tr⁡A<0 and det⁡A>0 if and only if R0<1. Thus E0 is locally asymptotically stable whenever R0<1 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 wheneverR0<1. Theoretically this shows that the virus is cleared from the body if R0<1.


### 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 form(13)dXdt=HX,ZdZdt=GX,Z,GX,0=0where X∈R denotes the number of uninfected cells and Z∈R2 denotes the number of actively infected cells and free virus particles, respectively. The infection free equilibrium (IFE) is now denoted by(14)U0=X0,0,X0=Tmax1-μrThe conditions H1 and H2 below must be met in order to guarantee global asymptotic stability:(i)
(H1) For dX/dt=H(X,0),X0 is Globally Asymptotically Stable (GAS)(ii)
(H2) G(X,Z)=PZ-G^(X,Z),G^(X,Z)≥0, for (X,Z)∈ΩwhereP=DZG(X0,0) is an M-matrix (the off-diagonal elements of P are nonnegative) and Ω is the region where the model makes biological sense. If system (13) satisfies conditions H1 and H2, then the following theorem holds.Theorem 2.
The fixed pointU0=(X0,0) is Globally Asymptotically Stable equilibrium of (13) provided that R0<1 and that assumptions (H1) and (H2) are satisfied.Proof.
LetX(t)=T(t), Z=(T⋆(t),V(t)), H(X,0)=rT1-T/Tmax-μT0, and G(X,Z)=PZ-G^(X,Z) where(15)P=αTmax1-ϑ-κβTmax1-ρωκ-c(16)G^X,Z=G1^X,ZG2^X,Z=00From (16) G1^(X,Z)=G2^(X,Z)=0 this implies that G^(X,Z)≥0. Therefore, E0 is globally asymptotically stable when R0<1.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, wheneverR0<1.


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

Theorem 3.
A positive endemic equilibriumEE exists provided R0>1.Proof.
The endemic equilibriumEE=(Te,Te⋆,Ve) satisfies(17)rTe1-TeTmax-1-ρβVeTe-α1-ϑTeTe⋆-μTe=0(18)1-ρβVeTe+α1-ϑTeTe⋆-κTe⋆=0(19)ωκTe⋆-cVe=0From (19) we have(20)Ve=ωκTe⋆cSubstituting (20) in (18) we get(21)Te=11-ρβω/c+1-ϑα/κ=Tmax1-μ/rR0Substituting Ve and Te in (17) we obtain(22)Te⋆=Tmax1-μ/rR0-1r-μκR02The endemic equilibrium (EE) is given as(23)EE=Tmax1-μ/rR0,TeR0-1r-μκR0,ωκTe⋆c Clearly Te⋆>0 if and only if R0>1.


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

Theorem 4.
The endemic equilibriumEE=(Te,Te⋆,Ve) is locally asymptotically stable whenever R0>1.Proof.
The Jacobian matrix of model (1) at EE is as follows:(24)JEE=r-2r-μrR0-α1-ϑTmax1-μ/rR0-β1-ρTmax1-μ/rR0R0-1r-μR0α1-ϑTmax1-μ/rR0-κ1-ρβTmax1-μ/rR00ωκ-cThe characteristic equation of (24) is in the form(25)λ3+a0λ2+a1λ+a2=0where(26)a0=κ+c-α1-ϑTmax1-μ/rR0-r-2r-μrR0a1=1-ϑTmax1-μ/rR0r-2r-μrR0+R0-1r-μR0+r-2r-μrR0c-κ+cκ-1a2=R0-1r-μR0+r-2r-μrR01-cκ The number of possible negative real roots of (25) depends on the signs of a0, a1, and a2. This can be established by applying Descartes Rule of Signs as used in [11].(27)Pλ=a1λ2+a2λ+a3According to this rule the number of negative real zeros of P(λ) is either equal to the number of sign changes of P(-λ) or less by an even number, as shown in Table 1.
From Table1 the maximum number of sign changes in P(-λ) is 2; hence the characteristic polynomial (27) has two negative roots. Thus(28)P-λ=-λ3+a0λ2-a1λ+a2=0has negative real roots. Hence for r>μ and if cases 1 to 8 are satisfied then the endemic equilibrium EE is locally asymptotically stable.Table 1
Roots of characteristic equation (27).
cases a 0 a 1 a 2 R 0 > 1 no. of sign changes no. of real -ve roots 1 - + - R 0 > 1 2 2,0 2 - - - R 0 > 1 0 0 3 + + - R 0 > 1 1 0 4 + - - R 0 > 1 1 0 5 - + + R 0 > 1 1 0 6 - - + R 0 > 1 1 0 7 + + + R 0 > 1 0 0 8 + - + R 0 > 1 2 2,0Therefore ifR0>1 and given a small number of free virus particles, each virus, in the entire period of its infectivity, will produce on average more than one infected cell, implying viral persistence.


