Modeling the Interplay Between Viral and Immune Dynamics in HIV: A Review and Mrgsolve Implementation and Exploration

ABSTRACT

Since its initial discovery, HIV has infected more than 70 million individuals globally, leading to the deaths of 35 million. At present, the annual number of deaths has significantly decreased due to 75% of HIV‐positive individuals being on antiretroviral therapy. Although there is no cure yet, available treatments extend life expectancy, enhance quality of life, and reduce transmission by maintaining viral load below the detection limit of 50 copies/mL, making the individual's levels undetectable and untransmittable. HIV has attracted considerable attention in the computational modeling area, with various models having been developed with different degrees of complexity in an attempt to explain the viral dynamics of the disease. It is important to note that no single model can fully incorporate and predict all the critical factors influencing the dynamics of the disease and its response to treatments. Since the number of published models is large, the purpose of this article is to review several relevant models found in the literature that describe biologically plausible scenarios of HIV infection, including key features of disease progression with or without treatment. A total of 15 models are described, with some implemented in the mrgsolve package in R Studio and shared for the benefit of the scientific community. The modeling framework concerning HIV infection aids in identifying the most impactful parameters within the system and their implications in the model outcomes. Insights provided by these models may help in confirming targets for current and novel therapies, thereby contributing to the exploration of new strategies.

1. Introduction

Since its discovery in the 1980s, human immunodeficiency virus (HIV), the causative agent of acquired immunodeficiency syndrome (AIDS), has infected more than 70 million people and killed more than 35 million [1]. It remains a worldwide health issue and is considered an epidemic. Currently, the number of deaths per year due to the virus has decreased from 1.3 million in 2010 to 650,000, likely due to the fact that 75% of HIV patients are currently on Antiretroviral Therapy (ART) [1]. However, there remains an unmet clinical for drugs that are both more convenient and efficacious. Moreover, despite years of research, an HIV cure remains elusive and will likely require a combination of drugs targeting pathways implicated in both viral and host immune biology. Developing such drugs and identifying an efficacious, multidrug regimen requires a detailed and quantitative understanding of complex disease mechanisms such as HIV cellular reservoirs and the emergence of resistant strains.

The HIV virus belongs to the lentivirus genus and the retroviridae family, characterized by having two copies of a single RNA strand. Key enzymes such as reverse transcriptase, protease, and integrase play crucial roles in infection and replication. Replication occurs via reverse transcription, leading to a high mutation rate, thus complicating the personalization of treatment and contributing to resistance to therapeutics [2]. HIV is primarily transmitted sexually but can also be transmitted through maternal–infant exposure, percutaneous inoculations, and blood contact.

Briefly, once the virus is transmitted and the host infected, HIV first stays in the mucosal tissues spreading later into the lymphoid organs. The main target of the virus is CD4⁺ cells, including T lymphocytes, monocytes, macrophages, and dendritic cells. Once a target cell is infected, the virus replicates and new virions are released from the infected cell [2, 3].

From a clinical perspective, the progression of the disease can be divided into four phases as follows [3]:

1. The eclipse phase: starting just after transmission and including infection of the first cells and spread to the lymph nodes. This phase is completely asymptomatic and normally undetectable and lasts up to 3 weeks.

2. The acute phase: plasma levels of HIV‐RNA peak (up to 10⁸ copies/mL) and subsequently decrease as a result of host immune system activity. Seroconversion occurs, which is the appearance of the detectable presence of HIV antibodies in the blood after the infection. This phase can be asymptomatic but also symptomatic with flu‐like manifestations (fever, lymphadenopathy, rash, myalgias, malaise) and can last from a few days to several weeks.

3. The chronic phase: the viral set point is established, a progressive loss of CD4⁺ T cells occurs, and the infection progresses slowly to AIDS if not treated. The most common symptom is chronic inflammation, which usually consists of inflammation of the lymphoid system leading to, for example, tissue fibrosis of the lymph nodes. However, systemic inflammation can also be observed. Plasma levels of HIV‐RNA in this phase can vary from thousands to millions of HIV‐RNA copies per mL.

4. AIDS: viremia is high and CD4⁺ T cells greatly decrease, compromising the immune response of the host, resulting in opportunistic infections and disorders such as pneumocystis pneumonia, candidiasis, tuberculosis, meningitis, and several types of cancer.

Both humoral and cell‐mediated immune responses are activated during HIV infection. The cell‐mediated response, governed by cytotoxic T cells (CTLs), serves as the primary defense mechanism in restricting virus replication by targeting infected cells. However, in untreated patients, CTLs ultimately become ineffective, leading to disease progression. Additionally, the virus employs mechanisms such as cell‐to‐cell transmission and viral reservoirs to evade them [4]. The humoral response to HIV triggers antibody production, targeting free virus particles. However, the importance of its contribution to virus control remains unclear [4, 5]. Lastly, macrophages play a crucial role in the immune response against HIV, since they are susceptible to infection and can exhibit a dual function of both clearing and producing viruses [4].

In the absence of treatment, the disease frequently results in the death of the patient within a 10‐year period, depending on individual characteristics and on the viral load of the infection. Although there is currently no cure, with the appropriate treatment the life expectancy and quality of life of infected patients has increased to a point in which AIDS is considered a chronic disease [3, 6].

2. Current Status of HIV Treatment and Computational Modeling

Although there are several ways of detecting the virus in the host, the preferred and most widely used biomarker is HIV‐RNA in the systemic circulation in combination with CD4⁺ T‐cell count. HIV‐RNA quantifies the viral load and CD4⁺ T cells represent the immunological state of the patient. Measurement of systemic drug concentrations is also considered to evaluate patient treatment compliance and therapeutic exposure [7]. In recent years, new measurement techniques have been developed that elucidate different infected cell populations comprising viral reservoirs [8].

Current HIV treatments aim to increase longevity and quality of life while reducing viral transmission [9]. Therapies aim to keep the viral load below 50 copies/mL. ART, including antiviral agents with overlapping or complementary mechanisms, and should start as soon as possible after a positive diagnosis. Factors influencing treatment selection include the patient's overall health status, disease progression (based on CD4⁺ T cell count and viral load), and treatment resistances [10].

While past treatment frequently consisted of monotherapy, the high rate of mutation and resistances led to the development of combination treatments as the modern standard of care [10]. The most widely used combination, highly active antiretroviral therapy (HAART), consists of three drugs with complementary mechanisms of action: two NRTI (Nucleoside reverse transcriptase inhibitor) drugs plus either a NNRTI (Non‐nucleoside reverse transcriptase inhibitor) drug or an II (Integrase inhibitor) drug administered orally and simultaneously [10]. In the case of treatment failure, other combinations should be explored to restore efficacy (i.e., undetectable viral load) [6, 9, 10]. For many ART patients, detectable viremia may transiently reemerge after viral control is achieved in so‐called “blips,” since they oscillate in time. While blips are poorly understood, they may result from the release of virus from reservoirs, mutations and resistances to treatments, or activation of immune cells [11, 12].

