# A Computational Approach to a Model for HIV and the Immune System Interaction

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## Abstract

**:**

## 1. Introduction

_{0}and discussed its stability based on the reproduction number. Wan et al. [10] discussed the behavior of the viral dynamical model and its solution. Wang et al. [11] addressed the global dynamics of HIV infection of CD$4+$ T-cells and analyzed the mathematical model. Srivastava et al. [12] presented drug therapy for HIV infection. Culshaw et al. [13] demonstrated the differential equation of HIV infection and discussed the modeling and dynamics of CD4+ T-cells.

_{0}> 1, the host cell is infected by HIV infection. They also found conditions for which the system is stable or unstable. Sarivastav and Chandra [23] proposed the dynamics of HIV and CD4+ T-cells in primary infection, also modeling for HIV infection. They analyzed stability during the infection state of the host cells of the body. They further discussed the local stability of the model and showed the results numerically. They determined the basic reproduction number R0 and discussed that when ${R}_{0}$, the host cells are free of disease; when ${R}_{0}>1$, the host cells are infected with the disease. Liu and Li [24] formulated an HIV infection model that depends on the age of infection, elapsed age of disease, and antiretroviral treatment. The model is concerned with two classes, infective (concentration of infection at birth) and AIDS (concentration of disease without birth), whose results give an approximate interval of optimal pulse and proportion of impulsivity. Ho et al. [25] discussed the rapid turnover of plasma virion and HIV-1 infection. They also demonstrated the treatment of individuals infected with HIV-1 infection, which causes AIDS. Perelson et al. [26] discussed the decay characteristics of HIV-1-infected compartments during combination therapy. Kuang [27,28] proposed the delay differential equation with applications in population dynamics. Ongun [29] implemented the Laplace Adomian decomposition method for solving a model for HIV infection of CD4+ T-cells. Yuzbasi [30] employed the numerical approach to solve the model for HIV infection of CD4+ T-cells. Khalid et al. [31] discussed the numerical solution of a model for HIV infection of CD4+ T-Cells. Merdan et al. [32] described the numerical solution of the model for HIV infection of CD4+ T-Cells. Attaullah et al. [33] studied the transmission and dynamical behavior of an HIV/AIDS epidemic model with a cure rate based on mathematical model. Ogunlaran and Noutchie [34] considered the HIV-infected model with two variables, uninfected CD4+ T-cells and incident term of the free virion. Ogunlaran and Noutchie [34] aimed to increase the concentration of uninfected CD4+ T-cells using minimal drug therapies and to stop the reproduction of infected cells. Boukari et al. [35] analyzed a discrete HIV infection model with a time delay and expressed the global stability of the model numerically. They used the backward Euler method and proved that ${E}_{0}$ is globally asymptotically stable when ${R}_{0}<1$. Li and Xiao [36] investigated the global dynamics of a virus’ immune system in order to establish the HIV load and structured treatment outages. They also addressed the global dynamics of the HIV elimination and infection cell growth model. Espindola et al. [37] investigated macrophages and their role in HIV infection. They also discussed how highly active antiretroviral therapy (HAART) affected HIV infection. Kinner et al. [38] examined the incidence of HIV, hepatitis B, and hepatitis C in older and younger adults and deduced that the incidence is lower in younger adults than in older adults. Angulo et al. [39] demonstrated that the main path of HIV-1 infection is transmitted from a mother to her child. They also found polymorphisms in human leukocyte antigen class-B (HLA-B) concerned with HIV-1 infection. Theys et al. [40] studied HIV-1 impact on host cells and their transmission. They also found a link between the evolution of the host cell and the fitness of the host cell. Hallberge et al. [41] established a developed stage of knowledge on the significance of HIV revelation between partners. They discussed that most HIV infection is transferred from one infected person (female) to another person (male) and the status of both are the main factor of this disease. Ransome et al. [42] analyzed the spread, cure, and prevention of HIV infection in social relationships. They realized that social capital is an important factor in HIV transmission from one person to another. Naidoo et al. [43] studied the care of tuberculosis (TB) and their class in those people that are already infected with HIV. Omondi et al. [44] considered a mathematical model of HIV infection and investigated the transmission between two kinds of different ages. They also showed that males are less infected than their female partners by this infection. Duro et al. [45] illustrated the CD4+ T-cell monitoring in HIV-infected people with the help of CD4+ T-cell counts. They also found the possibility of CD4+ T-cells being maintained during viral suppression by using the Kaplan–Meier technique. Mbogo et al. [46] considered the model of HIV infection with the help of the Stochastic approach and the probability of HIV when clear, which depends on drug cure rate and intracellular delay length, both of which play a vital role in HIV progression. Ghoreishi et al. [47] utilized the homotopy analysis method (HAM) to determine the solution of the HIV-infected model in the form of an infinite series. They used auxiliary parameters to adjust and control the convergence portion of the infinite series. Elaiw [48] presented the HIV model along with two categories of targeted cells, T-cells and macrophages, with an infection rate. They used Lyapunov and Lasalle principles to recognize the global stability of the infected and uninfected state. They computed the basic reproduction number “${R}_{0}$” and analyzed that when “${R}_{0}<1$” the uninfected state is globally asymptotically stable, and when “${R}_{0}>1$” the infected state is globally asymptotically stable. Ali et al. [49] computed the solution of the HIV-infected model by using the Adomian decomposition method (ADM) that illustrates the solution of ODE’s in terms of infinite series components. Yuzbasi and Karacayir [50] considered a model of HIV infection and determined the solution of the model by using the exponential Galerkin method (EGM). They used a technique of residual correction. The purpose of this technique is to reduce the error of the solution. They also showed his result numerically and compared them with numerous existing methods. Kirschner [51] studied HIV immunological dynamics employing mathematics. In the Chemotherapy of AIDS, Webb et al. [52] described the mathematical model for HIV treatment approach. Attaullah and Sohaib [53] implemented two numerical schemes, namely, continuous Galerkin–Petrov (cGP(2)) and Legendre wavelet collocation method (LWCM), for the approximate solution of the mathematical model which describes the behavior of CD4+ T-cells, infected CD4+ T-cells, and free HIV virus particles after HIV infection. They presented and analyzed the effect of constant and different variable source terms (depending on the viral load) used for the supply of new CD4+ T-cells from the thymus on the dynamics of CD4+ T-cells, infected CD4+ T-cells, and free HIV virus. Furthermore, they also solved the model using the fourth-order Runge–Kutta (RK4) method. They highlight the accuracy and efficiency of the proposed schemes with the other traditional schemes.