### 3.6. Global Stability Analysis of the Endemic Equilibrium

In this section we investigate the global stability of the endemic equilibrium using geometric approach, as developed by Li and Muldowney in [15]. For a brief description of this approach, see [6, 8]. Consider the autonomous dynamical system(29)y′=fxwhere f:Ω→Rn, Ω⊂Rn is an open set and is simply connected and y∈Ω, y↦f(x)∈Rn, f(x)∈C′(Ω). Let y⋆ be an equilibrium point, then y⋆ is said to be globally stable in Ω if it is locally stable in Ω and all trajectories in Ω converge to y⋆. In this method the equilibrium y⋆ is locally asymptotically stable provided the following conditions hold: (i)
(H1)Ω is simply connected(ii)
(H2) There exists a compact absorbing setK⊂Ω(iii)
(H3) Equation (29) has a unique equilibrium y⋆ in ΩLetP(y) be a n2×n2 matrix-valued function that is C′ on Ω and consider B=PfP-1+P∂f2/∂yP-1 where the matrix Pf is ∂Pij⋆/∂yf=dPij/dt and let the matrix J(2) be the second additive compound matrix of the Jacobian matrix J; that is, J(y)=(Jij), J(2) is a n2×n2 matrix, and in our case n=3; hence(30)J2=J11+J22J23-J13J32J11+J33J12-J31J21J22+J33Consider the Lonziskii measure μ of B with respect to a vector norm · in RN where N=n2(31)μB=limt→0+I+hB-1hIt is proved in [15] that if (H1), (H2), and (H3) hold and condition(32)q¯=limsupt→∞⁡supy0∈K⁡1t∫0tμBxs,x0ds<0is satisfied, then the unique equilibrium y⋆ is globally asymptotically stableLemma 5.
Assume that conditions (H1), (H2), and (H3) hold, theny⋆ is globally asymptotically stable in Ω provided that a function P(x) and a Lonziskii measure [16] μ exist such that condition (32) is satisfied.Theorem 6.
The endemic equilibriumEE is globally asymptotically stable in Ω if R0>1.Proof.
Consider the Jacobian of model (1)(33)J=r-2rTTmax-1-ρβV-1-ϑαT⋆-μ-1-ϑαT-1-ρβT1-ρβV+1-ϑαT⋆1-ϑαT-κ1-ρβT0ωκ-cand the second compound additive matrix of (33) is given as(34)J2=a+b-κddωκa-c-b0eb-κ-cwhere(35)a=r-2rTTmax-1-ρβV-α1-ϑT⋆-μb=αT1-ϑd=1-ρβTe=1-ρβV+αT⋆1-ϑ We define an auxiliary matrix function Q on Ω as(36)Q=diag⁡1T⋆,1V,1Vsetting T⋆,V>0 everywhere in Ω, and Q is smooth and nonsingular. Qf and QfQ-1 are given as(37)Qf=diag⁡-T⋆˙T⋆2,-V˙V2,-V˙V2(38)QfQ-1=diag⁡-T⋆˙T⋆,-V˙V,-V˙Vwhere T⋆˙=dT⋆/dt and V˙=dV/dt. Matrix QJ(2)Q-1 is given as(39)QJ2Q-1=a+b-κdVT⋆dVT⋆ωκT⋆Va-c-b0eb-κ-cThus the matrix M=QfQ-1+QJ(2)Q-1 as defined in (4.4) of [15] can be written in block form as(40)M=m11m12m21m22where m11=a+b-κ-T⋆˙/T⋆, m12=dV/T⋆,dV/T⋆, m21=ωκT⋆/V0, and m22=a-c-V˙/V-beb-c-κ-V˙/V. Let the vector norm · in R3≅R32 be chosen as(41)u,v,w=sup⁡u,v+wThe Lozinskii measure μ(M) with respect to · can be estimated as follows:(42)μM≤sup⁡g1,g2where(43)g1=μ1m11+m12(44)g2=μ1m22+m21m12 and m21 are operator norms associated with the linear mappings m12:R2→R and m12:R→R2, respectively, where R is endowed with the l1 vector norm in both cases. Specifically μ1(m11)=a+b-κ-T⋆˙/T⋆, m12=dV/T⋆, m21=sup⁡ωκT⋆/V,0, and μ1(m22)=-c-V˙/V+sup{a+e,-κ}. From model (1) we find(45)T⋆˙T⋆=1-ρβVT+1-ϑαT-κ(46)V˙V=ωκT⋆V-cand recalling expressions for a, b, d, and e we obtain(47)g1=r-2rTTmax-1-ρβV-1-ϑαT⋆-μ(48)g2=sup⁡r-2rTTmax-μ,-κSince r-2rT/Tmax-1-ρβV-1-ϑαT⋆-μ≤sup⁡r-2rT/Tmax-μ,-κ we find that g1≤g2, then (42) implies that μ(M)≤g2; thus(49)μM≤r-2rTTmax-μ+sup-1-ρβV-1-ϑαT⋆,0≤r-2rTTmax-μSince Tmax is the limiting value of T(t), then this implies that limsupt→∞T(t)≤Tmax; therefore expression (49) reduces to(50)μM≤r-2r-μIntegrating (50) we obtain(51)1t∫0tμMds≤1t∫0tr-2r-μds<0Hence the endemic equilibrium (EE) is globally asymptotically stable whenever R0>1.This implies that, regardless of any starting solution, the solution of the model will converge to EE wheneverR0>1. Immunologically, it means that, at any perturbation of the equilibrium point as a result of the introduction of the free virus particles, the model solutions will converge to the endemic state.


