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There have been many mathematical models aimed at analysing the in-vivo dynamics of HIV. However, in most cases the attention has been on the interaction between the HIV virions and the CD4
^{+} T-cells. This paper brings in the intervention of the CD8
^{+} T-cells in seeking, destroying, and killing the infected CD4
^{+} T-cells during early stages of infection. The paper presents and analyses a five-component in-vivo model and applies the results in investigating the in-vivo dynamics of HIV in presence of the CD8
^{+} T-cells. We prove the positivity and the boundedness of the model solutions. In addition, we show that the solutions are biologically meaningful. Both the endemic and virions- free equilibria are determined and their stability investigated. In addition, the basic reproductive number is derived by the next generation matrix method. We prove that the virions-free equilibrium state is locally asymptotically stable if and only if R
_{0} < 1 and unstable otherwise. The results show that at acute infection the CD8
^{+} T-cells play a paramount role in reducing HIV viral replication. We also observe that the model exhibits backward and trans-critical bifurcation for some set of parameters for
R
_{0}
< 1. This is a clear indication that having
R
_{0}
< 1 is not sufficient condition for virions depletion.

One of the most threatening retrovirus in the world is the Human Immunodeficiency Virus (HIV) that leads to Acquired Immunodeficiency Syndrome (AIDS). Unlike other viruses, HIV is encoded in ribonucleic acid (RNA) rather than deoxyribonucleic acid (DNA). HIV attacks the immune system, weakening it and eventually if not treated it makes the infected people highly vulnerable to various opportunistic infections. In absence of any HIV-management mechanism, infected people progresses to AIDS stage after 10 - 15 years [

Since the first case of HIV was reported in the early 1980’s, HIV/AIDS has been associated to the deaths of more than 35 million people while over 36.9 million are living with the virus worldwide, making it one of the worst menace in the recorded history [

According to [

In-vivo study for HIV dynamics have been done over the years, aimed at understanding the interaction mechanism of the body cells and the HIV virus. Such information has proved so valuable especially in the development of ARTs and in HIV management. In the recent years, mathematical models of various complexity level have been used to simulate and analyse such interactions unlike in the past where researches relied on clinical trials [^{+} T-cells and the HIV virions. These basic non-linear models were developed and used in the analysis of HIV dynamics and consequently, in estimating fundamental parameters that brought in new concepts of the disease processes and its progression [^{+} T-cells, free HIV virions and the already infected CD4^{+} T-cells. For instance, [^{+} T-cells, the already infected CD4^{+} T-cells and the free HIV virions. The model for instance, predicts adequately the disease progression from the early infection stage, asymptomatic stage, to full blown-AIDs and the viral load at the asymptomatic stage. However, great improvement on the model has been done and many other advanced models developed. [^{+} T-cells, macrophage cells and the viral load. The results indicate that the CD4^{+} T-cells play a very vital role as far as HIV virions replication is concerned. Similarly [

Other researches have sought to study the role played by the killer T-cells in preventing virions replication in the body. In particular, [^{+} T-cells in HIV dynamic model. The study developed a three-dimensional ordinary differential equations of the untreated model. The model showed the interaction between the non-infected CD4^{+} T-cells, infected CD4^{+} T-cells and the immune response. The model had a major shortcoming for its failure to incorporate the HIV virions. On other hand, [^{+} T-cells, virions, defense cells and ARTs. The results emphasized the importance of the CD8^{+} T-cells in fighting the HIV virions during acute infection. It has been established that the disease become more endemic due to exponential virions replication and the failure of the ARTs to reach all the cells. Therefore, the focus on how to reduce virions replication by targeting the defense cell is inevitable and it will play a big role in ensuring that the endemicity of the infection is reduced.

In this paper, we shall mainly focus on HIV dynamics at acute HIV infection stage without any focus on the disease progression to AIDS stage which may come as a result of not using ARTs. The motivation behind this simple initial infection model is the fact that most of the new strategies for HIV management in Kenya such as the pre-exposure antiviral treatment and post-exposure antiviral treatment targets the HIV virions at the early stages of HIV-infection. For this study to be successful we develop a non-linear, five-dimensional deterministic model for in-vivo dynamics of HIV with inclusion of the CD8^{+} T-cells.

HIV, like most viruses lacks the ability to replicate on its own and therefore, relies on a host for replication. Although, unlike all other viruses HIV is a retrovirus and hence carries the copies of its own RNA [^{+} T cells by attaching itself on the membrane of the cell. After the infection of the cells by the virus, it is important to note that the symptoms do not show immediately until their level reduces to about 200 cells per mm^{3}, and the viral load increases to 500 copies per ml [^{+} T-cell. Then it fuses with the harbour cell and releases an enzyme known as reverse transcriptase. Reverse transcriptase enzyme transform the genome of the HIV virus to a double-stranded HIV DNA from a single-stranded HIV RNA. The transcription process ensures that integration of the HIV virion into the host DNA. Once HIV is integrated in the cells DNA, it starts to manufacture long chains of HIV protein using their DNA. The HIV proteins are the support system for more HIV virions. These long chains of HIV proteins (immature and non-infectious) assembles closer the membrane of the CD4^{+} cells and bud out. The immature virions then release an enzyme called the protease, which cut the long HIV proteins RNA into smaller individual proteins. As the smaller HIV proteins come together with copies of HIV’s RNA genetic material, they form a new mature virus particle. Other cells can now be infected by the new HIV copies. This clearly shows that a single virion lead to the production of many other virions [

