Dynamics and Simulation of a Within-host HIV Infection Model of CD4⁺ T-Cells with Variable Source Term and Saturated Incidence

 

Chandrasekaran, E.1*, Ahmad Umar Abubakar2

1 Department of Mathematics, School of Science and Humanities, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai, India. 600062.

2 Department of Mathematics, School of Science and Humanities, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai, India. 600062.

Email:   drchandrasekarane@veltech.edu.in   

ahmadkaikai@gmail.com  

 https://orcid.org/0000-0001-9393-4184  

 https://orcid.org/0000-0001-5827-1872  
*Corresponding Author: drchandrasekarane@veltech.edu.in 

 

ABSTRACT

A nonlinear within-host HIV model of CD4+ T cells is analysed. The model incorporates a variable CD4⁺ T-cell source term and Holling type II saturated incidence. This accounts for immune exhaustion, finite receptor availability, and physiological limitations in T-cell proliferation, thereby extending classical mass-action formulations. We derive the basic reproduction number R0 and is shown to govern the dynamics of the virus. When R0<1, the infection-free equilibrium is globally asymptotically stable; when R0>1, the system exhibits a locally stable infected equilibrium representing a high level of infection. Local stability of the infection-free equilibrium is investigated via the Routh–Hurwitz criterion, while Lyapunov functional techniques are employed to establish global stability results. Existence and local stability of the infected equilibrium are also demonstrated. Numerical simulations produced clinically observed HIV dynamics. This shows that adding biologically inspired mechanisms improves the within-host HIV models' clinical interpretability and mathematical stability.

KEYWORDS: CD4⁺ T cells; HIV dynamics; Immune exhaustion; Saturated incidence; Stability analysis.

How to Cite: Chandrasekaran E., Ahmad Umar Abubakar, (2025) Dynamics and Simulation of a Within-host HIV Infection Model of CD4⁺ T-Cells with Variable Source Term and Saturated Incidence, European Journal of Clinical Pharmacy, Vol.7, No.1, pp. 9525-9533.

INTRODUCTION

Human immunodeficiency virus (HIV) continues to be a major global health challenge to date. Approximately 40 million people are living with the virus worldwide [1]. HIV specifically targets the CD4+ T cells (T cells), which are also known as T helper cells. The virus orchestrates adaptive immune responses, leading to progressive immunodeficiency leading to the development of acquired immunodeficiency syndrome (AIDS). To develop effective therapies and to forecast the disease progression, it is essential to understand the complex interplay of virus replication, T cell loss, and viral-immune interactions at the within-host level. Mathematical models have been critical in unravelling HIV pathogenesis, estimating biologically relevant parameters from clinical data, and assessing intervention strategies. The earliest deterministic within-host HIV model was presented in [2] and expresses the dynamics between healthy, infected, and HIV virions in the bloodstream. Various extensions have added other biological features, including time delays [3-5], cell-to-cell transmission [6-9], immune response [10-12], and antiretroviral treatment [14-16], which have greatly extended our knowledge of the HIV pathogenesis.

 

Even though classical models of mass-action (bilinear) incidence assumption have been successful in predicting underlying HIV resistance dynamics, they suffer from important biological constraints. A saturation of viral entry, immune clearance, and cell-virus binding are believed to have limited the maximal viral load; this is because there are only so many receptors for the virus to bind to (i.e., restricted receptor availability) or immunological resources in the form of cells (e.g., T cells; finite capacity), such that competition during binding results [16]. Most conventional within-host HIV models depend on mass-action infection. However, this assumption is biologically constraining. To address the limitations of these models, various authors have proposed models to capture this characteristic. Numerous studies have discussed SIR models with saturated incidence, as in [5], [17], [18]. This gave rise to improved HIV models with saturation, as studied by [19-25]. The production of new T cells has been observed to diminish with increasing viral load. Experimental evidence demonstrates that thymic output declines significantly with increasing viral burden, indicating immune exhaustion and lymphoid tissue damage [26]. Models that use non-variable source terms fail to capture the declining feedback between viral load and immunological regeneration, necessitating variable source term formulations wherein T-cell recruitment is mostly modelled as a decreasing function of viral load. Models with variable source terms have T cell generation as a decreasing function. Studies that worked on models with variable source terms include  [12], [13], [28-32], [35]. Several models have also incorporated multiple features, such as fusion effect, cure effect, and more, to address realism [14], [35-36].

