Mathematical Analysis and Clinical Implications of an HIV Model with Adaptive Immunity

In this paper, a mathematical model describing the human immunodeficiency virus (HIV) pathogenesis with adaptive immune response is presented and studied. The mathematical model includes six nonlinear differential equations describing the interaction between the uninfected cells, the exposed cells, the actively infected cells, the free viruses, and the adaptive immune response. The considered adaptive immunity will be represented by cytotoxic T-lymphocytes cells (CTLs) and antibodies. First, the global stability of the disease-free steady state and the endemic steady states is established depending on the basic reproduction number R0, the CTL immune response reproduction number R1z, the antibody immune response reproduction number R1w, the antibody immune competition reproduction number R2w, and the CTL immune response competition reproduction number R3z. On the other hand, different numerical simulations are performed in order to confirm numerically the stability for each steady state. Moreover, a comparison with some clinical data is conducted and analyzed. Finally, a sensitivity analysis for R0 is performed in order to check the impact of different input parameters.


Introduction
e human immunodeficiency virus (HIV) is a virus that gradually weakens the immune system since it targets the principal vital immune cells. It is considered as the main cause for several deadly diseases after the resulting acquired immunodeficiency syndrome (AIDS) is reached. With 36.7 million people living with HIV, 1.8 million people becoming newly infected with HIV, and more than 1 million deaths annually, HIV becomes a major global public health issue [1].
In the last decades, many mathematical models describing HIV dynamics were developed [2][3][4][5][6][7][8][9][10][11]. With the three main dynamics compartments that are free viruses, healthy CD4 + T cells, and infected CD4 + T cells, the first viral dynamics was presented and studied in [2]. Including the exposed cells as a new fourth compartment, a modified HIV viral model was tackled in [8]. More recently, the model describing HIV viral dynamics with another fifth compartment representing the cytotoxic T-lymphocytes (CTL) cells is formulated and studied in [9]. e authors study the global stability of the endemic states and illustrate the numerical simulations in order to show the numerical stability for each problem steady state. Notice that the adaptive immunity has two main arms that are cellular and humoral responses. e first one is mediated by CTL cells that play a crucial role in the infection by killing infected cells, while the second arm is mediated by the antibodies which are proteins that are produced by B cells and are specifically programmed to neutralize the viruses [12]. In this paper, we extend the recent work [9] by incorporating to the model this other main component of the adaptive immune response. e dynamics of the HIV infection model including these two arms of the adaptive immune response will be governed by the following nonlinear system of differential equations: With the initial conditions, x(0) � x 0 , s(0) � s 0 , y(0) � y 0 , v(0) � v 0 , w(0) � w 0 , and z(0) � z 0 . In this model, x, s, y, v, w, and z denote the concentration of uninfected cells, exposed cells, infected cells, free viruses, antibodies, and CTL cells, respectively. Susceptible host cells, CD4 + T cells, are produced at a rate λ and die at a rate d 1 x and become infected by virus at a rate k 1 xv/(x + v). e exposed cells die at a rate (d 2 + k 2 )s. e infected cells increase at rate k 2 s and decay at rate d 3 y and are killed by the CTL response at a rate pyz. Free viruses are produced by infected cells at a rate ay and decay at a rate d 4 v and are killed by the antibodies at a rate qvw. Antibodies develop in response to free viruses at a rate gvw and decay at a rate hw. Finally, CTLs expand in response to viral antigen derived from infected cells at a rate cyz and decay in the absence of antigenic stimulation at a rate bz. Note that this model (1), includes the saturated rate, called the saturated mass action [11], which describes better the rate of viral infection. Such HIV viral dynamics is illustrated in Figure 1. e model (1) extends the recent work [9] by adding a new compartment which is the adaptive immune response. In addition to the mathematical analysis of this new model, we will compare our simulations with some clinical data and we will perform a sensitivity analysis of our parameters. e rest of the paper is organized as follows. e analysis of the model is described in Section 2. In Section 3, we illustrate numerical simulations and compare the model solution to some clinical data. We conclude in the last section.

