A compound interest approach to HIV cure

In the era of antiretroviral therapy (ART), HIV-1 infection is no longer tantamount to early death. Yet the benefits of treatment are available only to those who can access, afford, and tolerate taking daily pills. True cure is challenged by HIV latency, the ability of integrated virus to persist within memory CD4T cells in a transcriptionally quiescent state and reactivate when ART is discontinued. Using a mathematical model of HIV dynamics, we demonstrate that treatment strategies offering modest but continual enhancement of reservoir clearance rates result in faster cure than abrupt, one-time reductions in reservoir size. We frame this concept in terms of compounding interest: small changes in interest rate drastically improve returns over time. On ART, latent cell proliferation rates are orders of magnitude larger than activation rates. Contingent on subtypes of cells that may make up the reservoir and their respective proliferation rates, our model predicts that coupling clinically available, anti-proliferative therapies with ART would result in functional cure within 2-10 years rather than many decades on ART alone.


Introduction
The most significant accomplishment in HIV medicine is the suppression of viral replication and prevention of AIDS with antiretroviral therapy (ART) [1]. However, HIV cure remains elusive due to viral latency, the ability of integrated virus to persist for decades within CD4 + T cells in a transcriptionally quiescent state. When ART is discontinued, latently infected cells soon reactivate, and virus rebounds [2,3]. Attempts to cure HIV by eradicating the latent reservoir (summarized in [4]) have been unsuccessful except in one notable example [5].
To investigate persistence of the latent reservoir, we use a mathematical model that distinguishes between latently and actively infected cells. After demonstrating analytically that infection of new cells does not meaningfully contribute to sustaining the latent reservoir in ART-treated patients, we separate the observed clearance half-life of 44 months [2] into one positive component rate (proliferation) and two negative component rates (death and reactivation). The reactivation rate defines the rate at which memory CD4 + T cells transition to an activated phenotype in which HIV initiates replication. We demonstrate the impact of changes to any of these rates via an analogy with the compound interest formula in finance: small changes in interest rates result in large overall gains or losses over time. In the absence of an intervention, proliferation and death rates far exceed the reactivation rate [6]. Thus, reduction of the proliferation rate will deplete the latent pool much more rapidly than a comparable increase in the reactivation rate as in latency reversal therapy ('shock-and-kill'). In keeping with the financial analogy, we call the strategy of continuous reduction of the proliferation rate compound interest cure. We conclude by proposing strategies to test this concept and by addressing theoretical reasons for its failure, including latency in various T cell subsets with different proliferation rates and potential toxicities of anti-proliferative drugs.  Table 1. Infection by HIV virus V of susceptible cells S mediated by ART of efficacy is given by β . The probability of latent infection is τ . The rate of activation from the latently infected (L) to active state (A) is ξ. Cellular proliferations and deaths are determined by α and δ for each compartment.