Given the complexity of both HIV and immune system dynamics, the development and use of predictive models represents an opportunity to understand from a quantitative point of view the role of interpatient variability, and the efficacy and toxicity of different treatments on disease progression. Models are developed and selected on the basis of their ability to describe the data as well as to predict novel outcomes from additional studies, with datasets rich in biomarkers typically requiring more complex models. Such models provide the possibility of optimizing both the preclinical and clinical development of combination therapies through the generation of clinical trial simulations and evaluation of simulation outcomes, as well as the ability to test hypotheses through simulation studies. Moreover, mechanistic models may support not only treatment, but also HIV prevention (e.g., pre‐exposure prophylaxis) and the development of cure strategies, as well.

HIV has received considerable attention from the mathematical modeling area, and models with different degrees of complexity have been developed [13, 14] across multiple phases of drug discovery and development. Figure 1 provides a schematic representation of the most relevant models that can be found in the literature regarding the viral dynamics in HIV infection. Several models appearing in the recent literature attempt to incorporate disease progression by considering different cells populations, infection processes, and immune system responses over time, thus providing a model that describes the three stages (Figure 1B). Furthermore, some models focus on the low persistence of the viremia under HAART and the characterization of the previously mentioned blips (Figure 1B). Table S1 sums up the main characteristics of some models that can be found in the literature.

FIGURE 1.

(A) Schematic representation of the model categories and models that are going to be described in the review. Models highlighted in red have been implemented using the mrgsolve package in R, and are available in Appendix S2. (B) Visual representation of the disease progression showing HIV‐RNA and CD4⁺ T‐cell count in plasma and the models that can explain the different processes of the infection. (1) Basic HIV viral dynamics model, (2) quiescent cell model, (3) chronic cell model, (4) latent cell model, (5) follicular dendritic cell (FDC) model, (6) two‐population model, (7) cell‐to‐cell and virus‐to‐cell transmission model, (8) cell response immunity model, (9) humoral response immunity model, (10) mutation and variability models, (11) from infection to AIDS model. Once therapy is administered and the models that can explain the low steady‐state viremia and the blips (12) Drug sanctuary model, (13) reduced efficacy by second population of cells model, (14) model for resistances and (15) T cell and latent cell activation models.

In the literature, modeling has been used in a variety of ways to address aspects of HIV such as agent‐based models of population spread [15], machine learning models based on genotype [16], or compartmental models for disease transmission [17]. In addition, many models have been developed focusing on the time‐dependent mechanisms of viral infection and associated host immune processes. The aim of this review is to gather and synthesize the existing literature on mathematical models that describe biologically plausible scenarios of HIV infection. This includes models representing various stages of disease progression, from acute infection to chronic disease, and encompassing key features such as viral dynamics, immune response, and treatment effects. Models are presented from the simplest (the basic viral dynamics model) to more complex models, adding different cell populations and characteristics. We present a simulation analysis for the effects of treatment strategies on HIV infection dynamics, exploring the impact of ART on HIV‐RNA levels, immune response and long‐term outcomes, and comparing this with the literature. To fulfill this objective, several models have been implemented in R with the mrgsolve package. Models were implemented using mrgsolve, with equations and parameters obtained from their respective publications. In order to facilitate reproducibility, the mrgsolve scripts for some of the described models are provided on Appendix S2.

3. HIV Viral and Pharmacodynamic Models

This review encompasses a total of 15 different models found in the literature, all of which stem from the basic HIV viral dynamics model and explore various scenarios of HIV infection. The models reviewed can be divided into four categories: (1) Models that include different populations of cells susceptible to infection during HIV infection, such as chronic cells, latent cells, quiescent cells, and follicular dendritic cells, along with different mechanisms of viral infection. (2) Models that describe the effect of the immune system on the virus, aiming to reduce viral levels. These models consider cellular immunity, humoral immunity, or both. (3) After therapy is administered, the virus has different mechanisms to evade its effects, such as viral sanctuaries or infected cells with varying drug susceptibilities. This section comprises models describing these mechanisms, as well as models that could explain viral blips, such as T cell and latent cell activation. Furthermore, a model that describes all three stages of disease progression is also included in this section. (4) Finally, a concluding section discusses additional features to consider, describing various characteristics that could be incorporated into any of the aforementioned models, such as saturation of viral entry to cells, time delays, and mutations and resistances. In Figure 1A, a schematic representation of these model categories can be found.

3.1. Methodology and Model Selection

A systematic review of the literature was conducted on PubMed and Google Scholar using search terms such as “HIV”, “mathematical model”, “viral dynamics”, “ODE system”, “treatment”, “mechanistic”, “deterministic”, etc. Stochastic models were excluded from the review, focusing solely on models based on ordinary differential equations (ODEs). Only HIV‐related viral dynamics models were considered, excluding models addressing other viruses. The selected models were organized by complexity, ranging from the simplest to the most intricate, based on factors such as the number of cell types and processes considered. The models were further classified into categories based on specific features, such as the types of cells represented, immune system interactions, and models focusing on ART. Since the number of models with similar base structure and minimal differences available on literature is high repeated models were not considered.

3.2. Pharmacodynamic Models for Reverse Transcriptase Inhibitors and Protease Inhibitors

While this review focuses on viral dynamics without treatment, in many cases therapies could be added in a straightforward manner. For example, reverse transcriptase inhibitors (RTIs) prevent the conversion of viral RNA into DNA by inhibiting the reverse transcriptase enzyme, thereby reducing the infectivity of the virus. A common approach to capture this effect is to link plasma drug concentration to viral dynamics by reducing the infection rate constant by a concentration‐dependent factor ε:

$$\epsilon \left(t\right)=I_{\mathrm{MAX}}\times \frac{C{\left(t\right)}^{\gamma }}{C{\left(t\right)}^{\gamma }+\mathrm{I}{\mathrm{C}}_{50}^{\gamma }}$$

where I _MAX is the maximum achievable inhibition, C(t) is the drug concentration at time t, IC₅₀ is the concentration at which 50% of the target process is inhibited, and γ controls the steepness of the dose–response curve. On the other hand, protease inhibitors (PIs) interfere with the final stages of viral replication by inhibiting the protease enzyme, leading to the production of defective, noninfectious viral particles. This effect can similarly be incorporated by reducing the viral production rate by a factor ε. Together, these modifications extend the standard HIV models by explicitly linking drug effects to viral and cellular dynamics, enabling predictions of therapeutic efficacy and supporting optimization of treatment strategies. While a full description of HIV therapy models is beyond the scope of this review, we provide equations illustrating these interactions in Appendix S1.