After infection the CD4^{+} T-cells sends a signal to the CD8^{+} T-cells. The CD8^{+} T-cells are aimed at destroying and killing the virions [^{+} T-cells need to be analyzed further. Although researchers suggest that CD8^{+} T-cells play a paramount role in host defense against the HIV virions nothing much has been done to show it especially during AIDS stage. One of the objectives of this paper is to present a realistic model that will analyze the importance of the CD8^{+} T-cells in destroying the HIV virions. The CD4^{+} T-cells play critical roles in controlling viral infections by prompting CD8^{+} T-cells to eliminate the free HIV virions. The

HIV virus life cycle is presented in

An in-host HIV dynamics model with the inclusion of the immune cells is formulated. We show the model is positively invariant. The basic reproduction number expression is derived using the next generation matrix method. We also do the analyses on the stability of the steady points of the model.

We shall put into consideration a mathematical model for the in-vivo interaction of the HIV virions and the immune system cells. The model is classified into five compartments. The following are the variables used in the model (1) the healthy CD4^{+} T-cells ( T ), the infectious HIV virions ( V ), the already infected CD4^{+} T-cells ( I ), the immune cells ( Z ), that is, CD8^{+} T-cells and the activated immune cells ( Z a ).

The healthy CD4^{+} T-cells are recruited at a constant rate λ T from the bone marrow and die naturally at a constant rate μ T . The healthy CD4^{+} T-cells are as a result of the interaction between the uninfected CD4^{+} T-cells and the virus at a rate χ . They die naturally at a rate μ I , they are also eliminated by the activated CD4^{+} T-cells at the rate α . In addition, the infected healthy CD4^{+} T-cells produces an average of ϵ V viral particles. The new mature virions produced will infect other CD4^{+} T-cells. The HIV virions population increases due to the budding of the infected CD4^{+} T-cells at a rate ϵ V and die at the rate μ V . The CD8^{+} T-cells, are generated from the thymus at a constant rate λ Z , and die naturally at a constant rate μ Z . Due to the presence of the HIV virions the CD8^{+} T-cells become activated at the rate β . The activated CD8^{+} T-cells are produced from the CD8^{+} T-cells the in the presence of the HIV virions, at rate β and die naturally at rate μ Z a . The interaction description can be summarized in

We present the model diagram in

From

d T d t = λ T − μ T T − χ T V , d I d t = χ T V − μ I I − α I Z a , d V d t = ϵ V μ I I − μ V V , d Z d t = λ Z − μ Z Z − β Z I , d Z a d t = β Z I − μ Z a Z a } (1)

Variable | Description |
---|---|

The concentration of the susceptible CD4^{+} T cells at any time t | |

The concentration of the infected CD4^{+} T cells at any time t | |

The concentration of infectious HIV virions at any time t | |

The concentration of the CD8^{+} T-cells at any time t | |

The population of the activated CD8^{+} T-cells at any time t |

Parameter | Description |
---|---|

The recruitment rate of the susceptible CD4^{+} T-cells per unit time. | |

The decay rate of the susceptible CD4^{+} T-cells. | |

The infection rate of the CD4^{+} T-cells by the virus. | |

The natural death rate of the infected CD4^{+} T-cells. | |

The HIV virions generation rate from the infected CD4^{+} T-cells. | |

The death rate of the infectious virus. | |

The rate at which the infected cells are eliminated by the activated CD8^{+} T-cells. | |

The recruitment rate of the CD8^{+} T-cells per unit time. | |

The death rate of the CD8^{+} T-cells. | |

The activation rate of the CD8^{+} T-cells due to the presence the infected CD4^{+} T-cells. | |

The decay rate at of the activated defence cells decay per unit time. |

Before commencing the steady-states analysis of the model (1), it is important to look at some properties to ensure existence of biologically meaningful solutions.

Before we analyze the model (1) it is paramount to prove that the key variables are non-negative implying that the model solutions will be positive for all t > 0 and must be bounded for all t > 0 in an invariant region. Invariant region is the area in which the model is well posed mathematically and has biological meaning.

Theorem 1. Let the initial values of the state variables be T ( 0 ) ≥ 0 , V ( 0 ) ≥ 0 , I ( 0 ) ≥ 0 , Z ( 0 ) ≥ 0 , Z a ( 0 ) ≥ 0 . Then show that, for every t > 0 the solution set Γ = { T ( t ) , V ( t ) , I ( t ) , Z ( t ) , Z a ( t ) } of the model (1) is non-negative and Γ is the invariant region.

Proof. Taking the first part of Equation (1) we have,

d T ( t ) d t ≥ − μ T T − χ T V T ( t ) ≥ T ( 0 ) e ∫ − ( μ T + χ V ) d t , (2)

Hence T is non-negative for all t > 0 .