ART decouples latent pool dynamics from ongoing infection
To model the latent reservoir, we expand the infected compartment of the original HIV model [7] to account for the latently infected cells L and actively replicating, infected cells A [6,8,9,10].
Our model is a set of ordinary differential equations (visualized schematically in the top panel of  Table 1 with further information in the Supporting Information (SI).
In particular, we adjust the infectivity parameter β to capture the effectiveness of ART therapy by defining β = β(1 − ) such that inert therapy implies = 0, and perfectly effective therapy implies = 1. If ART is perfectly effective, no virus is generated, and no new infections occur even when cells reactivate from latency. Then, the dynamics of the latent pool can be considered on their own, decoupled from the dynamics of the other cell types, and the only mechanisms changing the latent cell pool size are cell proliferation, death, and activation (shown in the lower panel of Fig. 1).
Perfectly effective ART is not strictly necessary to consider the latent pool on its own. Using the tools of dynamical systems mathematics, we can derive an expression for the viral set-point and calculate the critical therapy efficacy c in terms of the rates in Table 1 (Materials and Methods) [11]. When ART achieves the critical efficacy, viral production from reactivation infrequently creates another latent cell because the probability of latency is so low (τ 1). Therefore, we can consider the latent cells as their own system, decoupled from the other cell types. Put another way, the model does not preclude the possibility of some small, ongoing viral replication and infection, but rather demonstrates that this mechanism is unlikely to contribute to latent reservoir dynamics. [10,12].
The mathematical analysis is pertinent to discussion of viral evolution and ART escape mutations.
From a practical point of view, using parameters from Table 1, we find c ∼ 85%, considerably lower than empirically derived measures of ART potency [13,14]. Thus, because true ART efficacy is likely much larger than the necessary minimum c , we predict little de novo infection in patients suppressed on ART, consistent with the reported lack of viral evolution following years of ART [12,15,16]. We do not address the question of anatomic drug sanctuaries directly in the model.
However, to have an impact on the reservoir, a sanctuary would have to contribute enough ongoing infection to drive the average ART efficacy below critical levels and thus to permanent viral rebound, which does not occur on suppressive ART.
Above the critical efficacy, we can solve for the decoupled time-evolution of the latent pool (see Materials and Methods): in which we define θ L = α L − δ L − ξ as the net clearance rate for the latent cells and L 0 the initial size of the reservoir. Eq 1 implies that the clearance rate of infected cells is a function of their proliferation rate α L minus both death δ L and activation ξ rates. Experimental measurements indicate an average latent cell half-life of 44 months (θ L = −5.2 × 10 −4 per day) [2,17] and an  [2,17] net latent T cell clearance rate on ART α cm 0.015 day −1 [18,19] latent central memory T cell proliferation rate δ cm 0.0155 day −1 latent central memory T cell death rate α em 0.047 day −1 [18,19] latent effector memory T cell proliferation rate δ em 0.0475 day −1 latent effector memory T cell death rate α n 0.002 day −1 [18,19] latent naive T cell proliferation rate δ n 0.0025 day −1 latent naive T cell death rate ξ 5.7 × 10 −5 day −1 [8] activation rate α A 0 day −1 [14] active T cell proliferation rate δ A 1.0 day −1 [14] active T cell death rate τ 10 −4 day −1 [10] probability of latency given infection α S 300 cells/(µL-day) [20] susceptible T cell growth rate δ S 0.2 day −1 [14] susceptible T cell death rate β 10 −4 µL/(virus-day) [6] HIV infectivity off ART β β(1 − ) µL/(virus-day) [6] HIV infectivity on ART (with efficacy ∈ [0, 1]) π 10 3 virus/(cell-day) [8] viral production rate γ 23 day −1 [14] viral clearance rate average initial latent pool size L 0 of one-million cells [2]. A mathematical correspondence to the principle of continuous compound interest is now evident with L 0 as the principal investment and θ L as the interest rate. In the following discussion, we consider all four thresholds-henceforth referred to as Hill 1-yr, Hill cure, Pinkevych 1-yr and Pinkevych cure.   