## 3.1. Basic Reproduction Number

The basic reproduction numberR0 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 R0 for model (1) using the next generation matrix method as used in [13, 14]. Model (1) has two infected compartments T⋆ and V. Let fi be the rate of appearance of new infections in compartment i and νi as the transfer of individuals out of compartment i for the two compartments, respectively, and are given in partitioned form as follows:(2)f=1-ρβVT+α1-ϑTT⋆0and(3)ν=κT⋆-ωκT⋆+cVThe Jacobian of f and ν evaluated at the Infection Free Equilibrium E0=(Tmax,0,0) yields(4)F=α1-ϑT0β1-ρT000(5)V=κ0-ωκcwhere F is nonnegative and V is nonsingular. The basic reproduction number is thus given by R0=ρ(FV-1), where ρ(FV-1) is the spectral radius of the matrix FV-1. Hence(6)R0=Tmax1-μr1-ϑακ+Tmax1-μrωβc1-ρ


## 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 E0=(Tmax(1-μ/r),0,0).Theorem 1.
The infection free equilibriumE0 is locally asymptotically stable if and only if R0<1.Proof.
Evaluating the Jacobian of model (1) at E0, we obtain(7)JE0=μ-r-αTmax1-μr1-ϑ-βTmax1-μr1-ρ0αTmax1-μr1-ϑ-κβTmax1-μr1-ρ0ωκ-cand clearly(8)λ1=μ-ris 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 r>μ. The nature of the remaining roots of (7) can be determined from the reduced matrix:(9)A=αTmax1-μr1-ϑ-κβTmax1-μr1-ρωκ-cUsing Routh-Hurwitz stability criteria, matrix A in (9) will have negative real roots if and only if the tr⁡A<0 and det⁡A>0; thus(10)tr⁡A=R0-1-Tmax1-μr1-ρβωc-ckand(11)det⁡A=-ακTmax1-μr1-ϑ+βωcTmax1-μr1-ρ+1and using (6), (11) reduces to(12)det⁡A=1-R0 From (10) and (12), tr⁡A<0 and det⁡A>0 if and only if R0<1. Thus E0 is locally asymptotically stable whenever R0<1 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 wheneverR0<1. Theoretically this shows that the virus is cleared from the body if R0<1.