 

Although saturated incidence and variable CD4⁺ T-cell source terms have been studied separately, few works have analysed models that integrate both mechanisms. This work extends within-host HIV modelling by incorporating additional biological mechanisms. We present the impact of saturated incidence and variable source term on the behaviour of the proposed model. The source term captures immune exhaustion, the logistic constraint enforces finite lymphoid tissue capacity, and the Holling type II incidence reflects receptor saturation at high viral loads. This model analyses the population dynamics of T cells in the presence and absence of HIV. It studies infection patterns and predicts disease progression trajectories. It provides information regarding threshold conditions for viral clearance versus persistent infection and demonstrates that saturation mechanisms reduce viral overshoot and eliminate spurious oscillations compared to mass-action models, thereby offering more reliable predictions of viral load dynamics.

 

MODEL EQUATIONS

Mathematical Model

We present three state variables describing the model dynamics: the number of healthy T cells, infected T cells, and HIV virions, denoted as , , and , respectively. We propose the nonlinear system:

with

 

 

Figure 1: Diagram of the HIV model

The term  models’ immune exhaustion, which reflects the observed suppression of T-cell production at high virus concentration. This captures the negative feedback between viral burden and thymic output observed in clinical studies [26], [32], [34]. The nonlinear terms    and    are of Holling type II form and capture the saturation of viral entry and immune-mediated viral neutralization when the viral burden becomes large [16]. The parameter  measures the half-saturation constant: when . The infection and neutralization rates reach half of their maximum values. The logistic growth term  enforces a physiological bound on the total T-cell population.

Table 1: Model Parameters and Units

 

 

Positivity of Solutions

Theorem  For any initial condition satisfying (2.2), the solution  of (2.1) satisfies

 

Proof: For H: If  while , then   

Thus,  cannot decrease from zero. Similarly, for  and .

Given that the right-hand side is locally Lipschitz in , solutions with positive initial conditions remain positive for all .

Boundedness

Let . By finding the sum of the first and second equations of (2.1), we get

                    (2.3)

Using , , and , we obtain

(2.4)

where . The scalar comparison ODE  with

has a globally attracting equilibrium . Thus  is uniformly bounded.

For the virus,

 

So,  is bounded above by the solution  , converging to .

Theorem  All trajectories of (2.1) enter and remain in the compact positively invariant set

 

for some finite .

Proof: Define . Adding the first two equations of system (2.1) and cancelling the terms yields

 

Since , , and , we obtain

 

where . The scalar comparison ODE  has a unique positive equilibrium

 

By standard comparison,  for all .

For the virus, since and ,

 

The comparison ODE  has equilibrium , so  for all .

Therefore, the compact set  is positively invariant and attracts all solutions of system (2.1).

DiseaseFree Equilibrium

Setting  in (2.1) yields

(2.5)

which is quadratic in :

 

The positive root is

                    

The infection-free equilibrium is expressed as

 

STABILITY ANALYSIS

Basic Reproduction Number

The basic reproduction number  is derived via the next-generation matrix method [35]. Following this framework, we partition the state space into infected compartments  and uninfected compartments . The next-generation matrix is constructed with respect to the infected subsystem only, since H represents the susceptible population and does not directly contribute to disease transmission.

Let  denote the full state vector. For the stability analysis, we focus on the infected subsystem , with  treated as a dynamic but non-infected variable. System (1) is expressed as follows:

where  has the terms describing new infections and  collects remaining transitions. For system (1), we identify:

Linearizing at  gives the Jacobian matrices of  and  with respect to the infected subsystem :

The inverse of  is

The next-generation matrix method is

Hence, the basic reproduction number is

Local Stability of the Disease-Free Equilibrium

Theorem  The disease-free equilibrium  is locally asymptotically stable if  and unstable if .

Proof: We evaluate the Jacobian matrix of system (1) at

The characteristic polynomial is expressed as

One eigenvalue is

Since  is the positive root of (2.5), typical parameter regimes yield , ensuring .