Figure 2: Simulated comparisons of latent reservoir eradication strategies on standard antiretroviral (ART) treatment. The thresholds (shown by black lines) are discussed in the text. A) Onetime therapeutic reductions of the latent pool (L 0 ). B) Continuous therapeutic increases in the clearance rate (θ L ). Relatively small decreases in rates produces markedly faster times to cure than much larger decreases in the initial reservoir size. C-E) Latency reversal agent (LRA) and Antiproliferative (AP) therapies are given continuously for durations of weeks with potencies given by LRA , and AP respectively. The color bar is consistent between panels, and thresholds of cure are shown as solid black lines both on plots and on the color bar. C) Latency reversing agent therapy (LRA) administered alone requires years and potencies above 100 to achieve the cure thresholds. D) Anti-proliferative therapies (AP) administered alone lead to cure thresholds in 1-2 years provided potency is greater than 2-3. E) LRA and AP therapies are administered concurrently, and the reduction in the latent pool is measured at 70 weeks. Because the proliferation rate is naturally greater than the reactivation rate, increasing the AP potency has a much stronger effect than increasing the LRA potency.
2.3 Sustained mild effects on clearance rate deplete the reservoir more rapidly than large, one-time reductions in the reservoir The HIV cure strategy most extensively tested in humans is 'shock-and-kill' therapy: small molecules (called latency reversing agents or LRAs) activate latent cells to replicate and express HIV proteins, allowing immune clearance while ART prevents further infection [4,22]. Other strategies in development include therapeutic vaccines [23], viral delivery of DNA cleavage enzymes [24], and transplantation of modified HIV-resistant cells [25] informed by the 'Berlin patient' who was cured following a similar protocol [5]. Some of these therapies may manifest as one-time reductions in the number of latent cells. We simulate such instantaneous decreases using Eq. 1 in Fig. 2A. With θ L constant and a 100-fold reduction in L 0 , the Pinkevych 1-yr threshold is immediately satisfied, but the Hill 1-yr and Pinkevych cure still require 15 years of ART. Hill cure requires a 1000-fold reduction and over 10 subsequent years of ART.
In comparison to one-time treatments, continuous-time interventions are more promising. In On ART alone, the reservoir is depleted slowly with estimated cure occurring at roughly 70 years [2]. However, just a 3-fold increase in clearance rate achieves Hill-cure in fewer than 20 years. If a 10-fold sustained increase is possible, Hill cure requires only 5 years. Analogous to the 'miracle of compound interest,' increasing the clearance rate for an extended duration produces profound latency reduction.
2.4 Smaller reductions in proliferation rate achieve more rapid reservoir depletion than comparable relative increases in reactivation rate Latency reversing therapy can be modeled by Eq. 1 if treatment is assumed to be a continuous-time multiplication of the reactivation parameter (ξ). Simulations of the latent pool at various potencies and therapy durations in Fig. 2C indicate both Hill and Pinkevych cure thresholds require more than a 150-fold multiplication of ξ sustained for two or three years, respectively.
Experimental evidence [15,19,26,27] and theoretical analysis [6] demonstrate that proliferation of latently infected cells is critical for reservoir persistence. While two anti-proliferative drugs, mycophenolate mofetil (MMF) and hydroxyurea, have been used in the past as adjuncts to ART to further limit replication [28,29,30], anti-proliferative strategies have not been tested specifically to treat HIV latency [31]. We now justify mathematically how such therapies could have curative potential.
The latent cell proliferation rate is considerably larger than the reactivation rate (α L ξ). Thus, anti-proliferative therapies would clear the reservoir faster than equivalently potent latency reversal strategies. When the reservoir of CD4 + T cells harboring replication competent HIV is assumed to consist only of central memory cells (T cm ), a 10-fold reduction in α L(cm) reaches Pinkevych 1-yr, Hill 1-yr, Pinkevych cure, and Hill cure in 0.8, 1.6, 1.6, and 1.8 years, respectively (Fig. 2D).
The improvement in cure time in Fig. 2D (as opposed to equivalent 10-fold reduction of θ L in Fig. 2B) is possible because decreasing the proliferation rate means the net clearance rate approaches the death rate. The potency is relatively unimportant beyond reducing the proliferation rate by a factor of ten because the underlying death rate δ cm is actually the bound on the clearance rate. Fig. 2D also demonstrates substantial asymmetry of the contours over its y = x axis, conveying that outcomes improve more by extending duration than by equivalent increases in potency.
This can also be demonstrated mathematically (see SI). Fig. 2E further illustrates that the relative impact of anti-proliferative therapy is much greater than that of latency reversal therapy when the two therapies are given concurrently for just over a year.