## 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 form(13)dXdt=HX,ZdZdt=GX,Z,GX,0=0where X∈R denotes the number of uninfected cells and Z∈R2 denotes the number of actively infected cells and free virus particles, respectively. The infection free equilibrium (IFE) is now denoted by(14)U0=X0,0,X0=Tmax1-μrThe conditions H1 and H2 below must be met in order to guarantee global asymptotic stability:(i)
(H1) For dX/dt=H(X,0),X0 is Globally Asymptotically Stable (GAS)(ii)
(H2) G(X,Z)=PZ-G^(X,Z),G^(X,Z)≥0, for (X,Z)∈ΩwhereP=DZG(X0,0) is an M-matrix (the off-diagonal elements of P are nonnegative) and Ω is the region where the model makes biological sense. If system (13) satisfies conditions H1 and H2, then the following theorem holds.Theorem 2.
The fixed pointU0=(X0,0) is Globally Asymptotically Stable equilibrium of (13) provided that R0<1 and that assumptions (H1) and (H2) are satisfied.Proof.
LetX(t)=T(t), Z=(T⋆(t),V(t)), H(X,0)=rT1-T/Tmax-μT0, and G(X,Z)=PZ-G^(X,Z) where(15)P=αTmax1-ϑ-κβTmax1-ρωκ-c(16)G^X,Z=G1^X,ZG2^X,Z=00From (16) G1^(X,Z)=G2^(X,Z)=0 this implies that G^(X,Z)≥0. Therefore, E0 is globally asymptotically stable when R0<1.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, wheneverR0<1.


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

Theorem 3.
A positive endemic equilibriumEE exists provided R0>1.Proof.
The endemic equilibriumEE=(Te,Te⋆,Ve) satisfies(17)rTe1-TeTmax-1-ρβVeTe-α1-ϑTeTe⋆-μTe=0(18)1-ρβVeTe+α1-ϑTeTe⋆-κTe⋆=0(19)ωκTe⋆-cVe=0From (19) we have(20)Ve=ωκTe⋆cSubstituting (20) in (18) we get(21)Te=11-ρβω/c+1-ϑα/κ=Tmax1-μ/rR0Substituting Ve and Te in (17) we obtain(22)Te⋆=Tmax1-μ/rR0-1r-μκR02The endemic equilibrium (EE) is given as(23)EE=Tmax1-μ/rR0,TeR0-1r-μκR0,ωκTe⋆c Clearly Te⋆>0 if and only if R0>1.


## 3.5. Local Stability Analysis of the Endemic Equilibrium

Theorem 4.
The endemic equilibriumEE=(Te,Te⋆,Ve) is locally asymptotically stable whenever R0>1.Proof.
The Jacobian matrix of model (1) at EE is as follows:(24)JEE=r-2r-μrR0-α1-ϑTmax1-μ/rR0-β1-ρTmax1-μ/rR0R0-1r-μR0α1-ϑTmax1-μ/rR0-κ1-ρβTmax1-μ/rR00ωκ-cThe characteristic equation of (24) is in the form(25)λ3+a0λ2+a1λ+a2=0where(26)a0=κ+c-α1-ϑTmax1-μ/rR0-r-2r-μrR0a1=1-ϑTmax1-μ/rR0r-2r-μrR0+R0-1r-μR0+r-2r-μrR0c-κ+cκ-1a2=R0-1r-μR0+r-2r-μrR01-cκ The number of possible negative real roots of (25) depends on the signs of a0, a1, and a2. This can be established by applying Descartes Rule of Signs as used in [11].(27)Pλ=a1λ2+a2λ+a3According to this rule the number of negative real zeros of P(λ) is either equal to the number of sign changes of P(-λ) or less by an even number, as shown in Table 1.
From Table1 the maximum number of sign changes in P(-λ) is 2; hence the characteristic polynomial (27) has two negative roots. Thus(28)P-λ=-λ3+a0λ2-a1λ+a2=0has negative real roots. Hence for r>μ and if cases 1 to 8 are satisfied then the endemic equilibrium EE is locally asymptotically stable.Table 1
Roots of characteristic equation (27).
cases a 0 a 1 a 2 R 0 > 1 no. of sign changes no. of real -ve roots 1 - + - R 0 > 1 2 2,0 2 - - - R 0 > 1 0 0 3 + + - R 0 > 1 1 0 4 + - - R 0 > 1 1 0 5 - + + R 0 > 1 1 0 6 - - + R 0 > 1 1 0 7 + + + R 0 > 1 0 0 8 + - + R 0 > 1 2 2,0Therefore ifR0>1 and given a small number of free virus particles, each virus, in the entire period of its infectivity, will produce on average more than one infected cell, implying viral persistence.