Similarly for the infected CD4^{+} T-cells we have,

d I d t = χ T V − μ I I − α I Z a ≥ − ( μ I − α Z a ) I (3)

By integration and separation of variables Equation (3) gives,

I ≥ I ( 0 ) e ∫ − ( μ I + α Z a ) d t (4)

Hence I is non-negative for all t > 0 .

Similarly for the HIV virions we have,

d V d t = ϵ V μ I I − μ V V ≥ − μ V (5)

By integration and separation variables Equation (3) gives,

V ≥ V ( 0 ) e ∫ − ( μ V ) d t

Hence V is non-negative for all t > 0 .

For the CD8^{+}, part four of model (1) gives

d Z d t = λ Z − μ Z Z − β Z I ≥ − ( μ Z + β I ) Z d Z d t ≥ − ( μ Z + β I ) Z (6)

We separate variables and integrate both sides with respect to the corresponding variables as follow,

Z ≥ Z ( 0 ) e ∫ − ( μ Z + β I ) d t (7)

Hence Z is non-negative for all t > 0 .

Finally for the activated immune cells we have;

d Z a d t = β Z I − μ z a Z a ≥ − ( μ z a Z a ) d Z a d t ≥ − ( μ z a Z a ) (8)

By integration and separation of variables we get;

Z a ≥ Z a ( 0 ) e ∫ − ( μ z a ) d t (9)

Hence Z a is non-negative for all t > 0 . ,

Notably, all the state variables of system (1) have been proved to be non-negative. In addition, parameters of model (1) monitors cell population, hence they are also non-negative for all, t > 0 . Consequently the model (1) analysis is done in the region Γ that is biologically meaningful.

Theorem 2. Let T ( t ) ≥ 0 , V ( t ) ≥ 0 , I ( t ) ≥ 0 , Z ( t ) ≥ 0 , Z a ( t ) ≥ 0 . Then the solutions of T ( t ) , V ( t ) , I ( t ) , Z ( t ) , Z a ( t ) are bounded and the region Γ is positively invariant for all t ≥ 0 .

Γ = { ( T ( t ) , I ( t ) , V ( t ) , Z ( t ) , Z a ( t ) ) ∈ ℝ 5 , T + I ≤ λ T μ T , Z + Z a ≤ λ Z μ Z , V ≤ ε V μ I λ T μ T μ V + V 0 } (10)

Proof. The total population of the CD4^{+} T-cells, T + I = N 4 ( t ) , is clearly non-constant value. Hence the evolution equation representing the change in the population of the CD4^{+} T-cells is;

d N 4 ( t ) d t = λ T − μ T T − μ I I − α I Z a , d N 4 ( t ) d t ≤ λ T − μ T N 4 ( t ) (11)

By separation of variables method for solving differential inequality, Equation (11) becomes;

d N 4 ( t ) λ T − μ T N 4 ( t ) ≤ d t , ln ( λ T − μ T N 4 ( t ) ) − 1 μ T ≥ ln C + t (12)

Thus, Equation (12) reduces to;

ln ( λ T − μ T N 4 ( t ) ) − 1 μ T ≥ C e μ T t (13)

But

C = λ T − μ T N 0 (14)

Therefore, Equation (12) becomes;

N 4 ( t ) ≤ λ T μ T − ( λ T − μ T N 0 ) e − μ T t μ T (15)

Thus at any time t > 0 we have;

N 4 ( t ) ≤ max { N 0 , λ T μ T } (16)

Hence, all feasible solutions set for the CD4^{+} T-cells of the model (1) enters the region:

Γ T = { ( T ( t ) , I ( t ) ) ∈ ℝ 2 , N 4 ≤ max { N 0 , λ T μ T } } (17)

Similarly the total number of the CD8^{+} T-cells, Z + Z a = N 8 ( t ) , at disease free equilibrium are given by;

d N 8 ( t ) d t = λ Z − μ Z N 8 ( t ) (18)

By separation of variables method for solving differential inequality Equation (18) becomes;

d N 8 ( t ) λ Z − μ Z N 8 ( t ) ≤ d t (19)

Integrating Equation (19) we have

N 8 ( t ) ≤ λ Z μ Z − ( λ Z − μ Z N 0 c ) e − μ Z t μ Z (20)

Thus at any time t > 0 we have;

N 8 ( t ) ≤ m a x { N 0 c , λ Z μ Z } (21)

Hence, all feasible solutions set for the CD8^{+} T-cells of the model (1) enters the region;

Γ Z = { ( Z ( t ) , Z a ( t ) ) ∈ ℝ 2 , N 8 ( t ) ≤ max { N 0 c , λ Z μ Z } } (22)

Considering the V population of the model (1) we have’

d V ( t ) d t ≤ ε V μ I λ T μ T − μ V V , since I ≤ λ T μ T (23)

Integration gives

d V d t + μ V V ≤ ε V μ I λ T μ T V ≤ ε V μ I λ T μ T μ V + V 0 (24)

Hence V is bounded. Consequently the feasible solution for the model (1) is;

Γ = { ( T ( t ) , V ( t ) , I ( t ) , Z ( t ) , Z a ( t ) ) ∈ ℝ 5 , T + I ≤ λ T μ T , Z + Z a ≤ λ Z μ Z , V ≤ ε V μ I λ T μ T μ V + V 0 } (25)

All the state variables are positive and bounded. Consequently, from Equation (25), Γ is positively invariant of model (1). Hence, it is possible to study the dynamics of the HIV model (1) in Γ . ,

With theorem 2 we conclude that the model is valid and will remain so during the whole course of study if and only if the initial data are biologically meaningful. In addition it is evident that with time the number of virions will reduce to non- detectable level.