## 2. Main Objectives

## 3. Mathematical Formulation of HIV Model

## 4. Modified Formulation of HIV Model

#### 4.1. Uninfected Steady State

#### 4.2. Infected Steady State

#### 4.3. Reproduction Number

#### 4.4. Jacobian Matrix

#### 4.5. Stability Analysis

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

- (a)
- ${R}_{0}1$,
- (b)
- $\begin{array}{r}\left({\alpha}_{V}+\varphi -u+\frac{2{\beta}_{T}T(t)}{{T}_{max}}\right)\left(\varphi {\alpha}_{V}-\varphi {r}_{I}\psi \stackrel{\u02c9}{T}(t)-u{\alpha}_{V}-u\varphi +\frac{2{\beta}_{T}T(t){\alpha}_{V}}{{T}_{max}}+\right.\\ \left.\frac{2{\beta}_{T}\stackrel{\u02c9}{T}(t)\varphi}{{T}_{max}}+\frac{\psi \stackrel{\u02c9}{V}(t){\beta}_{T}\stackrel{\u02c9}{T}(t)}{{T}_{max}}\right)\left(-u\varphi {\alpha}_{V}+u{r}_{I}\varphi \psi \stackrel{\u02c9}{T}(t)+\frac{2{\beta}_{T}\stackrel{\u02c9}{T}(t)\varphi {\alpha}_{V}}{{T}_{max}}-\right.\\ \left.\frac{2{\beta}_{T}\stackrel{\u02c9}{T}(t{)}^{2}{r}_{I}\varphi \psi}{{T}_{max}}+\frac{\psi \stackrel{\u02c9}{V}(t){\alpha}_{V}{\beta}_{T}\stackrel{\u02c9}{T}(t)}{{T}_{max}}+{\psi}^{2}{r}_{I}\varphi \stackrel{\u02c9}{T}(t)\stackrel{\u02c9}{V}(t)+\frac{5\eta \psi {r}_{I}\varphi \stackrel{\u02c9}{V}(t)}{(1+\stackrel{\u02c9}{V}(t){)}^{2}}\right)>0\end{array}$