## 3.6. Global Stability Analysis of the Endemic Equilibrium

In this section we investigate the global stability of the endemic equilibrium using geometric approach, as developed by Li and Muldowney in [15]. For a brief description of this approach, see [6, 8]. Consider the autonomous dynamical system(29)y′=fxwhere f:Ω→Rn, Ω⊂Rn is an open set and is simply connected and y∈Ω, y↦f(x)∈Rn, f(x)∈C′(Ω). Let y⋆ be an equilibrium point, then y⋆ is said to be globally stable in Ω if it is locally stable in Ω and all trajectories in Ω converge to y⋆. In this method the equilibrium y⋆ is locally asymptotically stable provided the following conditions hold: (i)
(H1)Ω is simply connected(ii)
(H2) There exists a compact absorbing setK⊂Ω(iii)
(H3) Equation (29) has a unique equilibrium y⋆ in ΩLetP(y) be a n2×n2 matrix-valued function that is C′ on Ω and consider B=PfP-1+P∂f2/∂yP-1 where the matrix Pf is ∂Pij⋆/∂yf=dPij/dt and let the matrix J(2) be the second additive compound matrix of the Jacobian matrix J; that is, J(y)=(Jij), J(2) is a n2×n2 matrix, and in our case n=3; hence(30)J2=J11+J22J23-J13J32J11+J33J12-J31J21J22+J33Consider the Lonziskii measure μ of B with respect to a vector norm · in RN where N=n2(31)μB=limt→0+I+hB-1hIt is proved in [15] that if (H1), (H2), and (H3) hold and condition(32)q¯=limsupt→∞⁡supy0∈K⁡1t∫0tμBxs,x0ds<0is satisfied, then the unique equilibrium y⋆ is globally asymptotically stableLemma 5.
Assume that conditions (H1), (H2), and (H3) hold, theny⋆ is globally asymptotically stable in Ω provided that a function P(x) and a Lonziskii measure [16] μ exist such that condition (32) is satisfied.Theorem 6.
The endemic equilibriumEE is globally asymptotically stable in Ω if R0>1.Proof.
Consider the Jacobian of model (1)(33)J=r-2rTTmax-1-ρβV-1-ϑαT⋆-μ-1-ϑαT-1-ρβT1-ρβV+1-ϑαT⋆1-ϑαT-κ1-ρβT0ωκ-cand the second compound additive matrix of (33) is given as(34)J2=a+b-κddωκa-c-b0eb-κ-cwhere(35)a=r-2rTTmax-1-ρβV-α1-ϑT⋆-μb=αT1-ϑd=1-ρβTe=1-ρβV+αT⋆1-ϑ We define an auxiliary matrix function Q on Ω as(36)Q=diag⁡1T⋆,1V,1Vsetting T⋆,V>0 everywhere in Ω, and Q is smooth and nonsingular. Qf and QfQ-1 are given as(37)Qf=diag⁡-T⋆˙T⋆2,-V˙V2,-V˙V2(38)QfQ-1=diag⁡-T⋆˙T⋆,-V˙V,-V˙Vwhere T⋆˙=dT⋆/dt and V˙=dV/dt. Matrix QJ(2)Q-1 is given as(39)QJ2Q-1=a+b-κdVT⋆dVT⋆ωκT⋆Va-c-b0eb-κ-cThus the matrix M=QfQ-1+QJ(2)Q-1 as defined in (4.4) of [15] can be written in block form as(40)M=m11m12m21m22where m11=a+b-κ-T⋆˙/T⋆, m12=dV/T⋆,dV/T⋆, m21=ωκT⋆/V0, and m22=a-c-V˙/V-beb-c-κ-V˙/V. Let the vector norm · in R3≅R32 be chosen as(41)u,v,w=sup⁡u,v+wThe Lozinskii measure μ(M) with respect to · can be estimated as follows:(42)μM≤sup⁡g1,g2where(43)g1=μ1m11+m12(44)g2=μ1m22+m21m12 and m21 are operator norms associated with the linear mappings m12:R2→R and m12:R→R2, respectively, where R is endowed with the l1 vector norm in both cases. Specifically μ1(m11)=a+b-κ-T⋆˙/T⋆, m12=dV/T⋆, m21=sup⁡ωκT⋆/V,0, and μ1(m22)=-c-V˙/V+sup{a+e,-κ}. From model (1) we find(45)T⋆˙T⋆=1-ρβVT+1-ϑαT-κ(46)V˙V=ωκT⋆V-cand recalling expressions for a, b, d, and e we obtain(47)g1=r-2rTTmax-1-ρβV-1-ϑαT⋆-μ(48)g2=sup⁡r-2rTTmax-μ,-κSince r-2rT/Tmax-1-ρβV-1-ϑαT⋆-μ≤sup⁡r-2rT/Tmax-μ,-κ we find that g1≤g2, then (42) implies that μ(M)≤g2; thus(49)μM≤r-2rTTmax-μ+sup-1-ρβV-1-ϑαT⋆,0≤r-2rTTmax-μSince Tmax is the limiting value of T(t), then this implies that limsupt→∞T(t)≤Tmax; therefore expression (49) reduces to(50)μM≤r-2r-μIntegrating (50) we obtain(51)1t∫0tμMds≤1t∫0tr-2r-μds<0Hence the endemic equilibrium (EE) is globally asymptotically stable whenever R0>1.This implies that, regardless of any starting solution, the solution of the model will converge to EE wheneverR0>1. Immunologically, it means that, at any perturbation of the equilibrium point as a result of the introduction of the free virus particles, the model solutions will converge to the endemic state.