Remark 1. Suppose T ( 0 ) , I ( 0 ) , V ( 0 ) , Z ( 0 ) , Z a ( 0 ) > 0 be given. Then there exist a differentiable continuous function T , I , V , Z , Z a : [ 0, T f ] R such that ( T , I , V , Z , Z a ) is bounded and ( T , I , V , Z , Z a ) ( 0 ) = ( T ( 0 ) , I ( 0 ) , V ( 0 ) , Z ( 0 ) , Z a ( 0 ) ) .

Therefore, the model solutions will always be positive if the initial values for the state variables are non-negative for all t > 0 in the closed interval [ 0, T f ] .

For us to fully understand the dynamics of the five component HIV model we study its stability. In this model there exist two critical points. The critical points represent the case before the virions get to the body, that is virions-free equilibrium point that is, V = I = Z a = 0 , and when the virus persist in the body, that is, V ≠ 0, I ≠ 0 and Z a ≠ 0 .

Before infection by HIV virions, the model as represented by Equation (1), has a unique feasible HIV-free steady state solution to be referred to as the virions-free equilibrium (VFE). The virions-free equilibrium of the model (1) is given by:

E 0 ( T , I , V , Z , Z a ) = [ λ T μ T ,0,0, λ Z μ Z ,0 ] (26)

Researchers in the field of in-vivo HIV modelling aims at finding the optimal conditions that determine the spread of the HIV virions in the susceptible CD4^{+} T-cells. In order to do this, researchers consider the basic reproductive number ( R 0 ). According to [

_{0}

In this paper we shall adopt the next generation matrix method for the derivation of R 0 [

F = [ 0 χ λ T μ T 0 0 ] (27)

The matrix that represent the transfer between compartments at the disease-free equilibrium given respectively by,

V = [ μ I 0 − ε V μ I μ V ] (28)

The inverse of V is given by;

V − 1 = [ 1 μ I 0 ε V μ V 1 μ V ] (29)

The next generation matrix F V − 1 is given by;

F V − 1 = [ χ ε V λ T μ V μ T χ λ T μ T μ V 0 0 ] (30)

The eigenvalues of the matrix above are; χ ε V λ T μ V μ T and 0, therefore, the reproductive number (which is the largest eigenvalue) is given by;

R 0 = χ ε V λ T μ V μ T (31)

Increasing the infection rate would lead to a rise in the number of secondary infection this is clearly depicted from the reproductive number given in Equation (31). The only way of reducing this is by introducing ARTs that target the HIV lifecycle at the entry level of the HIV virions to the CD4^{+} T-cells, that is, during fusion stage and hence the recommended drugs are the Fusion Inhibitors. Similarly, increasing the rate at which free HIV virions are generated from the infected cells would lead to an increase in the reproductive number. This means that it is paramount to bring in treatment at the budding level or to introduce ARTs drugs that would help in ensuring that the HIV virions generated from the infected CD4^{+} T-cells are defective and non-infectious. The ARTs that can play that role are the protease inhibitors. The protease inhibitors (PIs) inhibits the release of the viral protease enzyme that ensures the maturity of HIV virions upon budding from the host membrane. Consequently, the virions produced by the infected cells after the introduction of PIs are defective and non-infectious [^{+} T-cells. According to [^{+} T-cells seek, destroy, and kill the cells infected by the HIV virions. This means that if the CD8^{+} T-cells are able to fight the virions by killing the infected CD^{+} T-cells the number of secondary infection would reduce and eventually the virions maybe eliminated from the body. This shows that during the initial HIV infection stage CD8^{+} T-cells are very important as far as fighting and reducing HIV virions replication is concerned. The most fundamental thing would for researcher to establish what happens to CD8^{+} T-cells at the chronic level. Do they still fight the virus? Or probably are the CD4^{+} T-cells so worn out that they are not able to alert the CD8^{+} T-cells?