Positivity and Boundedness.
For the problems dealing with cell population evolution, the cell densities should remain nonnegative and bounded. In this section, we will establish the positivity and boundedness of solutions of the model (1). First of all, for biological reasons, the parameters x 0 , s 0 , y 0 , v 0 , w 0 , and z 0 must be larger than or equal to 0. Hence, we have the following result: Proposition 1. For any initial conditions (x 0 , s 0 , y 0 , v 0 , w 0 , z 0 ), system (1) has a unique solution. Moreover, this solution is nonnegative and bounded for all t ≥ 0.
Proof. By the classical functional differential equations theory (see for instance [13], and the references therein), we can confirm that there is a unique local solution (x(t), s(t), y(t), v(t), w(t), z(t)) to system (1) in [0, t m ).
We have the following: this shows the positivity of solutions for t ∈ [0, t m ). For the boundedness of the solutions, then, we have where δ � min(d 1 , d 2 , d 3 , b). So, Similarly, let us consider therefore, where α � min(d 4 , h), then, where k 2 /(d 2 + k 2 ) is the proportion of the exposed cells to become productively infected cells, a/d 3 is the number of free virus production by an infected cell, and 1/d 4 is the average life of virus. From a biological point of view, R 0 stands for the average number of secondary infections generated by one infected cell when all cells are susceptibles. Depending on the value of this basic reproduction number R 0 ; in other words, depending on these three biological proportions, we will study the stability of the free-disease and the endemic equilibria. Indeed, it is easy to see that when R 0 > 1, system (1) has four of them. e first endemic equilibrium is E 1 � (x 1 , s 1 , y 1 , v 1 , w 1 , z 1 ), where We define the antibody immune response reproduction number by where 1/h is the average life of antibodies cells and v 1 is the number of free viruses at E 1 . For the biological significance, R w 1 represents the average number of the antibodies activated by virus when the viral infection is successful in the absence of CTL immune response.
Furthermore, we introduce the CTL immune response reproduction number given by where 1/b represents the average life of CTL cells and y 1 is the number of infected cells at E 1 . Hence, R z 1 represents the mean of CTL immune cells activated by an infected cell when the viral infection is successful in the absence of the antibody immune response. e second endemic equilibrium is where  with A � (abk 1 − λcd 4 We introduce the antibody immune competition reproduction number given by where We define the CTL immune competition reproduction number R z 3 of our model by (18) with 1/b represents the average life of CTL cells and y 3 is the number of infected cells at E 3 . Hence, R z 3 represents the average number of CTL immune cells activated by an infected cell when the viral infection is successful in the absence of the antibody immune response. e last endemic equilibrium is where We observe that the second endemic state E 2 � (x 2 , y 2 , v 2 , w 2 , z 2 ) exists when R z 1 > 1. We explain the existence of this endemic equilibrium E 2 as follows. We recall first that, in this state, both the free viruses and CTL cells are present. Assume that R 0 > 1, in the total absence of CTL immune response, the infected cell load per unit time is λd 4 (R 0 − 1)/(ad 1 + ak 1 (1 − (1/R 0 ))). Via the six equations of model (1), CTL cells are reproduced due to infected cells stimulated per unit time being (cλd 4 (R 0 − 1)/(ad 1 + ak 1 (1 − (1/R 0 )))) � cy 1 . e CTL load during the lifespan of a CTL cell is (cλd 4 ) > 1, we will have the existence of the endemic equilibrium E2. We observe also that the third endemic state We explain the existence of this endemic equilibrium E 3 as follows. We recall first that, in this state, both of the free viruses and antibodies are present. Assume that R 0 > 1, in the total absence of the antibody immune response, the viral load per unit time is Via the six equations of model (1), antibodies are reproduced due to free viruses stimulation per unit time is (gλ

Global Stability of the Disease-Free Equilibrium.
For the global stability of the disease-free equilibrium, we have the following result.

Proposition 2.
If R 0 ≤ 1, then the endemic point E f is globally asymptotically stable.
Proof. Let the following Lyapunov functional be e time derivative is given by According to LaSalle's invariance principle [14], we have lim +∞ v(t) � 0. e limit system of equations is We define Since the arithmetic mean is greater than or equal to the geometric mean, it follows that erefore, _ L ≤ 0, and the equality holds if x � x 0 and s � y � w � z � 0, which complete the proof. For the first endemic equilibrium E 1 , we have the following result.
Proof. Let the following Lyapunov functional be we have then On the other hand, we have Hence, us, this fact implies that Computational and Mathematical Methods in Medicine erefore, which implies that Since the arithmetic mean is greater than or equal to the geometric mean, it follows that and we know that R z 1 < 1 and R w 1 < 1, then _ L ≤ 0, and the equality holds when x � x 1 , y � y 1 , v � v 1 , w � w 1 , and z � z 1 . By the LaSalle invariance principle [14], the endemic point E 1 is asymptotically stable when R 0 > 1.
For the second endemic equilibrium E 2 , we have the following result. □ Proposition 4. If R 0 > 1, R z 1 > 1, and R w 2 ≤ 1, then the endemic point E 2 is globally asymptotically stable.
Proof. Let the following Lyapunov functional be then, we have We know that On the other hand, we have Computational and Mathematical Methods in Medicine is fact implies that Since the arithmetic mean is greater than or equal to the geometric mean, it follows that and we know that R w 2 < 1 which means that _ L ≤ 0, and the equality holds when x � x 2 , s � s 2 , y � y 2 , v � v 2 , w � w 2 , and z � z 2 . By the LaSalle invariance principle [14], the endemic point E 2 is asymptotically stable.
For the third endemic equilibrium E 3 , we have the following result. □ Proposition 5. If R 0 > 1, R z 3 ≤ 1, and R w 1 > 1, then the endemic point E 3 is globally asymptotically stable.
Proof. Let the following Lyapunov functional be (45) en, we have this fact implies that We know that Since the arithmetic mean is greater than or equal to the geometric mean, it follows that and we know that R z 3 < 1, then _ L ≤ 0, and the equality holds when x � x 3 , s � s 3 , y � y 3 , v � v 3 , w � w 3 , and z � z 3 . By the LaSalle invariance principle [14], the endemic point E 3 is asymptotically stable when R 0 > 1.
Finally, for the last endemic equilibrium E 4 , we have the following result. □ Proposition 6. If R 0 > 1, R z 3 > 1, and R w 2 > 1, then the endemic point E 4 is globally asymptotically stable.
Proof. Let the following Lyapunov functional be en, we have We know that    is fact implies that Since the arithmetic mean is greater than or equal to the geometric mean, it follows that which means that _ L ≤ 0, and the equality holds when x � x 4 , s � s 4 , y � y 4 , v � v 4 , w � w 4 , and z � z 4 . By the LaSalle invariance principle [14], the endemic point E 4 is globally asymptotically stable when R 0 > 1.

