# Modelling the Course of an HIV Infection: Insights from Ecology and Evolution

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{0}[1], but also in evolution, e.g., the adaptive dynamics framework [2]. Arguably, this was the first time that so many mathematical approaches have been mobilized to decipher the course of an infection.

^{+ }T-cells decreases because they are the primary target of the virus. The third phase or AIDS phase is characterized by a dramatic loss in CD4

^{+ }T-cells and a strong increase of viral load (Ho et al. [11], Coombs et al. [12] showed that viral titers increase in the AIDS phase and O’Brien et al. [13], Lyles et al. [14] confirmed this trend in longitudinal studies). Clinically, the onset of AIDS is defined as the time point at which the CD4

^{+ }T-cell count in the blood falls below 200 per µL. The AIDS phase also often (but not always) coincides with a shift in the virus population and the emergence of virus strains that are able to use CXCR4 co-receptors (instead of CCR5 coreceptors) and thus a wider range of immune cells become susceptible to the virus [15,16]. Because of the fragility of their immune system (low T-cell counts), patients suffer from a variety of opportunistic infections during the AIDS phase. Furthermore, within-host virus genetic diversity tends to decrease during this phase [10,17].

**Figure 1.**

**Typical course of an HIV infection.**The top panel (inspired by [10] and [17]) shows the diversity along with the type of HIV variant that dominates as in [18]. The diversity measure shown here is Tajima’s D, which compares the average pairwise distance of a set of sequences to the number of sites that are polymorphic [17,19]. The bottom panel shows the dynamics of the viral load, in red, and the CD4

^{+ }T-cells, in blue as in [3]. The three phases of an HIV infection are stressed with different colors.

^{+ }T-cell count in the early acute phase, its slight replenishment to a constant level in the asymptomatic phase and its decrease during the AIDS phase. We refer to these typical patterns as progression to AIDS in the following modeling context.

^{+}4 T cells (which they name the “runaway” hypothesis) is only appropriate for the early stages of an infection. The “runaway” hypothesis explains the massive loss of uninfected cells in the chronic phase by homeostatic compensation and/or immune activation of CD4

^{+ }T cells, which, as [23] phrase it, would “fuel the fire by generating new susceptible cells and thus more infection” . The problem, as proved by their simple model, is that this hypothesis can only account for a depletion in the range of months and not decades as observed in most HIV infected individuals (the memory CD4

^{+ }T cell pool reaches its equilibrium too rapidly). Yates et al. [23] conclude that other processes must be at play. More generally, modeling studies have been able to shed a new light on some of the details of the infection but still somehow fail to capture the whole course (and progression) of the infection. Note however that theoretical approaches share this failure with empirical approaches. Interestingly, the inability to explain the course of an HIV infection with a simple model has diverted a lot of the modeling effort towards more specified questions, in particular simple models to estimate parameters (especially related to drug treatments). The reason for this is probably that this is the area where within-host models have proved to be the most useful to clinicians.

## 2. Population Dynamics Models

^{+ }T-cell count) can be captured with a simple target-cell limitation model [25,27]. In particular, this model has led to the estimate of virus replication rates and proves that there is substantial virus replication during the chronic phase [9,28,29]. As shown in Box 1, the model captures the population dynamics of uninfected target cells, infected cells and the virus population with differential equations. The name “target cell limitation” comes from the fact that the production and natural death of target cells lead to an equilibrium level, similar to source-sink models in physics or in ecology.