Theorem 3. The virions-free equilibrium E 0 of the model (1) is locally asymptotically stable if R 0 < 1 and unstable if R 0 > 1

Proof. In this study we have a non-linear differential equations model hence we shall use linearization method by [

J = | − μ T 0 − χ T 0 0 0 − μ I χ T 0 0 0 ε V μ I − μ V 0 0 0 − β Z 0 − μ Z 0 0 β Z 0 0 − μ Z a | (32)

Substituting Equation (26) into Equation (32) we have;

J ( E 0 ) = | − μ T 0 − χ λ T μ T 0 0 0 − μ I χ λ T μ T 0 0 0 ε V μ I − μ V 0 0 0 − β λ Z μ Z 0 − μ Z 0 0 β λ Z μ Z 0 0 − μ Z a | (33)

The characteristic equation in Λ for Equation (33) is given by

Λ 5 − b 4 Λ 4 − b 3 Λ 3 − b 2 Λ 2 − b 1 Λ + b 0 = 0 (34)

where,

b 4 = − μ Z a − μ Z + μ V − μ I + μ T

b 3 = χ λ T μ T ε V μ I − λ T μ Z a − λ T μ Z − λ T μ I − λ T μ V − μ Z a μ Z − μ Z a μ I − μ Z a μ V − μ Z μ I − μ Z μ V − μ I μ V

b 2 = λ T χ λ T μ T ε V μ I + χ λ T μ T μ Z a ε V μ I + μ Z ε V μ I χ λ T μ T − λ T μ Z a μ Z − λ T μ Z a μ I − λ T μ Z a μ V − λ T μ Z μ I − λ T μ Z μ V − λ T μ I μ V − μ Z a μ Z μ I − μ Z a μ Z μ V − μ Z a μ I μ V − μ Z μ I μ V

b 1 = λ T μ Z a ε V μ I χ λ T μ T + λ T μ Z ε V μ I χ λ T μ T + T μ Z a μ Z ε V μ I χ λ T μ T + λ T μ Z a μ Z μ I − λ T μ Z a μ Z μ V − λ T μ Z a μ I μ V − λ T μ Z μ I μ V − μ Z a μ Z μ I μ V

b 0 = μ Z a μ Z μ I λ T ( ε V λ T μ V − μ V )

The eigenvalues for the Jacobian matrix are given by; Λ 1 = − μ T , Λ 2 = − μ Z , Λ 3 = − μ Z a ,

Λ 4 = − μ I μ T − μ V μ T + 4 χ λ T μ I μ T ε V + μ I 2 μ T 2 − 2 μ I μ T 2 μ V + μ T 2 μ V 2 2 μ T ,

Λ 5 = − μ I μ T + μ V μ T + 4 χ λ T μ I μ T ε V + μ I 2 μ T 2 − 2 μ I μ T 2 μ V + μ T 2 μ V 2 2 μ T

It is evident that Λ 1 , Λ 2 , Λ 3 and Λ 5 . However we need to determine the conditions that would guarantee that Λ 4 is also negative, since for local stability all the eigenvalues must be negative.

Suppose Λ 4 < 0 , we have:

− μ I μ T − μ V μ T + 4 χ λ T μ I μ T ε V + μ I 2 μ T 2 − 2 μ I μ T 2 μ V + μ T 2 μ V 2 2 μ T < 0 , ( μ I μ T + μ V μ T ) 2 > 4 χ λ T μ I μ T ε V + μ I 2 μ T 2 − 2 μ I μ T 2 μ V + μ T 2 μ V 2 , μ V μ T > χ λ T ε V (35)

Thus, from Equation (35) we have;

χ λ T ε V μ V μ T < 1 (36)

From (36) we deduce that R 0 = χ λ T ε V μ V μ T < 1 . Thus, the virions-free equilibrium is locally asymptotically stable. ,

To analyze the endemic equilibrium, this study adopt the assumption made by [

λ T * − μ T T * − χ T * V * = 0 , χ T * V * − μ I I * − α I * Z a * = 0 , ϵ V μ I I * − μ V V * = 0 , λ Z − μ Z Z * − β Z * I * = 0 , β Z * I * − μ z a Z a * = 0 } (37)

Hence, the endemic equilibria of the model (1) correspond to the non-negative solutions of the Equation (37). Therefore, we solve the system (37) in terms of Z a * and obtain the endemic equilibrium as;

T * = λ T μ v β ( λ Z − Z a * μ z a ) μ T μ v β ( λ Z − Z a * μ Z a ) + Z a * ϵ v μ z χ μ Z a μ T , I * = Z a * μ z a μ Z β ( λ Z − Z a * μ Z a ) , V * = ϵ V μ Z μ Z a μ I Z a * β μ v ( λ Z − Z a * μ Z a ) , Z * = λ Z − Z a * μ z a μ Z . } (38)

We then obtain the following cubic polynomial that describes the existence of the possible equilibria.

p ( Z a * ) = Z a * ( Φ 2 Z a * 2 + Φ 1 Z a * + Φ 0 ) = 0, (39)

where,

Φ 2 = − 4 β μ Z a μ Z ( μ Z a μ I + α λ Z ) , Φ 1 = 4 μ Z a μ Z μ T [ − R 0 μ I μ Z a ( μ I μ Z − β λ T ) − λ T β ( μ I μ Z a + α λ Z ) ] , Φ 0 = 4 R 0 μ Z a 2 μ Z 2 μ T 2 μ V 2 ( R 0 − 1 ) . } (40)

We re-write Equation (40) to ensure that Φ 2 > 0 as;

Φ 2 = 4 β μ Z a μ Z ( μ Z a μ I + α λ Z ) , Φ 1 = − 4 μ Z a μ Z μ T [ R 0 μ I μ Z a ( μ I μ Z − β λ T ) + λ T β ( μ I μ Z a + α λ Z ) ] , Φ 0 = 4 R 0 μ Z a 2 μ Z 2 μ T 2 μ V 2 ( 1 − R 0 ) . } (41)

From Equation (39), if Z a * = 0 , then we have disease-free equilibrium treated earlier in Equation (26). The solution to the following equation defines the existence of the possible endemic equilibrium.