**Proof.**

**Theorem**

**5.**

**Proof.**

## 5. The Numerical Methods

#### 5.1. The Continuous Galerkin–Petrov Method

#### The cGP(2) Method

#### 5.2. The Classical Explicit Runge–Kutta Method

#### Numerical Results and Discussions

## 6. Comparison between the Results of Proposed Method and other Classical Methods

## 7. Conclusions

- Increasing growth rate of healthy cells, (${\beta}_{T}$), shows a decreasing effect in the population dynamics of healthy cells, while showing an increasing effect in the population dynamics of infected cells and HIV particles. All the profiles showed a decaying oscillatory behavior.
- The healthy cells and infected cells show an increasing effect, while free virus distribution shows a decreasing behavior with an increase in the values of the virus death rate (${\alpha}_{V}$).
- It is noticed that the virus particles released by infected cells ($\varphi $) show significant variations in the population distributions of healthy cells, infected cells, and the virus. By increasing the value of “$\varphi $”, the healthy cells, infected cells, and the virus increases.
- The graphical trends illustrate increased decay in distributions of all dependent variables with an increase in the death rate of infected cells (${r}_{I}$).
- The decrease in the density of healthy cells, infected cells, and free HIV particles is observed by increasing “${\alpha}_{T}$”.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Table 1.**The explanation of parameters with their values (Yuzbasi and Karacayir [50]).

Variables | Description | Values |
---|---|---|

${T}_{0}$ | Concentration of healthy cells | $0.1{\mathrm{mm}}^{-3}$ |

${I}_{0}$ | Population of infected cells | $0.0{\mathrm{mm}}^{-3}$ |

${V}_{0}$ | Dynamics of free viruses | $0.1{\mathrm{day}}^{-1}{\mathrm{mm}}^{-3}$ |

$\eta $ | Supply rate of healthy cells | $0.1{\mathrm{day}}^{-1}{\mathrm{mm}}^{-3}$ |

${\alpha}_{T}$ | Natural death rate for healthy cells | $0.02{\mathrm{day}}^{-1}$ |

${T}_{max}$ | Maximum density of healthy cells population | $1500{\mathrm{day}}^{-1}{\mathrm{mm}}^{-3}$ |

$\psi $ | Infection rate of healthy cells | $0.0027{\mathrm{day}}^{-1}$ |

$\varphi $ | Virus particles released by infected cells | $0.3{\mathrm{day}}^{-1}$ |

${\alpha}_{V}$ | Virus death rate | $2.4{\mathrm{day}}^{-1}$ |

${r}_{I}$ | Death rate of infected cells | $10{\mathrm{mm}}^{-3}$ |

${\beta}_{T}$ | Growth rate of healthy cells | $3{\mathrm{day}}^{-1}$ |

t | Runge–Kutta | LADM-Pade [29] | Bessel Coll. N = 8 [30] | PIA(1,1) [31] | MVIM [32] |
---|---|---|---|---|---|

0.2 | 0.2088006789 | 0.2088072731 | 0.2038616561 | 0.2087295073 | 0.2088080868 |

0.4 | 0.4062136749 | 0.4061052625 | 0.3803309335 | 0.4059404993 | 0.4062407949 |

0.6 | 0.7643508145 | 0.7611467713 | 0.6954623767 | 0.7635790156 | 0.7644287245 |

0.8 | 1.4138702489 | 1.3773198590 | 1.2759624442 | 1.4119543417 | 1.4140941730 |

1.0 | 2.5911951903 | 2.3291697610 | 2.3832277428 | 2.5867690583 | 2.5919210760 |

t | DTM N = 6 [33] | EGM N = 3 [50] | EGM N = 4 [50] | EGM N = 5 [50] | cGP(2)-Method |