^{+ }T-cell count is not a problem per se because if its value is sufficiently low, it can be argued to correspond to the AIDS phase. The key issue is the speed at which the decrease in CD4

^{+ }T-cells occurs, which is difficult to reconcile with the slow progression of the disease (see the discussion above and [23]). Second, there is no AIDS phase, i.e., once the equilibrium densities of viral load and target cells are reached, they do not vary anymore. Furthermore, Bonhoeffer et al. [30] showed that such a model failed to explain the observed dynamics during drug treatment even in the asymptomatic phases: if there was only target-cell limitation, the virus load should be unaffected by the presence of drugs because the decrease in new infections is counterbalanced by the increased availability of susceptible cells. They concluded that another class of models is more likely to explain observations related to drug treatment. In this other class of models, the virus load does not stabilize because of the lack of cells to infect but because infected cells are actively killed by immune cells. Therefore, these models are referred to as “immune control models”. Note that these models in their simplest form also do not account for the increase of the viral load or the decrease in target cell density in the AIDS phase [25,27].

^{+ }T-cells that is interpreted as progression to the AIDS phase.

#### Box 1: The target-cell limitation model and a general scheme of HIV within-host models

_{T}and become infected at rate β. Upon infection, cells move into the I class and have a potentially increased death rate d

_{I}. Infected cells produce viruses at rate p. Viruses are removed from the system at rate c. Flow diagrams are a useful tool to illustrate these dynamics. By either solving the system of equations analytically when possible or using numerical methods, we can predict the behavior of densities of viruses and target cells. Stafford et al. [33] used this model to estimate the model parameters by fitting the model to viral load data of 10 HIV patients. The figure shows the model dynamics observed when using these estimates (see Table 1).

_{j}, j =1 ..., n) infecting m different target cell populations (T

_{i}, i =1 ..., m) leading to infected cells of type i with virus j (I

_{i,j}). The influx of new target cells is captured by a function b

_{i}(T

_{i}, I

_{ij}), their death by a function d

_{T}

_{i}(T

_{i}) and the infection of the cells and thus their transition into the infected class by a function β(T

_{i}, V

_{j}, I

_{ij}). Target cells of type i infected with viral strain j die a natural death according to d

_{I}

_{i}(I

_{ij}). However, this death rate can be increased by the immune response (S

_{I}(I

_{ij}, V

_{j}, T

_{i}) or ST (I

_{ij}, V

_{j}, T

_{i})). Type j viruses are produced from infected cells according to p(I

_{i,j}) and vanish by death, infection of target cells or different immune functions (Y

_{j}) captured by c(V

_{j}, T

_{i}, Y

_{j}). The change in the density of each of the cell and virus types is described by a separate differential equation:

**Table 1.**

**Mathematical notations used.**This table summarizes all the notations used with their biological description. For parameters, we provide a typical value when it has been estimated (these values are used to obtain the figures). v indicates a variable and f a function of several variables. Note that for Box 3, Nowak et al. [34] do not give units for their rates so we used a dimension week

^{−}

^{1 }(other dimension such as day

^{−}

^{1 }or year

^{−}

^{1 }did not make sense). We also have to point out that some of the parameters used here are still under debate (for instance, recent estimates of the virion clearance rate lead to rates of 5 to 500 day

^{−}

^{1 }[35]).

Symbol | Description | Value |
---|---|---|

NOTATIONS USED IN BOX 1 | Parameters from [33] | |

T | Density of susceptible target cells | v, T_{0} = 10 cells · µL^{−1} |

I | Density of cells infected by the virus | v, I_{0} = 0 cells · µL^{−1} |

V | Density of free viruses | v, V_{0} = 10^{−}^{6} virions · µL^{−1} |

b | Input rate of target cells | f, 0.17 cells · µL^{−1} · day^{−1} |

β | Infection rate of target cells by free viruses | 6.5 × 10^{−}^{4} µL · virion^{−1} · day^{−1} |

d_{T} | Death rate of uninfected cells | 0.01 day^{−1} |

d_{I} | Death rate of infected cells | 0.39 day^{−1} |

p | Virus production rate of infected cells | 850 virions · cell^{−1} · day^{−1} |

c | Clearance rate of free viruses | 3 day^{−1} |

NOTATIONS USED IN BOX 2 | Parameters from [21] | |

Q | Density of quiescent target cells | v, Q_{0} = 188 cells · µL^{−1} |

r | Proliferation rate of activated target cells | 1 day^{−1} |

T_{tot} | = Q + T + I, Total T-cell count | f, T_{0} = 12 cells · µL^{−1}, I_{0} = 0 cells · µL^{−1} |

T_{max} | Maximal T-cell number | 1200 cells · µL^{−1} |

V | Density of free viruses | v, V_{0} = 10^{−}^{6} virions · µL^{−1} |

α_{Q} | Activation rate of quiescent T-cells | 0.1 − 1 day^{−1} |

d_{Q} | Death rate of quiescent T-cells | 0.001 day^{−1} |

β | see above | 1.35 × 10^{−}^{3} µL · virion^{−1} · day^{−1} |

γ | Virus induced depletion rate of activated T-cells | 5.6 × 10^{−}^{3} µL · virion^{−1} · day^{−1} |

d_{I} | See above | 0.5 day^{−1} |

p | See above | 100 virion · cell^{−1} · day^{−1} |

c | See above | 3 day^{−1} |

NOTATIONS USED IN BOX 3 | Parameters from [34] | |

n | Number of virus strains in the host | v, n_{0} = 1 strain |

Y_{i} | Density of immune cells specific to virus strain i | v, Yi(0) = 1 cell · µL^{−1} |

Z | Density of non-specific immune cells | v, Z(0) = 1 cell · µL^{−1} |

r_{V} | Virus replication rate | 5 virion^{−1} · week^{−1} |

c_{1} and c_{2} | Activation rates of immune cells | 1 cell · virion^{−1} · week^{−1} |

u_{1} and u_{2} | Killing rate of immune cells by viruses | 1 virion^{−1} · week^{−1} |

k_{1} | Killing rate of viruses by specific immune cells | 5 cell^{−1} · week^{−1} |

k_{2} | Killing rate of viruses by non-specific immune cells | 4.5 cell^{−}^{1} · week^{−1} |

d_{1} and d_{2} | Baseline death rates of immune cells | 0 |

^{+ }T-cells by a virus-dependent production and proliferation rate of T-cells. Virions are produced by infected T-cells but also by another source of infected cells. This model accounts for the AIDS phase only by increasing the production rate of the non-T-cell based source during disease progression (which again seems to account for evolutionary changes in the virus population).

^{+ }T-cells alone. However, their activated T-cell model, in which the T-cells are quiescent but can be activated at a certain rate, and where only activated T-cells can be infected, can account for the progression to AIDS if the activation level and/or the viral infection rate increases over time. In their immune-control model, in which CTL effector cells can kill infected cells, progression to AIDS is achieved by changing the activation/proliferation rate over time. Further details about their activated T-cell model can be found in Box 2.

^{+ }and CD8

^{+ }T-cells and the rapid timescale of the turnover of activated CD4

^{+ }and CD8

^{+ }T-cells. In addition, they add antigenic stimulation, i.e., the transition from the resting to the activated class, using a simple random process. This model predicts the general trends in disease progression but also accounts for variability of disease outcomes observed amongst patients. The main drivers of these difference are the efficacy of anti-HIV cytotoxic T lymphocyte responses, the overall viral pathogenicity and additional (non-specified) random effects. In addition, this model is able to predict a variety of responses to anti-viral therapy.

#### Box 2: Activated T-cell model

_{Q}and die at rate d

_{Q}. Activated T-cells proliferate at maximal rate r but new cells are born into the quiescent T-cell class. Activated T-cells become infected at rate β. All the virus-induced depletion of activated CD4

^{+ }T-cells is incorporated into the virus induced depletion rate γ. Infection of T-cells and virus dynamics are the same as in Equation 1b and 1c. Thus the following system of differential equations describes the activated T-cell model with density dependent proliferation:

_{tot}/T

_{max}) acts as a density dependent regulation of the proliferation rate with total T-cell number T

_{tot}= Q + T + I and maximal T-cell number T

_{max}.

**A**shows that the model with constant parameters cannot predict the progression to AIDS. Theoretically, the maximum T-cell count, the additional depletion γ, the infection rate β and/or the activation rate of quiescent T-cells α

_{Q}can change over time. However, partial T-cell loss and thus the progression to AIDS is best explained in this framework with increasing activation rate α

_{Q}(

**B**and

**C**). In

**B**, the activation rate is increased by tenfold 400 days after initial infection and in

**C**, the activation rate increases linearly over time. All parameter values are taken from [21].

## 3. Evolution Models

^{+ }T-cell dynamics are not considered), the authors even manage to analytically derive the maximum number of virus strains that an immune system can control, whence the current name of this model: the diversity threshold model. The main factor which drives the progression to AIDS in this model is the asymmetry between the ability of viruses to infect cells and of the immune system to kill viruses: virions can infect all types of target cells but each type of immune system cell can only recognize one particular viral strain. Therefore, each CD4

^{+ }target cell has a very small chance to recognize its specific epitope but a high chance of becoming infected. This model by [34] changed the status of ecology and evolution of infectious diseases because for the first time it was argued that evolutionary dynamics could explain the clinical course of an infection. Furthermore, the model leads to patterns that match experimental observations quite well despite relying on only few simple assumptions. Note however that the timing of the onset of AIDS strongly depends on the initial conditions of the model [46].

#### Box 3: Diversity threshold model

_{1}and u

_{2}). The model is based on 2 n +1 equations (n being the number of HIV strains): one equation for the density of each virus strain (V

_{i}), one equation for the density of each clone of immune cells recognizing this strain (Y

_{i}) and a final equation for the density of non-specific immune cells that can target any virus strain (Z) (see Table 1 for further description of the parameters):

_{i}that a cell infected by strain i produces a new virus strain. The different panels differ in the mutation rates (

**A**: µ =1.2 × 10

^{−}

^{3 },

**B**and

**C**: µ =1.6 × 10

^{−}

^{3}) and the initial conditions (

**A**and

**B**have the same initial viral load and immune density whereas these are ten times smaller in

**C**). The sudden exponential increase in the viral load marks the onset of AIDS (note that the time scales differ in the panels).

**B**and

**C**). (iii) The diversity dynamics do not match those observed in vivo. However, this last point of criticism is probably the most fragile because diversity in this model is based on “strains” and not on genetic distances. In order to compare the model to data, it would be necessary to quantify HIV diversity by estimating the number of different epitopes that activate immune responses at a given time.

_{V}. Equations 2a and 2b are omitted and thus it follows p(I

_{ij}, V

_{j})= r

_{V}V

_{j}in Equation 2c. In contrast, immune dynamics are described in much more detail, i.e., the clearance term in Equation 2c equals (k

_{1}Y

_{i}+ k

_{2}Z) V

_{i}, where Y

_{i}and Z are each captured with a separate differential equation.

^{+ }T-cell loss was not associated with a particular trend in terms of genetic diversity. They also stressed that amino acid changes of the virus were consistent with epitopes being targeted by cytotoxic T lymphocytes, which they interpreted as evidence of adaptive evolution.

^{+ }T-cell infection rate) is captured by a separate differential equation. The implicit assumption made here is that virus replication rate increases over time. A theoretical model by Iwasa et al. [55] has shown that, in the absence of trade-offs, the “pathogenicity” of a virus (defined as the inverse of the equilibrium number of target cells) should increase over the course of an infection, provided that there is an accessible evolutionary trajectory. Studies have argued that HIV replication rate increases over the course of the infection [44] and recent data analysis using a predictive algorithm based on the virus sequence has confirmed this result [45].

^{+ }T-cell infection rate, Schenzle [54] includes target cells, infected cells and anti-viral activity in his model. Overall, his model can quantitatively describe T-cell depletion due to direct killing by HIV and persistent infection dynamics due to virus evolution. Different incubation periods to AIDS among individuals derive from sensible assumptions concerning the variation of model parameters. Stilianakis et al. [58] further study the effect of changing biologically relevant parameters in this model framework, like the increase in virus reproduction rate or the initial values of the basic reproduction number. These parameter changes can account for the different patterns of CD4

^{+ }T-cell decline among different patients.

^{+ }T-cell population into non-susceptible, susceptible and productively infected cells. The fraction of CD4

^{+ }T-cells entering the pool of susceptible cells is assumed to be time-dependent. This model has more biological relevance than the other two and explains the whole infection and AIDS phase very well.

^{+ }T-cells (i.e., an immune response acting against the virus). They show that there exist three critical values for the virus replication rate: the first (and lowest) value allows the virus to establish in the host, the second is the threshold that allows the virus to avoid being eradicated by the immune response and the third is the AIDS threshold (overcoming the immune response). Unfortunately, contrary to the setting by Ball et al. [60], there is no explicit evolutionary model for the virus replication rate and it is only assumed that the replication rate increases over time, which bears the same limitation as the models described above.

## 4. The Role of Stochasticity

^{+ }T cells, CD8

^{+ }T cells expressing different epitopes, B-cell immunity indirectly by an CD4

^{+ }helper dependent immune response and different virions. This biologically more realistic approach reflects the acute, asymptomatic and AIDS phase and confirms earlier findings concerning the diversity threshold from Nowak et al. [50].

## 5. Other Processes

^{+ }T-cells with different (randomly distributed) avidity to viruses. Note that viruses are not explicitly modelled but linked to the number of infected cells. With this model, the authors can show that avidity of the CD4

^{+}-regulated immune response is the main determinant for disease progression rather than the breadth of the immune response. In addition they can identify a link between the avidity of the best clones and the time until the onset of AIDS.

^{+ }and CD8

^{+ }T cells [32,103]. When infected by HIV virions, dendritic cells can disperse the virus to other parts of the body [104]. Ref [102] extended the ODE framework of viral dynamics to integrate these characteristics. This framework shows that dendritic cells drive the infection in the early stage of an HIV infection when CD4

^{+ }T-cell densities are low. In addition, failure of dendritic cell function is a significant driver of progression to the AIDS phase. Note that Iwami et al. [105] also modified their earlier framework [49] to incorporate dendritic cells. The originality of their study is that they use parameter estimates obtained from patient data in [33] and show that variability in these estimates account for variability in the time to AIDS. The downside of their approach is that they assume that one of the value of one parameters (the immune impairment rate) increases deterministically over time.

## 6. Discussion and Perspectives

**Figure 2.**

**Fraction of articles on HIV that involve theoretical biology.**The regression was highly significant (r=0.0027, p-value < 10

^{−}

^{3 }and adj-R

^{2 }=0.97). The data was collected on Web of Science on July 13, 2012. The articles on HIV were selected using the keywords Topic=(HIV) AND Topic=(virus OR immunodefic*) and there were 110, 064 hits. The restriction to theoretical articles was performed by adding the keyword AND Topic = (dynamics OR mathemat* OR computational) and there were 4, 277 hits.

**Table 2.**

**Overview of HIV dynamics models.**We list all the models described in the main text that focus on the course of an HIV infection. For each model, we indicate the number of CD4

^{+ }T-cell compartments, the number of virus strains (“v” means it varies as the virus evolves and the number of strains is then denoted by n

_{v}), whether the model includes a host anti-viral immune response (and if it does so which type of response) and whether it follows the entire infection and is able to reproduce the slow time scale of CD4

^{+ }T-cell decline. We split the CD4

^{+ }T-cell compartment into uninfected and infected compartments depending on whether the cells of this compartment are infected with viruses or not. If a paper includes more than one model, we list these models separately (“basic” stands for “basic model”, “act. T” for “activated T-cell model”, ‘im. con.’ for “immune control model” and “drug” for “drug model”). In the models where the number of viral strains are “NA”, the virus dynamics is assumed to be in quasi-steady state with the infected cells, i.e., the viral numbers are a function of the number of infected cells. In the models with “NA” numbers of CD4

^{+ }T-cell compartments, viruses are assumed to be generated at a constant, target-cell independent rate. If the model captures the progression to the AIDS phase, we list the driving force for disease progression. Here, “NA” indicates that a feature is not included in the model.

Model | Number of compartments: | Immune response | Dynamics captured: | Timescale of asymp- | Driver of disease progression | |||
---|---|---|---|---|---|---|---|---|

uninfected | infected | viral | CD4^{+} T-cells | virus load | tomatic phase | |||

CD4^{+} T-cells | strains | |||||||

POPULATION DYNAMICS MODELS | ||||||||

Perelson et al. [36] | 1 | 2 | 1 | NA | no initial peak but long-term increase in viruses anddecrease in T-cells | between 3 and 9 years | slow progression due to different T-cell compartments and initial parameter choice | |

Essunger and Perelson [37] | 3 | 4 | 1 | NA | no initial peak but long-term increase in viruses and decrease in T-cells | between 2 and 8 years | time-dependent viral production rate, initial viral peak observable for model extension allowing infection of resting cells | |

Perelson et al. [29] | 1 | 1 | 1 and 2 (drug) | NA | acute and asymptomatic phase | ∞ | only by changing parameters manually during simulations | |

Kirschner [38]; Kir-schner andWebb [39] | 1 | 1 | 1 | NA | no initial peak but long-term increase in viruses and decrease in T-cells | approx. 4 years | increasing the non-T-cell based viral production rate over time | |

De Boer and Perelson [21] | 1 (basic), 2 (act. T), 0 or 1 (im. con.) | 1 (basic), 1 (act. T), 1 (im. con.) | 1 | CD8^{+} (im cont only) | yes | yes | depending on parameter choice | different models described, progression to AIDS only achievable by changing the activation or proliferation rate in a special T-cell compartment (immune control model) or the viral infection rate (activated T-cell model) over time |

Kirschner et al. [31] | 2 | 4 | NA | NA | no | yes | < 10 years | slow but constant drop of CD4^{+} T-cells due to multi-compartment model |

Fraser et al. [40] | 2 | 2 | NA | CD8^{+} | yes | no | 4-14 years | slow and fast compartments and (random) antigenic stimulation |

Perelson [27] | 1 | 2 | 1 (basic) | NA | acute and asymptomatic phase | ∞ | only by changing parameters manually during simulations | |

Ribeiro et al. [41] | 2 | 6 | 3 | NA | yes | yes | approx. 4 years | two virus types using different coreceptors, rise in X4 type due to selection and dominance of X4 virus |

EVOLUTIONARY MODELS | ||||||||

Nowak et al. [34] | NA | NA | v | general and strain specific | NA | yes | 6 to 8 years | antigenic diversity threshold, asymmetry between viral infection and viral recognition |

Nowak and May [47] | NA | NA | v | general and strain specific | yes | yes | 6 to 8 years | diversity threshold and asymmetry, similar model as [34] |

Nowak et al. [50] | 1 | 1+ n_{v} | v | general and strain specific | yes | yes | approx. 8 years | diversity threshold and asymmetry |

Schenzle [54] | 1 | 1 | 1 | general | yes | yes | approx. 10 years | within-host evolution modelled by increasing CD4^{+} T-cell infection rate during the infection |

Stilianakis et al. [46] | NA | NA | v | general and strain specific | yes | yes | depending on initial conditions | diversity threshold and asymmetry, similar model as [34] |

Stilianakis et al. [58] | 1 | 1 | 1 | general | yes | yes | approx. 10 years | increasing CD4^{+} T-cell infection rate during the infection |

Regoes et al. [48] | 1 | 2n_{v} | v | strain specific | dynamics not shown | NA | adds target cell-limitation to [34] | |

Stilianakis and Schenzle [59] | 2 | 1 | 1 | general | yes | yes | approx. 10 years | increasing CD4^{+} T-cell infection rate and increasing susceptibility of CD4^{+} T-cells during the infection |

Ball et al. [60] | 1 | n_{v} | v | NA | yes | not shown | NA | target-cell limited model and virus diversification with trade-off between the virus replication rate and the death rate of an infected cell |

Sguanci et al. [66] | n_{v} | n_{v} | v | TNF | yes | yes | approx. 4 years | target-cell limited, transmission and death rates depend on TNF concentration |

Iwami et al. [49] | 1 | n_{v} | v | CD8^{+} | NA | NA | NA | AIDS begins when the number of virus strains exceeds a threshold |

Iwami et al. [105] | 1 | 1 | NA | CD8^{+} | not shown | yes | variable across patients | increase in immune impairment rate over time (as in [54]) |

Kamp [65] | NA | NA | v | general and strain specific | yes | not shown | NA | diversity threshold, increasing viral growth rate |

Alizon and Boldin [63] | 2 | 2 | v | NA | not shown | yes | approx. 10 years | trade-off between the virus replication rate and the death rate of an infected cell and cell heterogeneity |

Huang et al. [62] | 1 | 1 | 1 | general | yes | yes | 3 to 8 years | deterministic increase in virus replication rate |

STOCHASTICITY-DRIVEN MODELS | ||||||||

Tan and Wu [69] | 1 | 2 | 1 | NA | yes | yes | approx. 10 years | target cell proliferation rate is a decreasing function of viral load (as in [36]) |

Zorzenon dos Santos and Coutinho [79] | 1 | 2 | 1 | general | yes | yes | approx. 8 years | CA model; infected cells organize themselves into spacial structures |

Regoes and Bonhoeffer [16] | NA | NA | v | NA | yes | no | 5 to 30 years | emergence of mutants strains with different fitnesses |

Lin and Shuai [83] | 1 | 1 | 1 | CD8^{+}, B-cells indirectly | yes | yes | influenced by viral mutation rate | CA model; spatial structure, virus mutation (asymmetry) |

OTHER PROCESSES | ||||||||

Galvani [100] | 4 | 1 | 1 | CD8^{+}, B-cells | yes | yes | approx. 9 years | elevated production of new T-cell clones that accumulate deleterious mutation |

Korthals Altes et al. [101] | 1 | 1+n_{v} | NA | strain specific CD4^{+} | yes | no | 3 - 40 years | avidity of CD4^{+} T-cell response (the lower the avidity is the faster is progression to AIDS) |

Hogue et al. [102] | 2 | 1 | 1 | CD8^{+}, dendritic cells | yes | yes | dependent on parameter change | induced by change of one or more parameters related to infection and viral production or to immune effector functions |

^{+ }T-cells) is that they are more prone to programmed death (they tend to overexpress PD-1). The less the T-cells express PD-1, the better the infection is controlled. Finally, another point Douek et al. [119] mention is the importance of immune activation and depletion in the gut. This perturbation of the gut mucosa leads to translocation of microbial products into the main circulation of the body [120]. From a modeling point of view, this latter source of pathogenesis is quite different because it involves another actor (the gut flora) that is unrelated to the host, which also brings us back to co-infection frameworks. Integrating these facts into mathematical models of HIV infection might help to capture the different phases of disease progression. In addition, mathematical models might help to identify key factors in disease progression.

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Alizon, S.; Magnus, C. Modelling the Course of an HIV Infection: Insights from Ecology and Evolution. *Viruses* **2012**, *4*, 1984-2013.
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Alizon S, Magnus C. Modelling the Course of an HIV Infection: Insights from Ecology and Evolution. *Viruses*. 2012; 4(10):1984-2013.
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Alizon, Samuel, and Carsten Magnus. 2012. "Modelling the Course of an HIV Infection: Insights from Ecology and Evolution" *Viruses* 4, no. 10: 1984-2013.
https://doi.org/10.3390/v4101984