Φ 2 Z a * 2 + Φ 1 Z a * + Φ 0 = 0 , (42)

The two roots of the quadratic Equation (42) is given by;

Z a * = − Φ 1 ± Φ 1 2 − 4 Φ 2 Φ 0 2 Φ 0 (43)

Consequently, depending on the signs of Φ 2 , Φ 1 and Φ 0 the model (1) may have unique, two or no positive roots. We now analyze the three scenarios as follows.

Case 1:

If R 0 = 1 then Φ 0 = 0 , Φ 1 < 0 , Φ 2 > 0 ,

Φ 2 Z a * 2 − Φ 1 Z a * = 0 , Z a * ( Φ 2 Z a * − Φ 1 ) = 0 (44)

Therefore,

Z a 1 * = 0 , Z a 2 * = Φ 1 Φ 2 (45)

Z a 1 * = 0 represents the disease free case and Z a 2 * = Φ 1 Φ 2 represent a unique positive equilibrium point. This is the critical equilibrium point.

Case 2:

If R 0 > 1 then Φ 0 > 0 , Φ 1 < 0 , Φ 2 > 0 and using the signs rule Descartes, the sign of the coefficients of the quadratic Equation (39) changes once. So there is a unique positive equilibrium point, Z a 1 * > 0 . Consequently, there exist at least one endemic equilibrium point.

Case 3:

If R 0 < 1 then Φ 0 > 0 , Φ 1 < 0 and using the signs rule by Descartes, the sign of the coefficients of the quadratic Equation (31) changes twice. So there are two unique positive equilibria point. Consequently, Z a ∗ has two positive endemic turning points, implying that at R 0 < 1 there is a possibility for the model to exhibit backward bifurcation. So the existence of the threshold R c is assumed in the result.

Backward bifurcation plays a fundamental role in controlling and eradicating diseases. Backward bifurcation occurs in models that have multiple equilibria when R 0 < 1 . Consequently, having R 0 < 1 is important but not a sufficient indicator for the control and elimination of the infection [

R c = β λ T ( μ Z a μ I + α λ Z ) μ Z a μ I ( μ Z μ I − β λ T ) (46)

If R c > 1 in Equation (46) then we have a forward bifurcation and if R c < 1 then the model exhibit backward bifurcation. It is important to note that existence of a backward bifurcation with endemic equilibrium when R c < 1 is very important in epidemiological applications. Notably, it has very important consequences in the strategies and control policies designed for HIV viral eradication. From the epidemiological point of view reducing R 0 below unity is no longer a guarantee that the HIV virions will be eliminated completely or reduced to non-detectable level. In addition, this affects HIV virus control since the disease progresses even when R 0 < 1 . Furthermore, existence of backward bifurcation may result to a model that is globally unstable. From Equation (46) backward bifurcation is only possible if the rate μ I at which the infected CD4^{+} T-cells dies increases. From model (1) the infected cells may either die naturally or they can be destroyed and killed by the activated CD8^{+} T-cells. Another parameter of interest as far as backward bifurcation is concerned is the β , the rate at which the CD8^{+} T-cells are activated. Reducing β would lead to backward bifurcation. Biologically, bi-stability may lead to unexpected adverse consequences for ARTs and backward bifurcation may provide an explanation for several phenomena observed clinically among HIV patients.

It is paramount to carry out a deep discussion under epidemiological point of view. From the results it is evident that the bifurcation depends mainly on immunity of the infected person and Treatment, that is, efficacy of the ARTs. Use of ARTs as a way of managing an HIV persons may help in reducing the transmission rate. However, this may only be possible if the person adhere to the drugs. In the model the transmission rate is presented by χ . For the immunity of the infected person it is important to analyze the role played by the CD8^{+} T-cells. From (46) it can be seen clearly the immune cells plays a very vital role. Therefore, it is fundamental, for HIV eradication, to find ways in which the immunity of the infected person may be boosted. This may be done through proper diet or through use of prescribed medication. Thus it is very important to education people living with HIV/AIDS (PLWHAs) on proper nutrition and the availability on the drugs to boost the immunity.

From the point of HIV virions eradication public policy makers must work to ensure information education material (IEC) are available in all public places. They must also ensure that the drugs are accessible and available. This may play a major role in ensuring that the backward bifurcation scenario are avoided.