## 4. Numerical Simulation of the Model

In this section we perform a numerical simulation of model (1) using MATLAB with the parameter values given in Table 1. The primary purpose of the numerical simulation is to analyze the change in state of virus progression with time and also to outline the impact of the variation of treatment efficacy on the transmission dynamics of HIV. This is achieved by varying the parameter values ρ and ϑ while keeping the other parameters constant.


### 4.1. Effect of Variations in RTI and PI Treatment Efficacy on the Asymptotic Behaviour of the Equilibrium Points

The effect of variations in combined treatment efficacy on the stability of equilibria is investigated using the parameter values given in Table2; this is achieved by choosing four different initial conditions(i)
IC1:T1(t)=1000, T1⋆(t)=0, V1(t)=10-3(ii)
IC2:T2(t)=800, T2⋆(t)=10, V2(t)=0.001(iii)
IC3:T3(t)=900, T3⋆(t)=5, V3(t)=0.01(iv)
IC4:T4(t)=1000, T4⋆(t)=1, V4(t)=0.1Table 2
Parameters values for model (1).
Parameters units Description source T m a x 1500 cellsmm-3 MaximumCD4+ cell population level [9] r 0.03 cellsday-1 Production rate of [9] uninfected T cells μ 0.02 cellsday-1 Natural death rate of [9] uninfected T cells κ 0.24 cellsday-1 Death rate of actively [9] infected T cells c 2.4day-1 Shedding rate of virions [9] β 2.4 × 1 0 - 5 m m - 3 Viral infection rate [9] by free virions α 2.4 × 1 0 - 5 m m - 3 cellular infection rate [9] ω varies:ω≥0 Burst rate of actively - infected T cells ρ varies:0<ρ<1 Efficacy of RT Inhibitor - ϑ varies:0<ϑ<1 efficacy of Protease Inhibitor -and varying parametersρ and ϑ while keeping other parameters constant. From Theorems 2 and 6, the corresponding stabilities of model (1) are as follows:(1)
IfR0<1, the IFE is globally asymptotically stable.(2)
IfR0>1, the EE is globally asymptotically stable.(i)
ρ = 0.1 and ϑ=0.1. For this set of parameters we obtain R0=4.545>1. By Theorem 6 the endemic equilibrium is globally asymptotically stable and the states of the system converge to EE=(110.011,3.575,357.53), for the four initial conditions IC1-IC4, implying that the virus persists in the host. This shows that the low treatment efficacy of 0.1 cannot effectively combat the virus. This is illustrated by Figures 1, 2, 3, and 4. (ii)
ρ = 0.5 and ϑ=0.5. This set of parameters yields R0=2.525>1 and the system state approach EE=(198.02,4.98,498.31) for all the four initial conditions IC1-IC4. This means that EE exists and is globally asymptotically stable according to Theorem 6. Again despite the increase in treatment efficacy the virus persists in the host. Figures 1, 2, 3, and 4 illustrate this scenario. (iii)
ρ = 0.82 and ϑ=0.9. This treatment efficacy yields R0=0.101<1 and by Theorem 2E0=(1500,0,0) exists and is GAS for all initial values IC1-IC4, as can be observed from Figures 1, 2, 3, and 4 implying that the high treatment efficacy of over 0.82 helps in stifling viral replication within host.Figure 1
Effect of treatment efficacy onT(t),T⋆(t), and V(t).
(a) (b) (c)Figure 2
(a)-(c) The graph trajectory ofT(t),T⋆(t), and V(t) for ρ=0.1 and ϑ=0.1 with the initial conditions IC2-IC4. R0=4.545 and EE=(110.011,3.575,357.53) is GAS.
(a) (b) (c)Figure 3
(a)-(c) The graph trajectory ofT(t),T⋆(t), and V(t) for ρ=0.5 and ϑ=0.5 with the initial conditions IC2-IC4. R0=2.525>1 and EE=(198.02,4.98,498.31) is GAS.
(a) (b) (c)Figure 4
(a)-(c) The graph trajectory ofT(t),T⋆(t), and V(t) for ρ=0.9 and ϑ=0.9 with the initial conditions IC2-IC4. R0=0.101<1 and E0=(1500,0,0) is globally asymptotically stable.
(a) (b) (c)