The remaining two eigenvalues are obtained from

which yields   where,
By the Routh–Hurwitz criterion, both roots contain nonpositive real parts if and only if  and .

Since , stability requires  which corresponds to . If , then and  is unstable.

Lemma  If , then for all t sufficiently large.

Proof: As , equation  of system (2.1) reduces to

This is a concave-down quadratic satisfying . Its unique positive root is precisely  as given by equation (2.6). Since the parabola opens downward and is its positive root,  for all . Consequently  whenever , so cannot exceed asymptotically as the infection clears. Furthermore, for any  the variable source term satisfies , which only reduces T-cell recruitment relative to the limiting equation. By comparison, for all sufficiently large .

Global Stability of the Disease-Free Equilibrium

Theorem  If , then the disease-free equilibrium  is said to be globally asymptotically stable in the invariant region .

Proof. By Lemma 3.2, when , the healthy T-cell population satisfies  for all sufficiently large . Therefore, we construct a Lyapunov function for the infected subsystem , with  bounded by .

Define

 

Differentiating  along solutions of system (2.1) yields

 

Using the bound  in , we obtain

 

this can be rewritten as

Hence, if , then  for all . Moreover,  if and only if , which implies .

By LaSalle’s invariance principle, all trajectories in  converge to the infection-free equilibrium .

 

ENDEMIC EQUILIBRIUM

When , the infection-free equilibrium becomes unstable, and the system admits a positive infected equilibrium

 

corresponding to chronic infection.

Theorem  If , then system (2.1) admits at least one infected equilibrium , with all components positive.

Proof: At equilibrium, equations (2.1) give

       (4.1)

From the second equation,

 

Substituting (4.2) into the third equation,

 

Dividing by  and multiplying through by ,

 

Solving for ,

 

Define a continuous function  by substituting (4.2) and (4.3) into the first equilibrium equation of (4.1). When , one finds  if . As , the logistic and saturation terms ensure . By the intermediate value theorem, there exists  such that . The corresponding  from (4.2)–(4.3) are positive, proving existence.

Theorem  If , the endemic equilibrium  is locally asymptotically stable.

Proof: Let  be the endemic equilibrium. Define

 

which represent the effective infection and neutralization rates at equilibrium. The Jacobian matrix evaluated at  is , where

, ,

, , ,

, , ,

From the equilibrium equation for , we have

Subtracting  from both sides gives

Clearly,

 

 

The characteristic polynomial of  is

Where
Thus,

The coefficients and are given by combinations of the Jacobian entries:

Using the positivity of and the biological feasibility of parameters, it follows that all contributing terms satisfy the required sign conditions, yielding  and . Furthermore, direct computation shows that . Hence, all Routh–Hurwitz conditions are satisfied, and the endemic equilibrium is locally asymptotically stable 

 

NUMERICAL SIMULATION

To validate analytical results derived in the previous sections, we numerically integrate system (1). Parameter values are chosen based on published HIV modelling studies and clinical data to ensure biological realism. The initial conditions are fixed as , representing a healthy immune system with baseline CD4⁺ T-cell count close to  exposed to a very small viral inoculum. These initial conditions reflect the state of an individual shortly after exposure to HIV. For the HIV-free equilibrium, we employ the parameter values, , , , , , , , , , ,, . Under these conditions, , which theoretically predicts that the infection will not establish and the system will converge to the disease-free equilibrium . This threshold condition is confirmed by numerical integration over 500 days.

 

Figure 2: Dynamics of the model at infection-free equilibrium.

The system is capable of clearing viruses successfully, with healthy CD4+ T-cells remaining close to baseline while the infected cells and viral load exhibit a transient early increase before decaying exponentially to negligible levels, eventually converging smoothly and monotonically to the infection-free equilibrium. This clearance emerges from the interplay of three main processes, each driven by: viral saturation at high levels that features poor infection efficiency, sustained immune-mediated neutralization to deplete free virus from circulation, and logistic growth that constrains runaway infection by depleting the susceptible cell pool, and guarantees that the infection cannot be sustained when  falls below unity.