In summary if the backward bifurcation cannot be avoided, public policy makers have to particularly be careful since having R 0 < 1 does not guarantee that the viral load may get to non-detectable level, the disease might eventually progress to AIDS. However, from the numerical values used in model (1) results to a forward bifurcation as shown in the

Remark 2. The existence of a backward bifurcation shows that even if R 0 < 1 by some control measures, HIV may still persist. The control of HIV becomes more difficult.

Using the approach of [

Theorem 4. Suppose we can express model (1) as;

d X d t = H ( X , W ) d Z d t = G ( X , W ) (47)

such that,

G ( X , 0 ) = 0 (48)

where the column vector components of X ∈ R M denote the uninfected population and the components of W ∈ R n denote the infected population. Let E 0 = ( X * , 0 ) be the virions-free equilibrium for the system.

Then E 0 = ( X * , 0 ) is globally asymptotically stable if and only if:

1) The R 0 < 1 , that is, locally asymptotically stable.

2) d X d t = H ( X , 0 ) , X * is globally asymptotically stable.

3) G ( X , W ) = P G − G ^ ( X , W ) , G ^ ( X , W ) ≥ 0 for ( X , Z ) ∈ Ω H .

where P = D W G ( X * , 0 ) represents an M-matrix (the off diagonal elements of P are non negative) and Ω H is the feasible reqion for the model.

If model (1) satisfies the conditions mentioned above then the fixed point E 0 = ( X * , 0 ) is a globally asymptotic stable equilibrium of model system (1) provided that R 0 < R c . For model (1) the result is stated and proved in Theorem 5.

Theorem 5. The virions-free equilibrium point E 0 = ( X * , 0 ) is a globally asymptotically stable equilibrium of system (1) provided that R 0 < R c and the conditions (2) and (3) of Theorem 4 are satisfied.

Proof. From the system (1) we let X = ( T , Z , Z a ) and W = ( I , V ) , then we have;

H ( X , 0 ) = ( λ T − μ T T λ Z − μ Z − μ Z a Z a ) (49)

G ( X , W ) = P G − G ^ ( X , W ) (50)

where

P = ( − μ I χ λ T μ T ε V μ I − μ V ) (51)

and

G ^ = ( α Z a I 0 0 0 ) (52)

Since α Z a I ≥ 0 then, G ^ ( X , W ) ≥ 0 . In addition, the matrix P is an M-Matrix since all its off-diagonal elements are non-negative. This therefore, proves the global stability of the virions-free Equilibrium ( E 0 ) . That is,

X * = ( λ T μ T , 0 , 0 , λ Z μ Z , 0 ) is globally asymptotic stable equilibrium solution of d X d t = H ( X ,0 ) . Consequently, by Theorem 5, the disease free equilibrium of the model (1) is globally asymptomatically stable. ,

Theorem 5 implies that when R 0 < R c a small influx of free HIV virions into the body cells, will not lead to AIDS. The subsequent numbers of those infected cells will be less than that of their predecessors and eventually the disease maybe reduced to non-detectable level.

In order to observe the variables on the HIV model given in Equation (1) over a period of time, the study applied Matlab programming language. The initial values of the model were set as; T 0 = 1000 , I 0 = 10 , V 0 = 1 , Z 0 = 500 , Z a 0 = 10 . This section is aimed at investigating numerically the behaviour of each compartment on the onset of infection without any medical treatment. The values for the parameter are described in

^{+} T-cells reduces for the first three months and later the body immunity stabilizes and the number of the susceptible CD4^{+} T-cells increases. However, it fails to go back to the pre-infection stage. Clinicians have established that the depletion of CD4^{+} T-cells is a indication of HIV infection. Clinicians have established that the first few weeks after infection the virus is characterized by inflammatory response including extreme flu like symptoms such as swollen nodes, fever, sore

Parameters | Description | Value | Source |
---|---|---|---|

The recruitment rate of non-infected CD4^{+} T-cells produced per unit time | 10 cell/mm^{3}/day | [ | |

The rate at which the non-infected CD4^{+} T-cells decay | 0.01 day^{−1} | [ | |

The rate at which the CD4^{+} T-cells are infected by the virus | 0.000024 mm^{3} vir^{−1} day^{−1} | [ | |

The death rate of the infected CD4^{+}T-cells | 0.5 day^{−1} | [ | |

The rate in which HIV virions are generated from the infected CD4^{+}T-cells | 100 vir. cell^{−1} day^{−1} | [ | |

The death rate of the infectious virus | 3 day^{−1} | [ | |

The rate at which the infected cells are eliminated by the activated CD8^{+}T-cells | 0.02 day^{−1} | [ | |

The rate at which the CD8^{+} T-cells are produced per unit time | 20 cell/mm^{3}/day | [ | |

The death rate of the CD8^{+} T-cells | 0.06 day^{−1} | [ | |

The rate at which the CD8^{+} T-cells are activated by the presence of the virus and the infected CD4^{+} T-cells | 0.004 day^{−1} | [ | |