### 4.2. Effect of Variations in RTI and PI Treatment Efficacy on the CD4+ T Cell Population and Viral Load

From Figures1(a), 1(b), and 1(c), it is observed that variations in the treatment efficacy have significant effect on the number of CD4+ T cells. For instance, with a treatment efficacy of 0.1 at the onset of HIV infection, the CD4+ T cells undergo a sharp decline, while treatment efficacy of 0.5 only slows down CD4+ T cell depletion and takes longer (approximately 40 days) to reduce to zero. However with a drug efficacy of 0.9 the number of CD4+ T cells is kept above 700 mm-3. This demonstrates that treatment efficacy is directly proportional to the number of CD4+ T cells.Figures1(b) and 1(c) illustrate the effect of treatment efficacy on the number of actively infected cells and free virions, respectively. It is clear that from the time of infection if treatment efficacy is kept at a minimum of 0.1, then both actively infected cells and free virions replicate rapidly attaining a primary peak of 519 infected cells within the first 17 days, while the virions reach a peak of 5.074×104 within the same period of time. With an increase in treatment efficacy both infected cells and viral replication rates are significantly reduced; for instance, at 0.9 efficacy level the number of virus and infected cells approach zero within the first 90 days.


## 4.1. Effect of Variations in RTI and PI Treatment Efficacy on the Asymptotic Behaviour of the Equilibrium Points

The effect of variations in combined treatment efficacy on the stability of equilibria is investigated using the parameter values given in Table2; this is achieved by choosing four different initial conditions(i)
IC1:T1(t)=1000, T1⋆(t)=0, V1(t)=10-3(ii)
IC2:T2(t)=800, T2⋆(t)=10, V2(t)=0.001(iii)
IC3:T3(t)=900, T3⋆(t)=5, V3(t)=0.01(iv)
IC4:T4(t)=1000, T4⋆(t)=1, V4(t)=0.1Table 2
Parameters values for model (1).
Parameters units Description source T m a x 1500 cellsmm-3 MaximumCD4+ cell population level [9] r 0.03 cellsday-1 Production rate of [9] uninfected T cells μ 0.02 cellsday-1 Natural death rate of [9] uninfected T cells κ 0.24 cellsday-1 Death rate of actively [9] infected T cells c 2.4day-1 Shedding rate of virions [9] β 2.4 × 1 0 - 5 m m - 3 Viral infection rate [9] by free virions α 2.4 × 1 0 - 5 m m - 3 cellular infection rate [9] ω varies:ω≥0 Burst rate of actively - infected T cells ρ varies:0<ρ<1 Efficacy of RT Inhibitor - ϑ varies:0<ϑ<1 efficacy of Protease Inhibitor -and varying parametersρ and ϑ while keeping other parameters constant. From Theorems 2 and 6, the corresponding stabilities of model (1) are as follows:(1)
IfR0<1, the IFE is globally asymptotically stable.(2)
IfR0>1, the EE is globally asymptotically stable.(i)
ρ = 0.1 and ϑ=0.1. For this set of parameters we obtain R0=4.545>1. By Theorem 6 the endemic equilibrium is globally asymptotically stable and the states of the system converge to EE=(110.011,3.575,357.53), for the four initial conditions IC1-IC4, implying that the virus persists in the host. This shows that the low treatment efficacy of 0.1 cannot effectively combat the virus. This is illustrated by Figures 1, 2, 3, and 4. (ii)
ρ = 0.5 and ϑ=0.5. This set of parameters yields R0=2.525>1 and the system state approach EE=(198.02,4.98,498.31) for all the four initial conditions IC1-IC4. This means that EE exists and is globally asymptotically stable according to Theorem 6. Again despite the increase in treatment efficacy the virus persists in the host. Figures 1, 2, 3, and 4 illustrate this scenario. (iii)