Heterogeneity in reservoir cell types may necessitate prolonged anti-proliferative therapy
Recent studies indicate that the reservoir is heterogeneous, consisting of CD4 + central memory (T cm ), naïve (T n ), effector memory (T em ), and stem cell-like memory (T scm ) T cells with composition differing dramatically among patients [19,32,33]. This heterogeneity suggests the potential for differing responses to anti-proliferative agents. T cm homeostatic proliferation rates exceed the rates for T n but lag behind antigen-driven T em turnover rates (Table 1). T scm are assumed to proliferate at the same frequency as T n in our model based on similar properties. We simulate possible reservoir profiles in Fig. 3A-C. At least 7 years of treatment is needed for Pinkevych functional cure if slowly proliferating cells (T n and/or T scm ) comprise more than 20% of the reservoir.
In contrast, an increased proportion of T em has no clinically meaningful impact on time to cure.
Slowly proliferating cells are predicted to comprise the entirety of the reservoir within two years of 10-fold anti-proliferative treatment regardless of initial percentage T n or T scm (Fig. 3D&E).
Therefore, anti-proliferative strategies may face a challenge akin to the cancer stem cell paradox, whereby only the rapidly proliferating tumor cells are quickly expunged with chemotherapy. For   Table 1. The potency of the therapy is AP = 0.1 (i.e., each cell type i has proliferation rate equal to α i /10 with i ∈ [em, cm, n]). Plausible initial compositions of the reservoir (L i (0)) are taken from experimental measurements [19,32,33]. It is assumed that the HIV reactivation rate ξ is equivalent across all reservoir subsets. A-C) Plots of times to therapeutic landmarks on long-term ART and anti-proliferative therapy with heterogeneous reservoir compositions consisting of effector memory (T em ), central memory (T cm ), and naïve plus stem cell-like memory (T n +T scm ) CD4 + T cells. T em and T n +T scm percentages are shown with the remaining cells representing T cm . Times to one-year remission and functional cure are extremely sensitive to percentage of T n +T scm but not percentage of T em . D&E) Continuous 10-fold therapeutic decreases in all proliferation rates (α i ) result in Hill 1-yr/Pinkevych cure in D) 3.5 years assuming T n +T scm =1% and E) 6 years assuming T n +T scm =10%. The reservoir is predicted to become T n +T scm dominant within 2 years under both assumptions, thus providing an indicator to gauge the success of anti-proliferative therapy in a potential experiment.
example, tyrosine kinase inhibitors suppress proliferation of cancer cells in chronic myelogenous leukemia (CML). While many patients achieve 'undetectable minimal residual disease,' some patients relapse to pre-therapy levels of disease following cessation of therapy-perhaps due to slowly proliferating residual cancer cells [34].
The uncertainty in the reservoir composition tempers the results in Fig. 2. On the other hand, our model assumes that the HIV reactivation rate ξ is equivalent across all CD4 + T cell reservoir subsets. It is biologically plausible, though unproven, that HIV rarely or never reactivates from resting T n or T scm . Under this assumption, more rapid functional cure akin to Fig. 3D would be expected because theoretically this would allow cessation of ART despite long-term persistence of T n harboring latent HIV.