0.2 | 0.2116480000 | 0.2722229510 | 0.2345157340 | 0.1982953765 | 0.2088064964 |

0.4 | 0.4226850000 | 0.3065308713 | 0.4201803666 | 0.4183153468 | 0.4062347843 |

0.6 | 0.8179400000 | 0.7075440591 | 0.7255920466 | 0.7603331972 | 0.7644082444 |

0.8 | 1.5462110000 | 1.5297610198 | 1.4170402360 | 1.4077147917 | 1.4140090611 |

1.0 | 2.8540530000 | 2.6678673734 | 2.5916251711 | 2.5915947135 | 2.5915094589 |

t | Runge–Kutta | LADM-Pade [29] | Bessel coll. N=8 [30] | PIA(1,1) [31] | MVIM [32] |
---|---|---|---|---|---|

0.2 | 0.0000060318 | 0.0000060327 | 0.0000062478 | 0.0000060315 | 0.0000060327 |

0.4 | 0.0000131564 | 0.0000131591 | 0.0000129355 | 0.0000131530 | 0.0000131583 |

0.6 | 0.0000212206 | 0.0000212683 | 0.0000203526 | 0.0000212101 | 0.0000212233 |

0.8 | 0.0000301728 | 0.0000300691 | 0.0000283730 | 0.0000301480 | 0.0000301745 |

1.0 | 0.0000400314 | 0.0000398736 | 0.0000369084 | 0.0000399785 | 0.0000400254 |

t | DTM N = 6 [33] | EGM N = 3 [50] | EGM N = 4 [50] | EGM N = 5 [50] | cGP(2)-Method |

0.2 | 0.0000063666 | 0.0000091673 | 0.0000058251 | 0.0000059641 | 0.0000060325 |

0.4 | 0.0000139924 | 0.0000155229 | 0.0000134051 | 0.0000131340 | 0.0000131579 |

0.6 | 0.0000226514 | 0.0000228459 | 0.0000213405 | 0.0000212682 | 0.0000212231 |

0.8 | 0.0000332836 | 0.0000318486 | 0.0000301313 | 0.0000301754 | 0.0000301764 |

1.0 | 0.0000485399 | 0.0000421057 | 0.0000400369 | 0.0000400377 | 0.0000400364 |

t | Runge–Kutta | LADM-Pade [29] | Bessel Coll. N = 8 [30] | PIA(1,1) [31] | MVIM [32] |
---|---|---|---|---|---|

0.2 | 0.0618808474 | 0.0618799602 | 0.0618799185 | 0.0618796999 | 0.0618799087 |

0.4 | 0.0382961304 | 0.0383132488 | 0.0382949349 | 0.0382939096 | 0.0382959576 |

0.6 | 0.0237057031 | 0.0243917434 | 0.0237043186 | 0.0237016917 | 0.0237102948 |

0.8 | 0.0146813143 | 0.0099672189 | 0.0146795698 | 0.0146744145 | 0.0147004190 |

1.0 | 0.0091015791 | 0.0033050764 | 0.0090993030 | 0.0090905052 | 0.0091572387 |

t | DTM N = 6 [33] | EGM N = 3 [50] | EGM N = 4 [50] | EGM N = 5 [50] | cGP(2)-Method |

0.2 | 0.0618800000 | 0.0618823466 | 0.0618790041 | 0.0618799035 | 0.0618799805 |

0.4 | 0.0383090000 | 0.0383077329 | 0.0382950148 | 0.0382947890 | 0.0382950575 |

0.6 | 0.0239200000 | 0.0237055266 | 0.0237053683 | 0.0237046061 | 0.0237047074 |

0.8 | 0.0162120000 | 0.0146708169 | 0.0146798882 | 0.0146803810 | 0.0146804932 |

1.0 | 0.0160500000 | 0.0091056907 | 0.0091009339 | 0.0091008486 | 0.0091009447 |

**Table 5.**Comparison of absolute errors for T(t) of the cGP(2) and classical methods relative to the RK4-method.

t | LADM-Pade [29] | Bessel Coll. N = 8 [30] | PIA(1,1) [31] | MVIM [32] | DTM N = 6 [33] |
---|---|---|---|---|---|

0.2 | 0.000006594223280 | 0.004939022776720 | 0.000071171576720 | 7.40792327999 × 10^{−6} | 0.002847321123280 |

0.4 | 0.000108412464641 | 0.025882741464641 | 0.000273175664641 | 2.71199353589 × 10^{−5} | 0.016471325035359 |

0.6 | 0.003204043237096 | 0.068888437837096 | 0.000771798937096 | 7.79099629040 × 10^{−5} | 0.053589185462904 |

0.8 | 0.036550389916353 | 0.137907804716353 | 0.001915907216353 | 2.23924083647 × 10^{−3} | 0.132340751083647 |

1.0 | 0.262025429366243 | 0.207967447566243 | 0.004426132066243 | 7.25885633757 × 10^{−3} | 0.262857809633757 |

t | EGM N = 3 [50] | EGM N = 4 [50] | EGM N = 5 [50] | cGP(2)-Method | |

0.2 | 0.063422272123280 | 0.025715055123280 | 0.010505302376720 | 5.81760745974 × 10^{−6} | |

0.4 | 0.099682803664641 | 0.013966691635359 | 0.012101671835359 | 2.11093478030 × 10^{−5} | |

0.6 | 0.056806755437096 | 0.038758767937096 | 0.004017617337096 | 5.74299020841 × 10^{−5} | |

0.8 | 0.115890770883647 | 0.003169987083647 | 0.006155457216353 | 1.38812210279 × 10^{−4} | |

1.0 | 0.076672183033757 | 0.000429980733757 | 0.000399523133757 | 3.14268571135 × 10^{−4} |

**Table 6.**Comparison of absolute errors for I(t) of the cGP(2) and classical methods relative to the RK4-method.

t | LADM-Pade [29] | Bessel Coll. N = 8 [30] | PIA(1,1) [31] | MVIM [32] | DTM N= 6 [33] |
---|---|---|---|---|---|

0.2 | 8.21028439 × 10^{−10} | 2.15921028439 × 10^{−7} | 3.7897156100 × 10^{−10} | 8.2102843899 × 10^{−10} | 3.3472103 × 10^{−7} |

0.4 | 2.61393801 × 10^{−9} | 2.20986061988 × 10^{−7} | 3.4860619888 × 10^{−9} | 1.8139380112 × 10^{−9} | 8.3591393 × 10^{−7} |

0.6 | 4.76223967 × 10^{−8} | 8.68077603328 × 10^{−7} | 1.0577603329 × 10^{−8} | 2.6223966711 × 10^{−9} | 1.4307224 × 10^{−6} |

0.8 | 1.03710297 × 10^{−7} | 1.79981029736 × 10^{−6} | 2.4810297362 × 10^{−8} | 1.6897026384 × 10^{−9} | 3.1107898 × 10^{−6} |

1.0 | 1.57815845 × 10^{−7} | 3.12301584505 × 10^{−6} | 5.2915845057 × 10^{−8} | 6.0158450573 × 10^{−9} | 8.5084842 × 10^{−6} |

t | EGM N = 3 [50] | EGM N = 4 [50] | EGM N = 5 [50] | cGP(2)-Method | |

0.2 | 3.1354210284 × 10^{−6} | 2.0677897156 × 10^{−7} | 6.77789715609 × 10^{−8} | 6.6752196826462 × 10^{−10} | |

0.4 | 2.3664139380 × 10^{−6} | 2.4861393801 × 10^{−7} | 2.24860619888 × 10^{−8} | 1.4924805624754 × 10^{−9} | |

0.6 | 1.6252223967 × 10^{−6} | 1.1982239667 × 10^{−7} | 4.75223966710 × 10^{−8} | 2.4821350652576 × 10^{−9} | |

0.8 | 0.1.67578970 × 10^{−6} | 4.1510297361 × 10^{−8} | 2.58970263849 × 10^{−9} | 3.6558718785258 × 10^{−9} | |

1.0 | 2.0742841549 × 10^{−6} | 5.4841549427 × 10^{−9} | 6.28415494269 × 10^{−9} | 5.0422736406875 × 10^{−9} |

**Table 7.**Comparison of absolute errors for V(t) of the cGP(2) and classical methods relative to the RK4-method.

t | LADM-Pade [29] | Bessel Coll. N = 8 [30] | PIA(1,1) [31] | MVIM [32] | DTM N = 6 [33] |
---|---|---|---|---|---|

0.2 | 8.87201671 × 10^{−7} | 9.28901670999 × 10^{−7} | 1.14750167099 × 10^{−6} | 9.3870167099 × 10^{−7} | 8.47401671 × 10^{−7} |

0.4 | 1.71183628 × 10^{−5} | 1.19553719699 × 10^{−6} | 2.22083719699 × 10^{−6} | 1.7283719699 × 10^{−7} | 1.28695628 × 10^{−5} |

0.6 | 6.86040272 × 10^{−4} | 1.38452847400 × 10^{−6} | 4.01142847400 × 10^{−6} | 4.5916715259 × 10^{−6} | 2.14296871 × 10^{−4} |

0.8 | 4.71409543 × 10^{−3} | 1.74452298399 × 10^{−6} | 6.89982298399 × 10^{−6} | 1.9104677016 × 10^{−5} | 1.53068567 × 10^{−3} |

1.0 | 5.79650268 × 10^{−3} | 2.27607680900 × 10^{−6} | 1.10738768090 × 10^{−5} | 5.5659623191 × 10^{−5} | 6.94842092 × 10^{−3} |

t | EGM N = 3 [50] | EGM N = 4 [50] | EGM N = 5 [50] | cGP(2)-Method | |

0.2 | 1.499198329 × 10^{−6} | 1.8433016709 × 10^{−6} | 9.43901670998 × 10^{−7} | 8.6688500106069 × 10^{−7} | |

0.4 | 1.160246280 × 10^{−5} | 1.1156371969 × 10^{−6} | 1.34143719699 × 10^{−6} | 1.0728542867849 × 10^{−7} | |

0.6 | 1.765284740 × 10^{−7} | 3.3482847399 × 10^{−7} | 1.09702847399 × 10^{−6} | 9.9566078337957 × 10^{−7} | |

0.8 | 1.049742298 × 10^{−5} | 1.4261229840 × 10^{−6} | 9.33322984000 × 10^{−7} | 8.2106806348695 × 10^{−7} | |

1.0 | 4.111623191 × 10^{−6} | 6.4517680900 × 10^{−7} | 7.30476809001 × 10^{−7} | 6.3432370331871 × 10^{−7} |

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**MDPI and ACS Style**

Attaullah; Zeeshan; Tufail Khan, M.; Alyobi, S.; Yassen, M.F.; Prathumwan, D.
A Computational Approach to a Model for HIV and the Immune System Interaction. *Axioms* **2022**, *11*, 578.
https://doi.org/10.3390/axioms11100578

**AMA Style**

Attaullah, Zeeshan, Tufail Khan M, Alyobi S, Yassen MF, Prathumwan D.
A Computational Approach to a Model for HIV and the Immune System Interaction. *Axioms*. 2022; 11(10):578.
https://doi.org/10.3390/axioms11100578

**Chicago/Turabian Style**

Attaullah, Zeeshan, Muhammad Tufail Khan, Sultan Alyobi, Mansour F. Yassen, and Din Prathumwan.
2022. "A Computational Approach to a Model for HIV and the Immune System Interaction" *Axioms* 11, no. 10: 578.
https://doi.org/10.3390/axioms11100578