3.3. Different Cell Population Models

3.3.1. Basic HIV Viral Dynamics Model

Describing HIV viral dynamics and disease progression can be a difficult task considering the complexity of the interaction between the virus and the immune system of the host. To facilitate this, several (semi‐) mechanistic computational models have been developed. The Basic Viral Dynamics model of HIV, developed by Nowak and May [18], represents the first mathematical approach that was applied to describe HIV infection and treatment effects, serving as the starting point for the incorporation of additional components. This model tracks levels of uninfected T cells (T), the virus (V) that infects them, and the (ordinary) infected (I) cells that are produced. Figure 2A shows a schematic representation of the model presented.

FIGURE 2.

Diagrams of the different HIV viral dynamics models. (A) Basic model, (B) chronic cell model, (C) latent cell model, (D) quiescent cell model, (E) follicular dendritic cell model.

The structure of the model is based on the following assumptions [13, 18, 19]: only one type of target cell is considered and its production and degradation are controlled by the zero‐order rate constant λ (cell/μL/day), and the first‐order rate constant δ _T (day⁻¹), respectively. The production of infected cells depends on the concentration of uninfected target cells and viral load and is controlled by the second order rate constant β (mL/copy/day), representing virus infectiveness. Infected cells are the result of the balance between the production and death processes, the latter governed by the first‐order rate constant δ _I (day⁻¹). The death rate of infected cells is assumed to be higher than the rate corresponding to uninfected cells. Viral load is controlled by the rate of virion production from infected cells, $\rho$ (copies/cell/day), and by the clearance rate of virus, δ _V (day⁻¹). These processes are captured with the following set of ordinary differential equations (ODEs):

$$\frac{\mathit{dT}}{\mathit{dt}}=\lambda -{\delta }_{\mathrm{T}}T-\mathit{\beta VT}$$

$$\frac{\mathit{dI}}{\mathit{dt}}=\mathit{\beta VT}-{\delta }_{\mathrm{I}}I$$

$$\frac{\mathit{dV}}{\mathit{dt}}=\mathit{\rho I}-{\delta }_{\mathrm{V}}V$$

where ρ = N _v,I . δ _I stands for the viral release from infected cells as they die and N _V,I represents the number of free virions produced per infected cell (copies/cell). In a healthy individual, the level of uninfected cells (T ₀) is assumed to be in equilibrium ranging from 500 to 1500 cell/μL [20], whereas both viral load and number of infected cells are nil. At the time the infection occurs, the value reported for V ₀, the initial viral load, is 5 × 10⁴ copies/mL [19]. The model captures the acute phase resulting from an initially high viral replication rate and peak in viremia, and the subsequent drop in CD4⁺ T cells. Viral load rapidly decreases, and a small oscillatory pattern remains until it reaches a steady state. No effect of the immune system is explicitly incorporated, and yet the viremic decrease after the first peak is captured. However, the model cannot account for the slow but continuous decrease in CD4⁺ T cells or the increase in viral load over time.

3.3.2. Chronic Cell Population Model

In addition to T cells, HIV may also infect other subpopulations of cells during disease progression. These include cells with lower CD4 expression, such as long‐lived macrophages [19]. These cells may be less susceptible to cytopathic effects, and thus may harbor the virus for long time periods, constituting HIV reservoirs and posing a major obstacle to virus eradication [21].

Such phenomena may be included in an expanded model by adding a chronic cell population, as first suggested by Perelson [19, 22] Figure 2B. This modified model assumes that chronic cells are produced at a rate governed by β·α, where the parameter α, constrained between 0 and 1, represents the fraction of T cells that pass into a chronic state. Chronic cells also produce N _C virus (copies/cell) while dying at the rate δ _C (day⁻¹). With the addition of this new population of cells, the oscillatory pattern of the progression of the disease can be captured. Moreover, this model captures the apparent biphasic decay in plasma HIV‐RNA seen in clinical studies (Figure 3A). The following equation represents the new cell population of chronic cells. The full set of equations describing this new system can be found on Appendix S1.

$$\frac{\mathit{dC}}{\mathit{dt}}=\alpha \left(1-{\epsilon }_{\mathrm{RTI}}\right)\mathit{\beta VT}-{\delta }_{\mathrm{c}}C$$

FIGURE 3.

Graph representing the aforementioned, plasma virus levels are presented in (A) and CD4⁺ T cells in (B). Dashed vertical line indicates start of therapy. Parameters used: λ = 10⁴ cells/mL/day, β = 8 × 10⁻⁷ mL/copies/day, N _VI = 100 copies/cell, N _Vc = 4.11 copies/cell, δ _T = 0.01/day, δ _I = 0.7/day, δ _v = 13/day, δ _Q = 0.001/day, δ _c = 0.07/day, α _C = 0.195, α _L = 1.5 × 10⁻⁶, a = 0.01/day, δ _L = 0.004/day.

3.3.3. Latent Cell Population Model

An alternative approach is to incorporate a population of latently infected cells. Once the virus infects target cells, a portion of these enter a latent state with integrated HIV provirus [23, 24] and do not produce virus due to the absence of active forms of the transcriptional activators needed for the transcription of HIV‐1 [25, 26]. Latent cells can be reactivated upon antigenic stimulation and resume viral production. Latent cells constitute a viral reservoir and are almost indistinguishable from uninfected T cells. They are unaffected by HIV‐specific cytopathic effects [25, 27].

Latent cells represent a major obstacle to the eradication of the disease [28]. These cells have a remarkably long half‐life, persisting for several months or years even under treatment [19] and play an important role in the steady state or chronic phase of the infection. A schematic representation of the latent reservoir model presented by Callaway and Perelson [19] is shown in Figure 2C.

As was the case in the chronic cell model, in this model latent cells arise from infected cells with first‐order rate constant α·β (with α being the factor of differentiation) and δ _L, respectively, with reported values much lower compared to other cell types in the system [19, 23]. For example, it has been suggested that the proportion of latent cells is < 0.005% of the resting T cells [23]. The reactivation process follows a first‐order process characterized by the rate constant “a” upon antigenic stimulation (not modeled explicitly). The following equation shows how this new population is described and the full set of equations can be found in Appendix S1.

$$\frac{\mathit{dL}}{\mathit{dt}}=\alpha \left(1-{\epsilon }_{\mathrm{RTI}}\right)\mathit{\beta VT}-{\delta }_{\mathrm{L}}L-\mathit{aL}$$

Although the chronic and latent models appear to be very similar, the main difference between them lies in the fact that chronic cells actively produce virus whereas latent cells do not until reactivation. The latent model also elicits a biphasic decay upon therapy, with a steeper initial decay followed by a flatter dynamic. An extension of this model can be found in Appendix S1, focusing on the decay characteristics of the latent cell population.

3.3.4. Quiescent Cell Population Model

The previously presented models do not distinguish between naïve and active CD4⁺ T cell populations; all CD4⁺ cells are infected at the same rate β. However, cells activated by antigen exposure upregulate CD25 expression, thus potentially increasing the rate of viral production. The quiescent cell model [26] includes a population of quiescent cells (Q), from which activated cells are derived upon antigenic stimulation [27]. Upon activation, quiescent cells upregulate CD25 which directly correlates with HIV infection [29]. Callaway et al. [30] proposed a model in which these characteristics are included and this is shown schematically in Figure 2D [19].

Quiescent cells are assumed to be produced at rate $\lambda$ and activated at rate θ(V + B), where V is the number of viral particles and “B” represents another, unspecified antigen. Following activation, cells proliferate at rate “S,” and turn over at rate δ _Q. The complete equations for this model are given in Appendix S1.

$$\frac{\mathit{dQ}}{\mathit{dt}}=\lambda -{\delta }_{\mathrm{Q}}Q-\theta \left(V+B\right)Q$$

This model can capture intersubject variability in the viremic peak immediately following infection through differences in the activation rate constant. T‐cell dynamics are also better captured.

Figure 3 shows representative simulations of the previously mentioned four models, where it can be seen how the chronic and latent model show a biphasic decay of viremia compared to the monophasic decay of the basic and quiescent models. Biphasic decay has been described in clinical data, suggesting the improved reliability for these two models. However, all models predict that viral extinction can be achieved with a therapy of 100% efficacy, which is yet to be demonstrated clinically.

3.3.5. Follicular Dendritic Cell (FDC) Population Model

Follicular dendritic cells (FDCs) are located in the secondary lymphoid organs and are potentially crucial reservoirs for HIV due to their ability to bind HIV‐RNA or HIV‐antibody complexes. This binding occurs on the cell surface through interactions with complement receptors or Fc receptors [31]. In total, FDCs have been estimated to harbor up to 10¹¹ viral copies in untreated patients [32]. Furthermore, despite highly active antiretroviral therapy (HAART), the virus may persist on FDCs for extended periods [33]. This persistence presents a significant challenge for viral eradication [33, 34, 35, 36].

Callaway and Perelson [19] proposed a model assuming that free virus binds to FDCs. Bound virus is represented by the term V _b, with a binding rate “b” (day⁻¹) and a dissociation rate “u” (day⁻¹). FDCs and bound virus are cleared at a constant rate δ _b. Hence, in this model viral clearance is governed by two processes (i.e., as bound and free virus), as opposed to the previous cases. The equation representing bound virus is shown below, and the the full model is described in Appendix S1.

$$\frac{d{V}_{\mathrm{b}}}{\mathit{dt}}=\mathit{bV}-u{V}_{\mathrm{b}}-{\delta }_{\mathrm{b}}{V}_{\mathrm{b}}$$

Hlavacek et al. [33, 35, 36] further studied the effect of FDCs, expanding the previous model by considering multiple viral binding sites on FDCs [33, 37]. The model considers uninfected cells, infected cells, long‐lived infected cells, latent cells and the two types of viral particles: free virus and bound virus, in order to describe the persistence of virus under HAART and low steady‐state viremia [34]. A schematic representation of the model is shown in Figure 2E. In this case, the model is built under the assumption that each viral particle can bind surface receptors at up to “n” sites per viral particle. “R” refers to the number of receptors on the cells, the crosslinking constants are represented by the terms K _x and K _−x (representing the reactions on the surface of the cell that involve the addition or removal of a receptor to the virus–receptor complex). The full set of equations is described in Appendix S1.

The inclusion of FDC in the model may help explain the biphasic decay of the viremia under therapy, the maintenance of low virus levels with HAART, and could also be an explanation for blips, since free virus can dissociate from FDCs.

3.3.6. Long‐Lived and Latent Cell Population Model

Several analyses indicate that the decline of viremia under therapy may be multiphasic [22, 38]. The reason for this is as yet unknown, but may be due to the contribution of several cell populations with different turnover rates [22]. While CD4⁺ T cells are the main target for HIV, CD4⁺ receptors are also expressed on other types of cells such as monocytes and macrophages [39]. It is believed that macrophages are potential reservoirs of the virus, despite their low levels of CD4 expression. Compared with T cells, infected macrophages live longer but produce lower levels of virus [40]. In addition, macrophages are less susceptible to viral cytopathic effects [41]. Under untreated conditions, these cells likely do not contribute significantly to viral levels. However, they may become relevant sources of virus under therapy due to their relatively slower turnover [21]. Several models have been developed to address this scenario. For example, Hill [13] built a model incorporating long‐lived cells and latently infected cells, Figure 4A. The model treats macrophages as a viral reservoir but neglects the effect of macrophages on the virus. The model consists of uninfected CD4⁺ T cells, actively infected cells, and latent cells, as well as a second target cell population with an infectivity rate β ₂. The target cell populations also differ with respect to death rates and viral production. The full set of equations for this model can be found in Appendix S1. Simulations of this model are consistent with the observed persistence of low‐level viremia under HAART (Figure 5), even when treated with a 100% effective therapy.

FIGURE 4.

Diagrams of the different HIV viral dynamics models. (A) Two‐population model, (B) cell‐to‐cell transmission model, (C) cellular immunity model, (D) cellular immunity and humoral immunity model, (E) two compartment model.

FIGURE 5.

Simulation from the model of A. Hill 95 under administration of an 100% effective RTI. Vertical dashed line indicates start of treatment and horizontal dashed line refers to the value of 50 copies/mL. Parameters are: λ = 100 cells/μL/day, β = 10⁻⁷, ρ = 1000 virus/cell, δ _T = 0.1/day, δ _I = 1/day, λ ₂ = 0.01 cells/μL/day, β ₂ = 10⁻⁷ day/ (copies/mL), ρ ₂ = 100 copies/cell, δ _T2 = 0.01/day, δ _I2 = 0.1/day, δ _v = 25/day, α = 10⁻⁴, a = 4 × 10⁻⁴/day, δ _L = 10⁻⁴/day.

3.3.7. Cell‐to‐Cell Transmission Model

While the previously presented models assume that infection arises from virus to cell transmission, virus may also be transferred via cell‐to‐cell transmission [42] through virological synapses [43], which may facilitate viral spread [44]. Cell‐to‐cell transmission may hinder antiviral immunity [45] and impede neutralization by ART [46]. Cell‐to‐cell transmission is thought to become the dominant mechanism for cellular infection in later phases and thus should be taken into account when modeling viral dynamics [47].

Models including both transmission mechanisms have been previously developed [48]. For example, the model by Wang et al. [49] considers cell‐to‐cell transmission and a latent cell population, Figure 4B. Cell‐to‐cell transmission is represented by a second order process involving infected (I) and uninfected cells (T), and an infection rate constant ω (mL/cell/day). A fraction η of cell‐to‐cell‐infected cells directly become latent. The following equation shows the incorporation of the new cell‐to‐cell transmission mechanism in the model. The full model is included in Appendix S1.

$$\frac{\mathit{dI}}{\mathit{dt}}=\left(1-\alpha \right)\left(1-{\epsilon }_{\mathrm{RTI}}\right)\mathit{\beta VT}+\left(1-\eta \right)\omega \mathrm{IT}+\mathit{aL}-{\delta }_{\mathrm{I}}I$$

Cell‐to‐cell transmission contributes significantly to overall viral transmission, consistent with its presumed role as the predominant mode of spread for HIV‐1 [47, 49]. Furthermore, when therapy is included in the model, the cell‐to‐cell transmission mechanism helps maintain low‐level viremia, preventing virus extinction. This assumes that ART has no effect upon cell‐to‐cell transmission, although some evidence suggests that HAART may have an effect [50].

3.4. Immune Dynamics Models

3.4.1. Cell‐Mediated Immunity Models

The immune system plays a crucial role in protecting against pathogens. Human immunity against viruses can be categorized into two types: cell‐mediated immunity and humoral immunity [51]. Cell‐mediated immunity involves cytotoxic T lymphocytes (CTLs), which target and eliminate infected cells, while humoral immunity involves B cells producing antibodies that attack free virions [52].

Several models have been developed including the role of CTLs in viral dynamics [53]. In these models, CTLs (E) are recruited in proportion to infected cells with a rate constant “p” and die at rate δ _E. CTLs kill infected cells at a rate “m.” The expressions below show how this is described mathematically. The full set of equations can be found on Appendix S1.

$$\frac{\mathit{dI}}{\mathit{dt}}=\left(1-\epsilon \right)\mathit{\beta VT}-{\delta }_{\mathrm{I}}I-\mathit{mEI}$$

$$\frac{\mathit{dE}}{\mathit{dt}}=\mathrm{p}\mathit{IE}-{\delta }_{E}E$$

In the model developed by Guo and Qiu [54] CTLs are studied together with latent cells and cell‐to‐cell transmission (Figure 4C) in order to explain how the latent reservoir and the second mechanism of transmission helps in evading the CTLs and explores how CTLs together with ART help in decreasing virus levels. In Appendix S1, the full set of equations for this model can be found. The model predicts that CTLs play an important role in the control of virus together with ART, suggesting that strategies that could increase CTLs together with therapy should be found in order achieve better responses. Figure 6 shows how different concentrations of CTL, obtained by altering the CTL production rate constant ρ, affects virus levels when therapy is administered, with the best results being obtained when more CTLs are produced.

FIGURE 6.

Viremia levels (A) and CTL count (B) of the model from Guo and Qiu [54] testing the effect of different production rate constants “p” of CTLs on the system. Parameters used: λ = 10⁴ cells/mL/day, β = 2.4 × 10⁻⁷, N _VI = 2000 copies/cell, δ _T = 0.03/day, δ _I = 1/day, δ _E = 0.9/day, δ _v = 23/day, ω = 10⁻⁶/day, p = 0.01/day, m = 0.0024 α = 0.5, a = 0.01/day, δ _L = 0.001/day, ε = 0.89.

3.4.2. Humoral Response‐Mediated Immunity Models

B‐cell‐mediated humoral immunity and antibody production plays an important role in controlling HIV infection [52] and should be considered when modeling.

In line with this, the work of Wang and Zou [55] and Murase et al. [56] improved the basic viral dynamics model by incorporating the humoral immune response. The new population of B cells is denoted as Z, which is produced at a constant rate “p” depending directly on the amount of free virus and B cells. B cells attack and neutralize free virus at a constant rate “b _Z”. Finally, B cells are cleared at a constant rate δ _Z. The following expression shows how this is added to the system, and the full set of equations can be found in Appendix S1.

The model is built under the assumption that B cells directly attack the virus and not by the release of antibodies into the system. Furthermore, in reality, the B‐cell response requires time, thus, it should be considered at more advanced stages of disease progression [56].

Another model by Lin et al. [57] considers both cellular and humoral immune responses as well as cell‐to‐cell transmission. Figure 4D shows a representation of the model and the mathematical expressions can be found in Appendix S1.

The authors concluded that during chronic untreated infection, higher levels of infected cells and virus are obtained when neglecting the role of the immune system. However, when CTLs are activated, a noted decrease can be seen. Furthermore, when both responses are considered, the virus levels observed are at their lowest. This scenario helps to understand the stable low viremia during the chronic phase and further studies are needed in order to see how the depletion of the immune system could help in progression to AIDS.

3.5. Differential ART Efficacy and Blips Models

3.5.1. Drug Sanctuary by a Physical Barrier Model

Despite the advances in antiretroviral therapy (ART), the persistence of the virus in reservoirs remains a significant obstacle to preventing virus eradication. One factor contributing to persistence is the inadequate drug concentrations in specific sites which reduces efficacy in those areas [58]. Reduced drug efficacy in physiologically distinct sites such as the brain, testes, or gut [59, 60], termed drug sanctuaries, with a less sensitive relationship between drug effect and viral load has been observed. Callaway and Perelson [19] proposed a model with two compartments—one main and one drug sanctuary—allowing virus transport between them. Differences in drug efficacy between compartments may hinder eradication and contribute to viral blips or the emergence of resistance. A schematic representation of the model is provided in Figure 4E.

The proposed model [19, 61] is based on the previously explained chronic model, with the same parameters together with the addition of the second compartment and consequently a second population of everything. The transport of virus between the compartments is modeled as depending on diffusion constants D ₁ and D ₂ and the difference in virus concentration, that is, D ₁·(V ₂−V ₁) for flux into the second compartment. They further define an efficacy factor ʄ, which is a relative reduced efficacy in the sanctuary site. The following expression shows the virus population of the main compartment with the transport from the sanctuary. The full set of equations is available in Appendix S1.

$$\frac{d{V}_1}{\mathit{dt}}=N_{\mathrm{I}}{\delta }_{\mathrm{I}}{I}_1+N_{\mathrm{c}}{\delta }_{\mathrm{c}}{C}_1-{\delta }_{\mathrm{V}}{V}_1+D_1\left(V_2-V_1\right)$$

Virus cannot be eradicated due to the transport between compartments, even with 100% effective therapy. Normally, levels stay under the detection limit (50 copies/mL) but there are moments where they rise above it, serving as an explanation for the blips. Those blips decay in time, suggesting that they are more likely to occur at the start of treatment [62]. The aforementioned results can be seen in the graph shown in Figure 7.

FIGURE 7.

Graph showing the low steady‐state viremia of the main compartment, blips and how differences in the relative efficacy ʄ affect the system, as reduction is augmented the blips are longer in time and less pronounced. Parameters are: λ = 10⁴ cells/mL/day, β = 8 × 10⁻⁷, N _VI = 100 copies/cell, N _v,c = 4.11 copies/cell, δ _T = 0.01/day, δ _I = 0.7/day, δ _v = 13/day, δ _C = 0.07/day, α = 0.195, D ₁ = 0.1048/day, D ₂ = 19.66/day, ε = 1.

3.5.2. Co‐Circulating Two‐Population Model

One reason for heterogeneity in intracellular concentrations could be due to specific cell types (e.g., monocyte‐derived macrophages) that are less susceptible to the effect of some antiretrovirals [63, 64]. In this case, two types of target cells cocirculate but in a single compartment with the addition that, in one, the efficacy is reduced, now expressed as the relative efficacy ʄ. Similarly, to the sanctuary model, this model from Callaway and Perelson [19] considers a differential efficacy in these two cellular populations. However, unlike the previous model, there is only one virus population. It is assumed that both cell lines have different production rates (λ), infectivity (β) and death rate constants (δ _i) but once infected, the same values are assigned to both types. A visual representation of the model in shown in Figure S1A. The mathematical expressions in this case are similar to the previous case and the full set of ODEs can be found in Appendix S1.

Similar results as shown in Figure 7 from the previous model are obtained: a low steady‐state viral load is supported by the population of target cells in which antiretroviral efficacy is reduced and several blips above the detection limit are observed with these also decreasing over time. Similar models with additional features have been also studied in literature.

3.5.3. Target T‐Cell Activation Model

Several studies have investigated the nature of blips in recent years, with Di Mascio et al. [11] concluding that blips do not increase in frequency or amplitude, are not isolated events, and inter‐blip time intervals do not follow common probability distributions [65]. Various sources for the blips have been proposed, including increased viral production due to coinfection [66], leading to antigen‐driven T‐cell proliferation and a burst of viral production. Jones and Perelson [67] developed a model for immune expansion in the presence of pathogens, incorporating the effects of pathogens into the previously explained two co‐circulating population model. Figure S1B shows a schematic representation of the model.

In the model, target T cells have an activation factor denoted as g(A) which depends on the antigen concentration and follows a Type II functional response, that is, prey consumption by a predator rises as prey density increases, eventually reaching a maximum rate. The following expressions show how it is added to the system and the activation function g(A) in which “a _a” is the maximum T‐cell activation rate, K ₄ is the half‐saturation concentration of antigen that stimulates the CD4⁺ cell response, and A the pathogen.

$$\begin{array}{cc}\frac{d{T}_1}{\mathit{dt}}={\lambda }_1+g\left(A\right)T_1-\left(1-\epsilon \right){\beta }_{1}V{T}_1-{\delta }_{\mathrm{T}1}{T}_1,& g\left(A\right)=a_a\frac{A}{A+K_4}\end{array}$$

The rest of the mathematical expressions can be found in Appendix S1. This model was able to describe the blips under HAART with realistic amplitude and duration. Thus, this shows that pathogens can produce viral blips of typical size and duration with a suitable mechanism for generating robust low steady‐state viral loads. The model is also more realistic by capturing the response of CD8⁺ T cells and CD4⁺ T cells in the presence of a pathogen, modeling in a more detailed way the immune response when compared to previously mentioned models.

3.5.4. Latent Cell Reservoir Activation Model

Clonal expansion of latent cells [66] has been described as another mechanism for blips, with two different approaches. One hypothesis is that blips originate from asymmetric division of latently infected cells [68]. An alternative explanation is that latent cells could be reactivated due to a pathogen [69].

3.5.4.1. Asymmetric Division

Asymmetric division consists of two cells that follow different fates, a situation most commonly observed in stem cells [70] but evidence has also been found in T cells [68], hypothesizing that latent cell division could result in one latent cell and in one infected cell. This occurs when a pathogen enters the system and triggers some immune response, leading to the asymmetric division of latent cells that could result in a viral blip. Rong and Perelson [71] developed a model in which asymmetric division is considered. Figure S1C shows a simple diagram of the asymmetric division and the following mathematical expression describes the dynamics of the latent reservoir.

$$\begin{array}{c}\frac{\mathit{dL}}{\mathit{dt}}=\alpha \left(1-\epsilon \right)\mathit{\beta VT}-{\delta }_{\mathrm{L}}L+f\left(t\right)\left(-a_{L}L+2a_{\mathrm{L}}{p}_{\mathrm{L}}L\right)\hfill \end{array}$$

where p _L is the probability of remaining in a latent state and 1−p _L the probability of becoming active after cell division, 2p _L latently infected cells are generated after each division, meanwhile 2(1−p _L) activated cell are generated. f(t) is the antigen‐activation function and follows a basic on–off model, taking only two values, 0 for no activation and 1 for activation. The full set of equations can be found in Appendix S1.

The model is able to describe low levels of persistent viremia. Thus, with intermittent antigen encounter, stability is maintained and viral blips are reproduced with accurate duration and amplitude. However, it is currently unknown if latent T cells follow asymmetric division.

3.5.4.2. Programmed Expansion and Contraction

As explained above, T cells can proliferate on exposure to pathogens undergoing divisions that result in the formation of effector cells [72]. Accordingly, it is assumed that latently infected cells are CD4⁺ T memory cells and under antigenic stimulation they also proliferate. Jones and Perelson [69] developed a model which shows that latent cell activation is able to explain viral blips. The model comprises resting latently infected cells which can be stimulated and complete “I” divisions by the effect of a pathogen, while its dynamics are modeled in the same way as in the T‐cell activation model. Based on that work Rong and Perelson [73] developed a model under the assumption that latent cells experience a programmed expansion and contraction in response to a specific antigen. The following equations show how it is described, with L ₀ being the resting latent cells which are activated to L _a upon antigenic stimulation.

$$\frac{d{L}_0}{\mathit{dt}}=\alpha \left(1-\epsilon \right)\mathit{\beta VT}-{\delta }_{L}L-f\left(t\right)a{L}_0+\left(1-f\left(t\right)\right)p{L}_{\mathrm{a}}$$

$$\frac{d{L}_{\mathrm{a}}}{\mathit{dt}}=f\left(t\right)\left(a{L}_0+p{L}_{\mathrm{a}}\right)-\left(1-f\left(t\right)\right)\left(\sigma +\rho \right)L_{\mathrm{a}}-a_{\mathrm{L}}{L}_{\mathrm{a}}$$

When antigen is present, latent cells are activated at a constant rate “a” and proliferate at rate “p.” There is then a contraction phase in which activated cells die at rate σ and revert to a latent state at rate ρ. Lastly, activated cells die at rate a _L. f(t) denotes the activation function which can be found in Appendix S1 with the rest of the equations from the model. The model [73] is able to generate viral blips with consistent amplitude and duration while maintaining the stability of the latent reservoir and low‐level viremia under HAART.

3.5.5. From Infection to AIDS Model

While the models introduced so far can describe some characteristics of the HIV infection process, none of them are capable of describing all of AIDS disease progression. A further model by Freda Wasserstein‐Robins [74] attempts to capture the three stages of the infection. It describes the dynamics of different T‐cell subpopulations and infected macrophages, and takes into account the immune recognition of infected cells, antibody production, adaptation and change in virus tropism, thymic aging and time‐varying parameters. Figure S1D depicts the dynamics of the system with all the population of cells and their relationships.

Cells are divided into CD4⁺ and CD8⁺ T cells, HIV‐specific and non‐HIV‐specific, effector and noneffector types. CD4⁺ cells are also divided into actively infected, latently infected and noninfected. The model further includes macrophages, which can be infected, and HIV‐specific antibodies. Noneffector cells (CD4⁺ and CD8⁺) are produced by the thymus and their production decreases with aging. A baseline stimulation of noneffector cells is assumed due to environmental antigens. However, HIV viral epitopes stimulate the production of specific CD4⁺ and CD8⁺ cells. Latently infected cells can also be stimulated by antigens which results in daughter infected cells that can be reactivated to produce virus. Also, noneffector latent cells can be reactivated. In contrast, cytotoxic CD8⁺ cells remove infected cells while antibodies remove virus.

To sum up the dynamics of the model: The virus is produced at rates N or p _im, stimulates CD4⁺ T cells and is removed by macrophages (k _m) or antibodies (k _ab). CD4⁺ cell subsets proliferate (p ₄) and differentiate (r ₄), specific CD4⁺ effector cells can stimulate Ab production, and infected cells are removed by CD8⁺ cells at rates K₈ and δK₈. Latently infected cells, actively infected cells and infected macrophages are also removed by the CD8⁺ cells that they stimulated previously. The corresponding mathematical expressions can be found in Appendix S1.

The model is able to describe the documented course of HIV infection including the first viral peak corresponding to the acute phase, establishment of the chronic phase in which the slow progression is described and finally the last peak of high viremia and low CD4 + T cells of the AIDS state. No treatment was incorporated into the model, but it could be informative and useful for future considerations.

3.6. Other Features to Consider

While the main characteristics to consider when modeling HIV dynamics have already been discussed, there are other features worth mentioning. In the next section, we will briefly describe four features we consider to be particularly important.

3.6.1. Mutations and Treatment Resistances

HIV exhibits a high degree of variability and mutations and can successfully evade the immune response of the host, mainly by avoiding the effect of CTLs [75]. Several attempts have been made to tackle this issue by modeling escape mutants from CTLs [53, 76]. The following equations show a simple example of modeling this by adding different viral epitopes [53]:

$$\begin{array}{cc}\frac{\mathit{dT}}{\mathit{dt}}=\lambda -{\delta }_{\mathrm{T}}T-T{\sum }_{i=1}^m{\beta }_i{V}_i& \frac{\mathit{dZ}}{\mathit{dt}}=\rho I_i{Z}_i-{\delta }_{\mathrm{Z}}{Z}_i\end{array}$$

where “i” is the viral epitope of the respective mutant, and Z _i is the respective concentration of CTLs against that mutant.

Another reason for treatment failure is the appearance of drug resistances [77], which can be due to two reasons: the transmission of a drug‐resistant mutant or the mutation generated during treatment [78, 79], with the most likely event being the transmission of a resistant mutant. Rong et al. [80] developed a model considering a wild‐type sensitive strain and a resistant strain, in which “s” denotes the sensitive population and “r” the resistant population, while, “u” is the rate at which the sensitive virus becomes drug‐resistant. The following ODEs show how this feature can be added.

$$\begin{array}{cc}\frac{\mathit{dT}}{\mathit{dt}}=\lambda -{\delta }_{\mathrm{T}}T-T{\beta }_{\mathrm{s}}{V}_{\mathrm{s}}-T{\beta }_{\mathrm{r}}{V}_{\mathrm{r}}& \frac{d{T}_{\mathrm{r}}}{\mathit{dt}}=\mathit{uT}{\beta }_{\mathrm{s}}{V}_{\mathrm{s}}+T{\beta }_{\mathrm{r}}{V}_{\mathrm{r}}-{\delta }_{\mathrm{Tr}}{T}_{\mathrm{r}}\end{array}$$

3.6.2. Time Delays

Most of the previous models assume that cells start producing virus right after infection without taking into account intracellular processes [81], or that the immune response also starts immediately, ignoring that adaptive immune responses need time to exert their effect upon controlling viral replication [82]. However, time delays affect the dynamics of HIV. In modeling, a delay between viral entry and latent infection or between viral entry and viral production [83] is frequently considered, and multiple delays can be included. Several models have already incorporated different delays in different processes [84].

3.6.3. Logistic Growth of T Cells

Focusing attention on the dynamics of T cells, it can be seen that all the models assume that CD4⁺ T cells are produced at a constant rate λ. It would be more realistic to assume that these cells follow a logistic growth in which there is a maximum number of T cells [85]. Several modeling studies has been conducted in HIV T‐cell dynamics considering the logistic growth of target cells [86]. The following equation shows how this is implemented in a model where T _max is the maximum number of T cells and “r” is the rate constant at which multiplication occurs [85].

$$\mathit{rT}\left(1-\frac{T}{T_{\max }}\right)$$

3.6.4. Saturation of Infectivity

Normally, it is assumed that the infection process is governed by the mass‐action principle and is expressed as βVT. Nevertheless, some studies in other pathogens suggest that the infection rate can be sigmoidal since infectivity can be saturated [87]. It has been suggested that viruses such as HIV‐1 could behave similarly, with infectivity saturating at high concentrations, giving infection rate [88]:

$$\frac{\mathit{\beta VT}}{1+\mathit{sV}}$$

With “s” being the saturation index constant, several models can be found in the literature that consider saturation of infectivity [89, 90].

These are some of the most relevant characteristics that can be included into the models when capturing the dynamics of HIV infection. Table 1 summarizes all the previously described parameters with the respective reference papers from which parameter values can be obtained.

TABLE 1.

Table summarizing the most relevant parameters from the explained models.

Parameter	Description	Units	Ref.
λ	Synthesis rate of CD4+ T cells	cell/μL/day	[18]
β	Rate of viral infectivity	mL/copies/day	[18]
δ T	First‐order rate constant of death of uninfected T cells	day−1	[18]
δ I	First‐order rate constant of death of infected T cells	day−1	[18]
N V,I	Number of virions produced per infected cell	copies/cell	[18]
δ v	First‐order rate constant of death of virus	day−1	[18]
δ C	First‐order rate constant of death of chronic T cells	day−1	[22]
N V,C	Number of virions produced per chronic cell	copies/cell	[22]
α	Rate of cells that enter the chronic/latent state	dimensionless	[19]
a	Reactivation rate of latent cells	day−1	[19]
δ L	First‐order rate constant of death of latent T cells	day−1	[19]
ε	Drug efficacy	dimensionless	[19]
B	Represents a non‐HIV antigen	dimensionless	[30]
θ	Rate of activation of quiescent cells	dimensionless	[30]
S	Proliferation of quiescent cells	cell/μL/day	[30]
δ Q	First‐order rate constant of death of quiescent T cells	day−1	[30]
b	Binding rate of virus to FDCs	day−1	[33, 34, 35, 36]
u	Dissociation rate of virus from FDCs	day−1	[33, 34, 35, 36]
δ b	Clearance of virus bounded to FDCs	day−1	[33, 34, 35, 36]
K x and K −x	Crosslinking constants of virus receptor binding on FDCs	day−1	[33, 34, 35, 36]
R	Number of receptors on FDC cells	dimensionless	[33, 34, 35, 36]
n	Number of binding sites per viral particle	dimensionless	[33, 34, 35, 36]
ω	Cell‐to‐cell infection rate constant	mL/cell/day	[49]
η	Rate of cells that enter latent state in cell‐to‐cell transmission	dimensionless	[49]
p 1	Synthesis rate constant of CTLs	day−1	[54]
δ E	First‐order rate constant of death of CTLs	day−1	[54]
m	Rate constant of the killing of infected cells by CTLs	day−1	[54]
p 2	Synthesis rate constant of B cells	day−1	[57]
δ Z	First‐order rate constant of the death of B cells	day−1	[57]
b Z	Rate constant of virus neutralization by B cells	day−1	[57]
D 1 and D 2	Rate constant of virus transport between compartments	day−1	[61]
ʄ	Factor by which efficacy is reduced	dimensionless	[61]
A	Activation factor for T‐cell activation	cell/mL/day	[67]
a a	Maximum T‐cell activation rate	day−1	[67]
K 4	Half‐saturation concentration of antigen that stimulates the CD4+ cell response	mL−1	[71]
pL	Probability of remaining in latent state	dimensionless	[71]
f(t)	Antigen‐activation function	dimensionless	[73]
N	Number of virions produced per infected cell	copies/cell	[74]
p im	Number of virions produced per infected macrophage	copies/cell/day	[74]
k m	Rate constant of virus removal by macrophages	copies/day	[74]
k ab	Rate constant of virus removal by Ab	copies/day	[74]
p 4	Proliferation rate constant of CD4+ T‐cell subsets	day−1	[74]
r 4	Differentiation rate constant of T cells	day−1	[74]
K 8	Rate constant of removal of infected cells by CD8+	mL/cell/day	[74]
δ	Coefficient of reduced immune recognition of T4	dimensionless	[74]
τ	Time delay	day−1	[84]
r	Rate constant at which multiplication happens	day−1	[85]
T max	Maximum number of T cells	cell/μL	[85]
s	Saturation index constant	cells mL/day	[91]

Note: Parameters, a brief description, units and reference paper where the parameters can be found with values are shown.

4. Conclusions

In order to faithfully describe the viral dynamics of HIV, models should be able to simulate the three stages of disease progression. Moreover, once therapy is administered and viremia levels decrease, models should also be capable of portraying the low‐level replication and low, sustained viremia under ART, with virus eradication being unlikely. Further features of relevance include is the ability to capture viral blips, emergent mutations and the appearance of treatment resistance. This article reviews some of the most relevant models from the literature with regard to their ability to describe disease progression and treatment outcomes and their inclusion of various biological characteristics and processes. The model developed by Wasserstein‐Robbins [74] best aligns with reality, since it considers different populations of target cells, effector cells, antibodies, and different processes and is able to describe the three main stages of the disease. However, it is worth noting that the model presented is quite complex and has not been tested under ART. To improve the predictive capability and scope of the model, additional features could be included such as the previously mentioned FDC population, which would yield insight into viral reservoirs and their eradication, as well as viral mutations and resistances, as described in the “other features” section.

Essential aspects of a model for HIV addressing disease progression and the effects of ART include actively infected cells, long‐lived cells and latent cells, as in the model presented by Hill [92] in which the eradication of virus is not achieved even when therapy is 100% effective. However, each of the models presented captured various important biological processes relevant for viral spread and the maintenance of low viremia levels under therapy. Of key interest is the origin of clinically observed viral blips. This review included models hypothesizing various mechanisms such as different cells and compartments serving as viral reservoirs including latent cells [93] and follicular dendritic cells [33, 35, 36] which can describe low, steady‐state viremia and the release of viral particles leading to blips.

Some of the models reviewed seem not be as relevant as the others, since outcomes do not seem to be markedly improved, as is the case with quiescent cells [30] or saturation of the infectivity process, suggesting that these processes might not be crucial and could be omitted, thus reducing the number of parameters. Lastly, models that consider differential efficacies depending on the compartment [61] or cell type [19] seem to correctly describe low viral loads and they may help to explain the appearance of resistances and mutations. Such models could be applied in drug development by including predictions of drug concentration in the “sanctuary sites” using, for example, a physiologically based pharmacokinetic (PBPK) model.

Models such as the ones presented in this review can help to address questions concerning the roles and relative importance of various mechanisms and cell types. For example, novel therapies may seek to target cellular reservoirs of HIV or enhance the killing potential of effector cells. By utilizing the calibrated models, numerical simulation studies could be used to both predict the impact of such interventions on HIV viral dynamics, as well as to identify quantitative efficacy targets for novel drugs, for example, the degree of CD8 expansion or reservoir depletion necessary to achieve the desired clinical outcome. Frequently, multiple investigational drugs are combined, for example, to suppress viral replication while also enhancing cellular killing or reactivating latent cells. Accurately estimating the impact of combined therapies is prohibitively difficult without quantitative, dynamical models such as those reviewed here. Thus, such models may help to identify the parameters that most affect viral and host immune cell dynamics, enabling effective development of treatments and possibly an HIV cure.

Conflicts of Interest

N.V.M., S.G., and J.S.F. are employees of Gilead Sciences, Foster City, CA, United States.

Supporting information

Appendix S1

Appendix S2

Figure S1

Table S1