To examine infected dynamics, we use  , , , , , , , , , ,,  . This yields a basic reproduction number of , which theoretically predicts that the infection-free equilibrium becomes unstable and the system will evolve toward a persistent infection state corresponding to an infected equilibrium  with all components strictly positive.

 

Figure 3: Dynamics of the model at the infected equilibrium

The system then moves to the decaying and converging periods after the acute phase, and finally reaches the endemic equilibrium. With this increased viral load, a healthy population of T cells declines from the baseline through the inhibition of CD4 recruitment and thus imprints immune exhaustion. Saturation feedback processes, harnessing the immune system and limiting infection, constrain viral growth so that infected cells and virus load decline from their maximum levels. The system finally settles down at a moderate level of CD4⁺ depletion and continues to support some amount of viral replication, which is a biologically sound representation for chronic infection without unbounded growth or sustained oscillations.

To demonstrate saturation's stabilizing role, we compare viral dynamics across varying α values while maintaining . The mass-action model () exhibits threefold higher viral overshoot, pronounced persistent oscillations, and elevated endemic burden compared to saturated formulations. Increasing saturation strength progressively reduces acute viral peaks, accelerates convergence, and lowers chronic viral loads. The baseline saturated model () produces smooth monotonic convergence without spurious oscillations, while stronger saturation () further dampens overshoot and expedites equilibration.

 

Figure 4: Effect of saturation parameter α on viral dynamics

This systematic trend demonstrates that saturation captures finite receptor availability, yielding biologically realistic dynamics absent in mass-action formulations. The saturated framework thus provides essential improvements, reduced viral overshoot, elimination of nonphysical oscillations, and superior numerical stability, which justify its adoption despite increased mathematical complexity.

 

Conclusion

A nonlinear within-host HIV infection model incorporating immune exhaustion, Holling type II saturated virus–cell interactions, and logistic regulation of T-cells has been developed and rigorously analysed. The proposed model is mathematically well posed, nonnegative, and uniformly bounded. Stability analysis confirmed that the infection-free equilibrium is locally and globally asymptotically stable. Numerical simulations support the theoretical results and reproduce biologically realistic acute-to-chronic infection dynamics. Comparative studies indicate that saturation is an important factor in maintaining stability by lowering viral overshoot, getting rid of nonphysical oscillations that come with mass-action formulations, and improving numerical robustness by limiting infection pressure at high viral loads. In general, the proposed model strikes a balance between analytical tractability and biological realism. It also provides a flexible base for future additions that include spatial effects, immune heterogeneity, therapeutic interventions, and random influences.

 

ACKNOWLEDGEMENT

The authors declare no external funding for this research.

 

References

1. “HIV.” Accessed: Feb. 09, 2026. [Online]. Available: https://www.who.int/data/gho/data/themes/hiv-aids

2. A. S. Perelson, “Modeling the Interaction of the Immune System with HIV,” in Mathematical and Statistical Approaches to AIDS Epidemiology, vol. 83, C. Castillo-Chavez, Ed., in Lecture Notes in Biomathematics, vol. 83. , Berlin, Heidelberg: Springer Berlin Heidelberg, 1989, pp. 350–370. doi: 10.1007/978-3-642-93454-4_17.

3.  R. V. Culshaw and S. Ruan, “A delay-differential equation model of HIV infection of CD4+ T-cells,” Math. Biosci., vol. 165, no. 1, pp. 27–39, May 2000, doi: 10.1016/S0025-5564(00)00006-7.

4. K. A. Pawelek, S. Liu, F. Pahlevani, and L. Rong, “A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data,” Math. Biosci., vol. 235, no. 1, pp. 98–109, Jan. 2012, doi: 10.1016/j.mbs.2011.11.002.

5. J.-Z. Zhang, Z. Jin, Q.-X. Liu, and Z.-Y. Zhang, “Analysis of a Delayed SIR Model with Nonlinear Incidence Rate,” Discrete Dyn. Nat. Soc., vol. 2008, no. 1, p. 636153, Jan. 2008, doi: 10.1155/2008/636153.