Reservoir size, reservoir composition, and efficacy of anti-proliferation agents predict time to HIV cure
We performed a single parameter sensitivity analysis (Fig. S3) and identified that the latency probability τ , ART efficacy above the critical efficacy , and reactivation rate of HIV ξ did not vary the kinetics of the HIV reservoir using concurrent ART and anti-proliferative therapy. Alternatively, low initial reservoir size L 0 , low percentage of naïve cells in the reservoir L n (0)/L 0 , and high reservoir clearance rate −θ L decreased time to cure.
We also performed an uncertainty analysis in which multiple therapeutic parameters were ran-  to MMF [28]. Our model estimates a 1-2 order of magnitude reduction in reservoir size after 17 weeks of treatment, which in turn would correspond with a median of a 4-8 week delay in HIV reactivation, consistent with their findings (see SI for parameter selections) [8].

Discussion
Several authors have studied similar systems of equations to represent the latency dynamics of HIV [6,8,9,10]. Here, we focus on cure strategies and contrast their respective times to functional cure. We demonstrate that minor reductions in memory CD4 + T cell proliferation rate would exhibit powerful reductions in the latent pool when therapy duration is extended over time. We call this proposed strategy compound interest cure due to the correspondence with financial modeling.
Our results are relevant because the only HIV cure strategy currently being tested in humanslatency reversal therapy ('shock-and-kill')-does not capitalize on the advantages of a compound interest strategy. Promising latency reversing agents are typically dosed over short time-frames based on concern for toxicity. Furthermore, even if these agents exert a large relative impact on the reactivation rate of memory CD4 + T cells, we predict the reduction in the reservoir will be insignificant given the small size of the natural reactivation rate (orders of magnitude smaller than the proliferation and death rates).
The potential of the compound interest approach is enhanced by the existence of licensed medications that limit T cell proliferation, including mycophenolate mofetil (MMF) and azathioprine.
In line with our suggestion that duration is more important than potency, these drugs are dosed over time periods of months or years. MMF is commonly used to treat several rheumatologic conditions and to prevent rejection in solid organ transplant. The most frequent side effects reported are gastrointestinal symptoms and increased risk of infection though the latter risk is obscured by concurrent use of high-dose glucocorticoids [35]. MMF has been given to a small number of HIV patients suppressed on ART for 17 [28] and 24 weeks [29]. No opportunistic infections or adverse events were observed in either study, and CD4 + T cell counts did not decrease significantly during therapy. We hypothesize that whereas MMF decreases proliferation of existing CD4 + T cells, it does not suppress bone marrow replenishment of these cells (see SI). Finally, MMF did not coun-teract the effects of ART [28,29], and thus we do not expect viral drug resistance to develop or ongoing viral evolution to occur on anti-proliferative therapy. However, despite these reassuring findings, future studies of HIV infected patients on anti-proliferative agents will require extremely close monitoring for drug toxicity and immunosuppression.
Our model suggests that slowly proliferating cells in the reservoir could present a barrier to rapid eradication of latently HIV-infected cells. Additional limitations could include insufficient drug delivery to anatomic sanctuaries, certain cellular subsets that are unaffected by treatment, and cytokine-driven feedback mechanisms that compensate for decreased proliferation. These challenges might be countered by combining anti-proliferative agents with other cure therapies. Avoidance of nucleoside and nucleotide reverse transcriptase inhibitors, which may enhance homeostatic proliferation, could provide an important adjunctive benefit [31,36,37].
The anti-proliferative approach is attractive because it is readily testable without the considerable research and development expenditures required for other HIV cure strategies. Anti-proliferative approaches require minimal potency relative to latency reversal agents, and T cell anti-proliferative medications are well studied mainstays of organ rejection prevention. Therefore, we propose trials with anti-proliferative agents as an important next step in the HIV cure agenda.

Decoupling the latent reservoir dynamics
HIV dynamics have been described mathematically in many fashions [38]. We describe a model (schematic in Fig. 1) that includes susceptible and latently or actively infected T cells. We follow the concentrations [cells/µL] of susceptible CD4 + T cells S, latently infected cells L, actively infected cells A, and plasma viral load V [virions/µL]. The system of ordinary differential equations tracks the states and is expressed (using the over-dot to denote derivative in time) where we have defined the dimensionless latency factor f L = 1 − τ (ξ/θ L + 1) to encapsulate all latent pool dynamics. Solving this equation, the critical drug efficacy c can be derived such that the viral equilibrium is unstable (V * < 0): If it is assumed that ART also affects π in Eq. S10, i.e. π → π(1− ) this decreases the critical efficacy (SI). Above the critical efficacy reactivation from latency is the only mechanism that produces virus. After a reactivation, a transient small viral load exists, but the small probability (τ ) of new latent infections from this small amount of virus allows us to ignore any increase in latent cells. Thus, the latent state can be considered independently, and the transition from latency to activation represents another pathway to removal from the latent pool. We express the dynamics of latency decoupled from virus with the two-equation system as shown in the shaded bottom panel of Fig. 1: A similar model has been studied recently in Conway and Perelson [10] who also show that transitions back to latent cells from replicating active cells are negligible. In another manner, we demonstrate the ability to ignore transitions back from active cells in simulation (Fig. S1&S2).
The two equation system is linear, and we solve for the latent dynamics as where the initial number of latent cells is L 0 and the total clearance rate θ L = α L −δ L −ξ for the latent cells corresponds with the experimentally known net clearance rate. We use the correspondence of Eq. 6 to the formula for continuous compound interest throughout the paper.

Modeling multiple T cell subsets
It has been demonstrated that several subsets of CD4 + T cells may contribute to the reservoir.
We reviewed existing data to find the rates of proliferation for each subset as well as estimates of the total fractional makeup of the reservoir [19,32,33]. The typical groupings are central memory (T cm ), effector memory (T em ), and naïve (T n ) cells. Deuterated glucose measurements by Macallan et al. found turnover rates as 4.7% per day for T em compared with 1.5% per day for T cm and only 0.2% per day for T n [18]. Transitional memory T cells (T tm ) have also been described to represent a transition from T cm to T em [19]. For the model we consider T tm to have equivalent proliferation rates to central memory cells. Similarly, we characterize the recently described stem cell-like memory CD4 + T cells (T scm ) as T n given their likely slow turnover rate [33]. Chomont et al. determined the percentage of each cellular subset from 31 aviremic individuals [19]. They find mean contributions of T cm +T tm to be 86%, whereas T em are 13% and Tn roughly 1-2% though there is a wide range of values amongst patients. Buzon et al. report larger percentages of naive cells up to 15% and include T scm at 24%, T em at 26%, and T cm at 22% [33].
To model the heterogeneity, we make the latent pool L → L i a vector with a dimension i for each T cell subset. The differential equation for the decoupled latent dynamics is theṅ We ignore transitions among subsets, as the natural composition of the reservoir is reasonably stable over time [19]. The solution then with diagonal The percentages discussed in the preceding paragraph determine the initial makeup of the reservoir L i (0). We simulate the dynamics of a heterogeneous reservoir with 10% potent anti-proliferative therapy AP = 10 in Fig. 3 using in which the index i ∈ [cm, em, n] indicates the T cell subsets with rates from Table 1.

Important model parameters
The parameters that affect our model most critically are those that contribute to the total clearance of latent cells θ L . The half-life of latently infected cells has been determined experimentally using the quantitative viral outgrowth assay [2,17]. We use a proliferation rate for α L obtained using in vivo labeling of CD4 + T cells with deuterated glucose [18]. Importantly, these estimates depend on the length of the labeling period and the model used to interpret the measurements [40]. Thus, we perform uncertainty and sensitivity analyses (see SI) to explore a larger range of values. In Fig. 2, we assume that the majority of latently infected CD4 + T cells are in fact central memory T cells (T cm ), and thus α L = α cm = 0.015 per day. The importance of reservoir proliferation has been supported experimentally [19,16] and predicted theoretically [6,41]. In [6], α L parameter values were chosen between 0.011 and 0.03 per day, consistent with our model. The activation rate ξ is several orders of magnitude smaller than the proliferation rate. Luo et al. estimated HIV model parameters from structured treatment interruptions in the AutoVac trial [14]. Using those estimates, Hill et al. found that on average, 57 CD4 + T cells per day transition from latency to the activated state and assuming 1 million latent cells then, ξ = 5.7 × 10 −5 per day [8]. We calculate the death rate for each of i cell types δ i in Table 1, using the respective proliferation, activation and net clearance rates [18]. The remaining parameters are necessary here only to calculate the critical efficacy. They are tabulated in Table 1 for completeness and are discussed in further detail in the SI.

Acknowledgments
We We compare our theoretical treatment using anti-proliferative agents in conjunction with ART with previous experimental results, and discuss how anti-proliferative therapy can deplete HIV-infected CD4 + T cells without depleting all CD4 + T cells from the body.
6 Appendix I: Additional model discussion and calculations 6

.1 HIV dynamical equations including latency and therapy
The SLAV model for HIV latency is briefly described in the main body and schematized in  Table 1 of the main body. The system isṠ where the over-dot denotes time derivative. We use the variables described in Table S1 throughout the paper. Virus production rate, due to constant leakage or burst upon cell death γ Clearance rate of virus θ L = α L − δ L − ξ Net clearance rate of the latent reservoir, assumed the same between T cell subtypes Table S2: List of parameters and descriptions. Dimensions and values are found in Table 1 of the body; confidence intervals are included in Table S7 then solving the second equation for L * leads to

Equilibrium solutions leading to decoupled equations for the latent pool
so that we can factor and cancel V * and rewrite the bracketed term as f L = 1 − (1 + ξ/θ L )τ so that we can solve for the viral set-point equilibrium value of the susceptible cells From here, we solve for the viral set-point equilibrium viral load concentration, and the other equilibria follow, leading to the set of equilibrium solutions: where f L = 1 − (1 + ξ/θ L )τ . We refer to this factor as the latency factor because it encapsulates all the latent dynamics. When τ 1, we have f L ∼ 1 which is true for parameter values from the literature (see Table 1).
We can also calculate the basic reproductive ratio R ART 0 of HIV on ART quickly using the equation for the viral equilibrium. When the basic reproductive number is less than 1, the viral equilibrium point is unstable. We can rewrite and thus identify the combination of parameters where we include the superscript to remind the reader that the value is not the natural value, but the value with β . This expression for the basic reproductive number can be checked using the next-generation matrix method for example [42]. This expression is fairly independent of the activation rate. For example, if LRA treatment is not very potent and if LRA treatment provides a huge increase in the activation parameter ( LRA ξ > α L − δ L ), aside from the point when they are actually equal and the function is undefined, the expression approaches (using the Binomial expansion because α L − δ L is a negative number, as activation increases which is the typical value for the viral dynamics model without a latent compartment. This is understandable because when reactivation is instantaneous, the latent model acts as a model without latency. This is excellent for clearance of the reservoir, but it is unclear how possible it is to have such a potency.

Critical drug efficacy
The critical drug efficacy c is then calculated for the situation where V * < 0 [6]. This leads us which is precisely the drug efficacy for which R ART 0 < 1. Thus when therapeutic effectiveness is above this threshold, the virus will decay and never reach a stable set point. We could also consider a therapeutic effect in the parameter that controls viral production from active cells. This could be implemented by changing π → π = π(1 − ) for example. If this choice is made instead of the change in the infectivity the equilibrium analysis is identical. If both parameters are changed we arrive at the second critical efficacy which is in general lower than the previous calculation (i.e., c < c ), the critical efficacy provided by a drug only affecting the infectivity.

Stability analysis
By linearizing our system of equations, we can comment on the stability of the equilibrium points.
We begin by defining our state variables as a vector x = [S, L, A, V ] T such that we can express the system of ODEs as F(x) = d dt x. We have calculated the equilibrium solutions above, and thus can Taylor expand our function around the equilibrium: By construction the first term is zero, because it is the equilibrium point, and calling a small ∆x = x − x * 1, then terms of O(∆x 2 ) are negligible. Because the derivative function is linear and using x = x * + ∆x we can rewrite and because x * is not a function of time we have finally the linear equation 4]) is referred to as the Jacobian. The eigenvalues of the Jacobian describe how perturbations near equilibrium behave. Notably, if all eigenvalues λ j of the Jacobian have negative real components Re(λ j ) < 0, perturbations decay back to equilibrium and the equilibrium is deemed stable. Importantly, a stable set-point equilibrium only exists when < c . As mentioned in the body, for realistic parameters the critical efficacy is c ∼ 85%. Above the critical efficacy of ART, no stable viral set-point equilibrium exists, and the viral-free equilibrium becomes stable.

Understanding how perturbations to the viral-free equilibrium behave gives insight into what hap-
pens when a small bit of virus is introduced. We evaluate the Jacobian at the viral-free equilibrium We can see immediately that one of the eigenvalues is −δ S , and by noting that τ β α S /δ S is very small relative to the other rates, we see that another eigenvalue is θ L . In fact, all eigenvalues (when calculated numerically) above the critical efficacy have negative real parts, so we know the equilibrium is stable. Furthermore, the timescale to return to equilibrium t * is due to the largest (least negative) eigenvalue by t * = 1/ max λ j [43]. Numerically, it turns out that this timescale is on the order of 1/θ L , demonstrating that for the equilibrium to truly return to virus-free, all the latent cells must be cleared.
The mathematical analysis gives the condition for true eradication, and it is possible to make a heuristic argument for the dynamics of the latent cells. Soon after initiation of ART, virus only exists when long lived latent cells are activated, and these small activation events are quickly removed by ART. Furthermore, the few infections that do recur also have a minute probability of becoming latent (due to the magnitude of τ ). All of this results in the observation that ongoing infection is ignorable in the latent reservoir [10]. We can now focus on the active and latent cells in the two equation system as shown and visualized in the bottom of Fig. 1 of the paper. S15)

Solving the decoupled equations: the compound interest formula
The system Eq. S15 is linear and trivially solvable. Solving the equation using the initial number of latent cells L 0 yields We will now define the total clearance rate θ L = α L − δ L − ξ for the latent cells. This definition is important to make correspondence with the experimentally known parameter of the net clearance rate first measured by Siliciano et al. [2]. We have now It is clear that when θ L < 0 the latent cell pool is cleared exponentially. Equivalently, if the latent cell proliferation rate α L is greater than the sum of the reactivation rate ξ and the latent cell death rate δ L , the latent reservoir L grows indefinitely. We call the exponential decay expression the compound interest model because of the correspondence with financial modeling. The principal investment is analogous to the initial size of the latent pool, and the interest rate analogous to the clearance rate of the latent pool.

The differential equation for the active cells iṡ
with solution that can be computed by multiplying both sides by exp(δ A t) and identifying the total . We have then A plot of the trajectory of the latent pool with the drug efficacy above and below the critical efficacy is shown in Fig. S4. As expected by the stability analysis, above the critical efficacy the decoupled solution (the compound interest formula) matches the full solution perfectly. The full solution is achieved by solving the set of equations Eq. S10 using Matlab's ode23s solver. Compound interest Full SLAV model Figure S4: Plots of the latent pool clearance below and above the critical efficacy c ∼ 85%. Above the critical value, the compound interest formula L = L 0 exp(θ L t) matches the full ODE solutions exactly.

Impact of duration and rates on clearance time
It was claimed and demonstrated graphically in Fig. 2D of the body that it is better to increase the duration of therapy to deplete the latent pool than to increase the potency of the therapy equivalently. We can show this formally as well. If we examine the remaining fraction of latent cells over time we can study the effect of multiplying any rate by a factor r or multiplying the duration of time by factor d. The ratio of the percent remaining after doing either of these multiplications tells us which procedure is more valuable. For example if we multiply the reactivation rate only then for the rate changed L (r) to be smaller than the duration changed L (d) we must have This means that the factor multiplying the rate must obey and note if we set d = r, we find that d < 1 because (α L − δ L ) is a negative number. Therefore, the decrease in the latent pool due to multiplying the duration of therapy will always be larger than an equivalent multiplication of the rate (unless the r < 1, which is a nonsensical proposition equivalent to making therapy less effective).

The minimal impact of allowing active to latent transitions
In some modeling works a transition from active to latent cells has been included [6,44]. If we add that term to the model as a rate φ the equations for the latent and active pool (decoupled from the virus and susceptible as before) arė ? =0.81 Figure S5: Solutions of Eq. S18 compared to the compound interest model L = L 0 exp(θ L t). Only for unrealistically large values of the return to latency rate φ does the solution deviate from the exponential decay.

Composition of the reservoir: modeling T-cell subsets
It has been demonstrated that several subsets of CD4 + T cells may contribute to the reservoir [19,32,33]. We reviewed the existing data to find the rates of proliferation for each subset, as well as estimates of the total fractional makeup of the reservoir.
The typical T cell phenotypes considered to make up the reservoir are central memory T cm , effector memory T em , and naïve T n cells. Transitional memory T tm have also been described to represent a transition from T cm → T em . For the model we consider T tm to have equivalent proliferation rates to T cm . Similarly, we characterize newly recognized stem-cell-like memory CD4 + T cells (T scm ) as T n given their likely slow turnover rate. Of note, these are conservative estimates that would not favor anti-proliferative therapy.
We include the T cell diversity into the decoupled model by breaking the differential equation for the latent pool down into three differential equations, one for each subtype L i with i ∈ [cm, em, n]. We ignore transitions between types because on average the composition of the reservoir is reasonably stable over time and thus our model is the systeