A simple but effective in-vivo virion/immune-cell model underlying analyses of the progression and treatment of HIV in infected individuals is based on a system of three ordinary differential equations describing the dynamics of uninfected immune system cells (x), infected immune system cells (y), and free virions (v) [1,2]. The model is based on immune cells (often interpreted as CD4 cells) that are produced at a constant rate but die at a rate proportional to the density of these cells in the blood. On infection of one of these immune cells by an HIV virion, the model assumes that the infected cell will die at a characteristic rate releasing a specified ‘burst-size’ number of new virions. These virions then go on to infect uninfected immune cells at a rate proportional to the densities of virions and uninfected immune cells in the blood. With an appropriate set of rate and burst-size parameters, this model is able to capture virion and cell densities during the initial acute and later chronic stages of the infection (Fig. 1a). With the addition of a fourth differential equation that, as described below, can be interpreted as modelling the drawdown of a reservoir of stem cells, also interpreted as a process of immunosenesence , in which the size of the reservoir is set either by a tolerance to an accumulation of genetic errors  or the loss of telomere function , the final full-blown AIDS stage of the infection can also be captured by the model.
In particular, in the three differential equations that underlie many current in-vivo HIV immune cell models, uninfected cells are produced at a constant rate λ, infected cells at a rate βxv (transmission process with rate parameter β), and virions at a rate ky, whereas δx, ay, and uv are, respectively, the rates at which uninfected cells, infected cells and virions are removed from the system (death processes): k/a represents the burst size. This basic model and its elaborations do not account for immune cell production limitations caused by accumulated genetic errors  and the loss of telomere function  that ultimately results in immune senescence . These factors can be modelled by assuming a finite reservoir of hematopoietic stem cells  represented by the variable z(t) that decreases steadily with time t. A simple elaboration to the basic model that accounts for this is:
The basic model [1,2] is the first three equations with constant λ and u. The additional reservoir equation implies a drawdown rate c per uninfected cell production unit λ that we now assume depends on the size of the stem-cell reservoir z(t) and the level x(t) of uninfected cells. We will also assume now that the virion clearance rate u depends on z(t). These dependencies can be phenomenologically described as follows. First, we expect λ (x, z) to be a decreasing function of x because production should increase when uninfected cell counts are low. Initially, for non-depleted values of z, there should be little effect on either the maximum rates λ0 or u 0 until the reservoir z(t) approaches exhaustion, at which point λ (x, z) reduces to 0 and u(z) to its background rate γ u 0 (0 < γ < 1 to account for the loss of macrophage activity in removing virions). The extent to which z must be reduced for these rates to be at half their maximum values depends on many factors. Simple forms appropriate to the resolution of the model are:
accounts for increasing production rates as the uninfected cell population drops.
This extended model, in addition to replicating the acute and chronic infection stages of a typical HIV infection profile, captures the senescing immune system state at the onset of full-blown AIDS (Fig. 1) . The parameters used in Figure 1 are illustrative, because considerable variability exists among individuals in the intensity of infection at different stages of infection and in the time from infection to the onset of full-blown AIDS; the latter, for example, is known to decrease with age, with estimates [7,8] indicating a mean of 14–15 years at approximately the age of 10 years to about 8 years at approximately the age of 50 years. This decrease is expected under the assumption of a continuously drawndown hematopoietic stem-cell reservoir, because the reservoir equation implies z(0) decreases with the individual's age at the start of infection. It is also expected that the progression rate to full-blown AIDS is increased by factors that draw down the reservoir, such as previous parasite loads, chronic disease, and co-infections .
The first three equations in the four equation model presented here are too simple to capture many subtle phenomena associated with the acute and chronic phases of HIV infection, such as the persistence of low levels of plasma virus during potent antiretroviral therapy or the accumulation of mutations in immune cells over time. Multicell-type models employing additional differential equations are needed to investigate such phenomena [4,9]. Despite numerous elaborations to acute/chronic phase models represented by the first three equations, however, a basic model that captures all three in-vivo phases of HIV/immune cell dynamics (acute, chronic, collapse) using a single set of relatively simple differential equations has yet to be proposed. The four equation model presented here fulfils this role, although it will also need to be elaborated when used to address questions regarding the most effective design of HAART regimens or issues relating to the emergence of strains resistant to such interventions [9–11]. In addition, the fourth equation stresses our need to investigate and characterize more fully the exhaustibility of an individual's immunological system, particularly with regard to assessing its state as a hematopoietic stem-cell reservoir as a result of changes in lymphatic organs and bone marrow structure with age and history of infections. Finally, if immune system senescence is caused by the kind of extractive process modelled here, the results presented in Figure 1c have a profound bearing on the appropriate time to start antiretroviral treatment in HIV-positive individuals, particularly in individuals with co-infections such as Mycobacterium tuberculosis.
This work was funded by a Grant from NIH. I would like to thank John Hargrove, Brian Williams, Travis Porco, Maria Sanchez, and Russell Vance for comments that have improved this Research Letter.
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