A stochastic model of latently infected cell reactivation and viral blip generation in treated HIV patients

Abstract

Motivated by viral persistence in HIV+ patients on long-term anti-retroviral treatment (ART), we present a stochastic model of HIV viral dynamics in the blood stream. We consider the hypothesis that the residual viremia in patients on ART can be explained principally by the activation of cells latently infected by HIV before the initiation of ART and that viral blips (clinically-observed short periods of detectable viral load) represent large deviations from the mean. We model the system as a continuous-time, multi-type branching process. Deriving equations for the probability generating function we use a novel numerical approach to extract the probability distributions for latent reservoir sizes and viral loads. We find that latent reservoir extinction-time distributions underscore the importance of considering reservoir dynamics beyond simply the half-life. We calculate blip amplitudes and frequencies by computing complete viral load probability distributions, and study the duration of viral blips via direct numerical simulation. We find that our model qualitatively reproduces short small-amplitude blips detected in clinical studies of treated HIV infection. Stochastic models of this type provide insight into treatment-outcome variability that cannot be found from deterministic models.

Figure 1. Model schematic.

Latently infected cells (L) divide, die, and become activated with rates and respectively. Productively infected cells (T*) die at rate and produce virus (V) continuously, at rate . Free virions are cleared at rate and infect healthy cells at rate , reduced by drug treatment of efficacy . A fraction of newly infected cells become latently infected cells and the rest become productively infected cells.

Figure 2. Initial probability distribution on latent reservoir size.

We take an initial mean viral load of 25 c/mL and parameters given in Tables 1 and 2. Production rates have units . Initial distributions assuming initial mean viral load of 35 c/mL (Table 3) are qualitatively similar (not shown).

Figure 3. Latent reservoir extinction probability over time.

Parameters given in Tables 1 and 2. Production rates have units .

Figure 4. Reductions in latent reservoir lifetime with improving drug efficacy.

(A) Percent mean reduction in latent reservoir lifetime with improving drug efficacy . (B–D) Corresponding latent reservoir extinction distributions with improving drug efficacy for fraction of newly infected cells becoming latently infected (B) , (C) , (D) . Parameters: Tables 1 and 2 with .

Figure 5. Transient and permanent viral extinction.

We plot the probabilities that the viral load is zero (transient viral extinction) and that the latent reservoir is zero (permanent viral extinction) as a function of time. Parameters: Tables 1 and 2, with .

Figure 6. Viral load probability distributions for initial mean viral load of 25 c/ml.

(A–C) Distribution functions are plotted at 6 month intervals for parameters given in Tables 1 and 2, and (A) , (B) , (C) . Insets: enlargement of probability distribution curves above the detection level, ; a log scale is used to better distinguish the curves. As time advances the distributions move from right to left. (D) Blip probability plotted against time. The curves in (D) are computed by integrating the probability density functions from (A–C) over viral loads exceeding 50 c/mL.

Figure 7. Viral load probability distributions for initial mean viral load of 35 c/ml.

(A–C) Distribution functions are plotted at 6 month intervals for parameters given in Tables 1 and 3, and (A) , (B) , (C) . Insets: enlargement of probability distribution curves above the detection level, ; a log scale is used to better distinguish the curves. As time advances the distributions move from right to left. (D) Blip probability plotted against time. The curves in (D) are computed by integrating the probability density functions from (A–C) over viral loads exceeding 50 c/mL.

Figure 8. Maximum viral load under immune system activation.

(A) Maximum mean viral load for different multiplicative increases in target cell populations and activation rates . Dashed lines indicate target cell multiplier 100 (vertical) and activation rate multiplier 5 (horizontal). (B) Maximum mean viral load (symbols) one standard deviation (shaded area) depending on activation rate multiplier, for target cell multiplier 100 (along vertical line in (A)). (C) Maximum mean viral load (symbols) one standard deviation (shaded area) depending on target cell multiplier, for activation rate multiplier 5 (along horizontal line in (A)). Parameters: Tables 1 and 2 with , and drug efficacy .

Figure 9. Blip durations depend on initial blip amplitude.

(A) Mean blip durations (symbols) standard deviation (shaded area), computed over 10000 simulations, plotted as a function of the initial viral load measurement (initial blip amplitude). Production rates have units . (B) Frequency plots of time distributions of detectable viral load given initial measurements of 60–90 c/mL, computed over 10000 simulations. Parameters: Tables 1 and 2; latent reservoir size 1 per cells; in (B).

Figure 10. Realizations of Gillespie simulations showing viral load evolution.

(A–C) show sample viral load evolutions and the associated histogram of durations until the viral load is below 50, over 10000 simulations, given an initial viral load measurement and latent reservoir size. (A) Initial viral load measurement of 60 c/mL with latent reservoir size 1 per cells; (B) initial viral load measurement of 80 c/mL with latent reservoir size 1 per cells; (C) initial viral load measurement of 60 c/mL with latent reservoir size 1.5 per cells. Parameters: Tables 1 and 2, for .

Figure 11. Blip durations depend on initial latent reservoir size.

Mean blip durations (symbols) 1 standard deviation (shaded area), computed over 10000 simulations, plotted as a function of the initial latent reservoir size. (B) Frequency plots of time distributions of detectable viral load given initial latent reservoir sizes of 1–1.5 cells per . Parameters: Tables 1 and 2; initial viral load measurement of 60 c/mL; in (B).