Comparison with the Clinical Data.
First, define the following objective function: where v(t i ) represents the virus concentration at time t i using the mathematical model (1) and v(t i ) represents the virus concentration clinical data at time t i [18]. e numerical simulations are performed and compared to three patients' data picked from [18]. e data were from the University of Washington study [7] and from the Aaron Diamond AIDS Research Center (see Table 2).
In Figure 7, the dots show the evolution of the infection during the first 120 days for the first patient [18], while the solid curve represents the numerical simulation of our suggested model. e error between the numerical simulation and the clinical data is approximately J ≈ 2.378 × 10 − 1 which indicates that the numerical simulation is a good approximation of the clinical data. Figures 8 and 9 show a comparison between the clinical data (dots) and the mathematical model (solid line), and the error is approximately J ≈ 8.43 × 10 − 2 and J ≈ 1.64 × 10 − 1 , respectively. ese three results indicate that our mathematical model can fit the clinical data of different patients for the first days of observations. However, the limit of our model is to predict a long time behavior of the infection disease.

Sensitivity Analysis.
Using the method outlined in [19], we perform a sensitivity analysis using partial rank correlation coefficients (PRCC) to identify the main drivers of the basic reproduction number R 0 . Parameters were tested within the ranges given in Table 1.    In Figure 10, we observe that a and k 1 is highly positively correlated with R 0 . However, d 4 has a strong negative correlation with R 0 . e other parameters k 2 , d 2 , and d 3 present a weak correlation with R 0 . From the biological point of view, the sensitivity analysis shows that an increase of production rate of the virus by infected cells a or an increase of the infection rate k 1 leads to an increase of the basic reproduction number R 0 . However, an increase in the clearance rate of virus d 4 leads to a significant decease of the basic reproduction number R 0 .

Conclusion
In this paper, we have presented and studied a mathematical model describing HIV viral infection with saturated rate in the presence of the adaptive immune response. is adaptive immunity is represented by CTL immune response and antibodies. By using suitable Lyapunov functionals, the global stability of each equilibrium has been established. More precisely, the disease-free equilibrium is globally asymptotically stable when the basic reproduction number is below unity (R 0 ≤ 1). Also, the endemic steady state E 1 is globally asymptotically stable when R 0 ≥ 1, R z 1 ≤ 1, and R w 1 ≤ 1. In presence of the adaptive immune response governed by competition between CTL and antibody responses, system (1) admits three infection steady states. e first infection steady state E 2 is with only the presence of CTL response which is globally asymptotically stable if R z 1 ≥ 1 and R w 2 ≤ 1. e second infection steady state E 3 is with only the presence of the antibody response which is globally asymptotically stable if R w 1 ≥ 1 and R z 3 ≤ 1. e third infection steady state is E 4 with the activation of the antibodies and the CTL response at the same time. In this case, this equilibrium E 4 is globally asymptotically stable when R w 2 ≥ 1 and R z 3 ≥ 1. In addition, different numerical simulations are performed in order to confirm the theoretical findings and to show that the adaptive immune response is responsible to reduce the viral load, increase the uninfected cells, and decrease the infected cells. Moreover, a comparison with some clinical data shows that our suggested model can be considered as a good approximation of the clinical tests especially for the first days of observation.