6. A. S. Perelson, D. E. Kirschner, and R. De Boer, “Dynamics of HIV infection of CD4+ T cells,” Math. Biosci., vol. 114, no. 1, pp. 81–125, Mar. 1993, doi: 10.1016/0025-5564(93)90043-A.

7. L. Wang and M. Y. Li, “Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells,” Math. Biosci., vol. 200, no. 1, pp. 44–57, Mar. 2006, doi: 10.1016/j.mbs.2005.12.026.

8. A. S. Perelson and P. W. Nelson, “Mathematical Analysis of HIV-1 Dynamics in Vivo,” SIAM Rev., vol. 41, no. 1, pp. 3–44, Jan. 1999, doi: 10.1137/S0036144598335107.

9. P. Essunger and A. S. Perelson, “Modeling HIV Infection of CD4+ T-cell Subpopulations,” J. Theor. Biol., vol. 170, no. 4, pp. 367–391, Oct. 1994, doi: 10.1006/jtbi.1994.1199.

10. K. Hattaf and N. Yousfi, “Optimal Control of a Delayed HIV Infection Model with ImmuneResponse Using an Efficient Numerical Method,” ISRN Biomath., vol. 2012, pp. 1–7, Nov. 2012, doi: 10.5402/2012/215124.

11. Denise Kirschner, “Using Mathematics to Understand HIV Immune Dynamics,” vol. 43, no. 2, 1996.

12. [Attaullah, Zeeshan, M. Tufail Khan, S. Alyobi, M. F. Yassen, and D. Prathumwan, “A Computational Approach to a Model for HIV and the Immune System Interaction,” Axioms, vol. 11, no. 10, p. 578, Oct. 2022, doi: 10.3390/axioms11100578.

13. “A model for treatment strategy in the chemotheraphy of aids.”

14. H.-F. Huo, R. Chen, and X.-Y. Wang, “Modelling and stability of HIV/AIDS epidemic model with treatment,” Appl. Math. Model., vol. 40, no. 13, pp. 6550–6559, Jul. 2016, doi: 10.1016/j.apm.2016.01.054.

15. A. L. Hill, D. I. S. Rosenbloom, F. Fu, M. A. Nowak, and R. F. Siliciano, “Predicting the outcomes of treatment to eradicate the latent reservoir for HIV-1,” Proc. Natl. Acad. Sci., vol. 111, no. 37, pp. 13475–13480, Sep. 2014, doi: 10.1073/pnas.1406663111.

16. S. Bonhoeffer, R. M. May, G. M. Shaw, and M. A. Nowak, “Virus dynamics and drug therapy,” Proc. Natl. Acad. Sci., vol. 94, no. 13, pp. 6971–6976, Jun. 1997, doi: 10.1073/pnas.94.13.6971.

17. W. R. Derrick and P. Van Den Driessche, “A disease transmission model in a nonconstant population,” J. Math. Biol., vol. 31, no. 5, pp. 495–512, May 1993, doi: 10.1007/BF00173889.

18. W. Liu, H. W. Hethcote, and S. A. Levin, “Dynamical behavior of epidemiological models with nonlinear incidence rates,” J. Math. Biol., vol. 25, no. 4, pp. 359–380, Sep. 1987, doi: 10.1007/BF00277162.

19. Attaullah, K. Zeb, and A. Mohamed, “The Influence of Saturated and Bilinear Incidence Functions on the Dynamical Behavior of HIV Model Using Galerkin Scheme Having a Polynomial of Order Two,” Comput. Model. Eng. Sci., vol. 136, no. 2, pp. 1661–1685, 2023, doi: 10.32604/cmes.2023.023059.

20. N. Bairagi and D. Adak, “Global analysis of HIV-1 dynamics with Hill type infection rate and intracellular delay,” Appl. Math. Model., vol. 38, no. 21–22, pp. 5047–5066, Nov. 2014, doi: 10.1016/j.apm.2014.03.010.

21. N. S. Rathnayaka, J. K. Wijerathna, and B. G. S. A. Pradeep, “Stability Properties and Hopf Bifurcation of a Delayed HIV Dynamics Model with Saturation Functional Response, Absorption Effect and Cure Rate,” Int. J. Anal. Appl., vol. 23, p. 92, Apr. 2025, doi: 10.28924/2291-8639-23-2025-92.

22. N.S. Rathnayaka, J.K. WIJERATHNA, B.G. Pradeep, and P.D. Silva, “Stability of a delayed HIV-1 dynamics model with Beddington-DeAngelis functional response and absorption effect with two delays,” Commun. Math. Biol. Neurosci., 2022, doi: 10.28919/cmbn/7702.

23. O. A. Odebiyi, J. K. Oladejo, S. Wasiu Omotayo, A. A. Taiwo, and O. W. Ayanrinola, “Stability and Sensitivity Analysis of HIV/AIDS Model with Saturated Incidence Rate,” TRANSPUBLIKA Int. Res. EXACT Sci., vol. 4, no. 2, pp. 68–86, Mar. 2025, doi: 10.55047/tires.v4i2.1650.

24. S. Ruan and W. Wang, “Dynamical behavior of an epidemic model with a nonlinear incidence rate,” J. Differ. Equ., vol. 188, no. 1, pp. 135–163, Feb. 2003, doi: 10.1016/S0022-0396(02)00089-X.

25. Q. Sun and L. Min, “Dynamics Analysis and Simulation of a Modified HIV Infection Model with a Saturated Infection Rate,” Comput. Math. Methods Med., vol. 2014, pp. 1–14, 2014, doi: 10.1155/2014/145162.

26. L. Rong, M. A. Gilchrist, Z. Feng, and A. S. Perelson, “Modeling within-host HIV-1 dynamics and the evolution of drug resistance: Trade-offs between viral enzyme function and drug susceptibility,” J. Theor. Biol., vol. 247, no. 4, pp. 804–818, Aug. 2007, doi: 10.1016/j.jtbi.2007.04.014.

27. R. Jan and A. Jabeen, “Solution of the HIV Infection Model with Full Logistic Proliferation and Variable Source Term Using Galerkin Scheme,” Matrix Sci. Math., 2020.

28. Attaullah and M. Sohaib, “Mathematical modeling and numerical simulation of HIV infection model,” Results Appl. Math., vol. 7, p. 100118, Aug. 2020, doi: 10.1016/j.rinam.2020.100118.

29. Attaullah, R. Drissi, and W. Weera, “Galerkin time discretization scheme for the transmission dynamics of HIV infection with non-linear supply rate,” AIMS Math., vol. 7, no. 6, pp. 11292–11310, 2022, doi: 10.3934/math.2022630.

30. A. Khurshaid, “Model of HIV Infection Dynamics with Varying Source Parameters for the Rate of T-Cell Supply Through Using Contionus Galerkin-Petrov Time Discretization Scheme,” Biomed. J. Sci. Tech. Res., vol. 50, no. 1, May 2023, doi: 10.26717/BJSTR.2023.50.007901.

31. S. Boulaaras, R. Jan, A. Khan, A. Allahem, I. Ahmad, and S. Bahramand, “Modeling the dynamical behavior of the interaction of T-cells and human immunodeficiency virus with saturated incidence,” Commun. Theor. Phys., vol. 76, no. 3, p. 035001, Mar. 2024, doi: 10.1088/1572-9494/ad2368.

32. Attaullah, S. Alyobi, and M. F. Yassen, “A study on the transmission and dynamical behavior of an HIV/AIDS epidemic model with a cure rate,” AIMS Math., vol. 7, no. 9, pp. 17507–17528, 2022, doi: 10.3934/math.2022965.

33. J. N. BHAGYA and K. S. HEMANTA, “On dynamics of a mathematical model for HIV infection with fusion effect and cure rate,” Commun. Math. Biol. Neurosci., 2020, doi: 10.28919/cmbn/5008.

34. A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard, and D. D. Ho, “HIV-1 Dynamics in Vivo: Virion Clearance Rate, Infected Cell Life-Span, and Viral Generation Time,” Science, vol. 271, no. 5255, pp. 1582–1586, Mar. 1996, doi: 10.1126/science.271.5255.1582.

35. P. Van Den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Math. Biosci., vol. 180, no. 1–2, pp. 29–48, Nov. 2002, doi: 10.1016/S0025-5564(02)00108-6.