ρ = 0.82 and ϑ=0.9. This treatment efficacy yields R0=0.101<1 and by Theorem 2E0=(1500,0,0) exists and is GAS for all initial values IC1-IC4, as can be observed from Figures 1, 2, 3, and 4 implying that the high treatment efficacy of over 0.82 helps in stifling viral replication within host.Figure 1
Effect of treatment efficacy onT(t),T⋆(t), and V(t).
(a) (b) (c)Figure 2
(a)-(c) The graph trajectory ofT(t),T⋆(t), and V(t) for ρ=0.1 and ϑ=0.1 with the initial conditions IC2-IC4. R0=4.545 and EE=(110.011,3.575,357.53) is GAS.
(a) (b) (c)Figure 3
(a)-(c) The graph trajectory ofT(t),T⋆(t), and V(t) for ρ=0.5 and ϑ=0.5 with the initial conditions IC2-IC4. R0=2.525>1 and EE=(198.02,4.98,498.31) is GAS.
(a) (b) (c)Figure 4
(a)-(c) The graph trajectory ofT(t),T⋆(t), and V(t) for ρ=0.9 and ϑ=0.9 with the initial conditions IC2-IC4. R0=0.101<1 and E0=(1500,0,0) is globally asymptotically stable.
(a) (b) (c)


## 4.2. Effect of Variations in RTI and PI Treatment Efficacy on the CD4+ T Cell Population and Viral Load

From Figures1(a), 1(b), and 1(c), it is observed that variations in the treatment efficacy have significant effect on the number of CD4+ T cells. For instance, with a treatment efficacy of 0.1 at the onset of HIV infection, the CD4+ T cells undergo a sharp decline, while treatment efficacy of 0.5 only slows down CD4+ T cell depletion and takes longer (approximately 40 days) to reduce to zero. However with a drug efficacy of 0.9 the number of CD4+ T cells is kept above 700 mm-3. This demonstrates that treatment efficacy is directly proportional to the number of CD4+ T cells.Figures1(b) and 1(c) illustrate the effect of treatment efficacy on the number of actively infected cells and free virions, respectively. It is clear that from the time of infection if treatment efficacy is kept at a minimum of 0.1, then both actively infected cells and free virions replicate rapidly attaining a primary peak of 519 infected cells within the first 17 days, while the virions reach a peak of 5.074×104 within the same period of time. With an increase in treatment efficacy both infected cells and viral replication rates are significantly reduced; for instance, at 0.9 efficacy level the number of virus and infected cells approach zero within the first 90 days.


## 5. Conclusion

In this paper, a mathematical model for within host HIV infection with virus-to-cell and cell-to-cell treatment has been formulated and analyzed. Target cell production has been modelled by a logistic incidence rate. The global dynamics have been shown to be dependent onR0, in which for values of R0<1 the infection free equilibrium is globally asymptotically stable, and for R0>1 the endemic equilibrium is globally asymptotically stable. The global asymptotic stability of IFE and EE has been proved using the method by Castillo-Chavez and the geometric approach, respectively. Numerical simulations have shown that combined treatment has significant effect on the dynamics of HIV infection, in which high levels of treatment efficacy for both RTI and PI treatment was observed to stifle viral replication. This observation is in agreement with the findings of [9] which showed that if R0>1 before treatment then the virus will increase as would be the number of infected cells, but after treatment if R0<1, then T⋆ and V would both decline.



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