The rate at which the activated defence cells decay | 0.004 day^{−1} | [ |

throat, rashes, muscles and joint pains and headache. This takes place up to the forth week. In this phase the natural immune response changes to “allergy-like” immune response replica of a mild anaphylactic reaction. Due to these changes the viral replication is high, infection of the CD4^{+} T-cells is high and the activation of the CD8^{+} T-cells and B-lymphocytes rises. As a result, the amount of CD4^{+} T-cells falls very drastically. Later, due to the immune response, new CD4^{+} T-cells are generated rapidly by the thymus to replace the already infected ones and hence their level rise again as depicted by the

In ^{+} T-cells, replications take place and new HIV virions are produced. The high number of infectious virions attaches themselves to the membrane of the CD4^{+} T-cells infecting them. The cycle continues and more virions are produced hence more CD4^{+} T-cells are infected. This explains why the number of the infected CD4^{+} T-cells increases rapidly for the first 2 months as depicted by the ^{+} T-cells Adaptive Immune response. Consequently, the adaptive immune response sets in and kill most of the infected CD4^{+} T-cells causing a drastic fall on the number of infected CD4^{+} T-cells to almost nil as depicted in the ^{+} T-cells. As a result to this the infected CD4^{+} T-cells count rises but at a slower rate. After 300 days the level start to rise again.

^{+} T-cells. However, after about three days the infected CD4^{+} T-cells burst releasing infectious HIV virus. This explain why at acute stage of infection, large number of HIV virions are produced in the

patients body. With time the HIV virions continue to replicate and infecting more CD4^{+} T-cells. This explains the decline on the level of the susceptible CD4^{+} T-cells as depicted in ^{+} T-cells this is contrarily to the work done by [^{+} T-cells), a process called target cell limitation. Consequently, due to the incerase on the number of the infected CD4^{+} T-cells, a signal is sent to the CD8^{+} T-cell and consequently the cells are activated to kill the infected cells. This helps in reducing the level of the viral load in the body. The number of the CD4^{+} T-cells count begins to increase during this point, though it may never return to the pre-infection levels. It may be paramount for the patient to begin ARTs during this stage. The virus level cannot reduce to non-detectable level since it is very difficult to control the HIV virions free in circulation when not attached to the CD4^{+} T-cells.

In ^{+} T-cells during initial infection stage. From ^{+} T-cells reduces during for the first three months. This may be due to the fact that a big number of the CD8^{+} T-cells die within few weeks, leaving a reservoir of CD8^{+} memory T-cells which

are HIV-specific which persist, irrespective of the presence of antigen or CD4^{+} T-cells. In addition, many of the cells get activated to fight the virus. However, after three months the number increases gradually, this is because of the reduction of the viral load and the infected CD4^{+} T-cells.

In ^{+} T-cells. The number of the activated cells rises after the first 3 days. This correspond to the time in which the infected CD4^{+} T-cells start to increase. Most of the cells are activated to kill the infected T-cells and consequently control the viral replication. According to [^{+} T-cells plays an important role in the initiation and persistence of CD8^{+} T-cells responses. The CD8^{+} T-cell activation can lead to a number of immune responses such as antibody production, activation of phagocytic cells and direct cell killing. Therefore, the best immune response for different types of diseases is implemented by natural mechanism. CD8^{+} T-cells have been shown to express CD4^{+} T-cells receptors on their surface after activation through the T-cell receptor, allowing infection by HIV. Some researchers such as [^{+} T-cells get destroyed late in infection. From ^{+}T-cells get activated to fight the virus. This explain the exponential rise.

In this paper, we have presented an in-vivo HIV dynamics model with inclusion

of the CD8^{+} T-cells. We first showed that the key variables of the model were non-negative and bounded to ensure that it is biologically relevant. We have computed the expression for the basic reproductive number R 0 and the equilibria of the model. It is evident that the rate of infection greatly influences the basic reproductive number. The mathematical analysis of the model showed the existence of virions-free equilibrium. In addition, the system exhibits backward and trans-critical bifurcation under some restriction on parameters. This shows that having R 0 < 1 is not enough to eradicate the HIV-virions to non-detectable level. Numerical analysis were done to give more insight regarding the model. The results clearly show the introduction of HIV virions in the body without the use of ARTs does not mean that the disease is likely to persist in the body. The body have a way of reducing the HIV virions to very low level after three months of infection. This is in agreement with the biological mechanism of the HIV-cells interaction. However, as much as it is so in this study, the parameters were not varied; this means that the behavior might slightly be different between individuals. Furthermore, the simulations for the model have showed the importannce of the CD8^{+} T-cells in fighting the HIV virions. At the primary phase of HIV infection, there is an increase in the level viral load and a reduction in the population of the CD4^{+} T-cells, which reduces after three months due to the presence of the killer cells. However, as much as this study has only established the role played by the immune cells at the acute infection other researches such as [

The corresponding author acknowledge the financial support from the DAAD and the National Research Fund from the Government of Kenya.

Ngina, P.M., Mbogo, R.W. and Luboobi, L.S. (2017) Mathematical Modelling of In-Vivo Dynamics of HIV Subject to the Influence of the CD8^{+ }T-Cells. Applied Mathematics, 8, 1153-1179. https://doi.org/10.4236/am.2017.88087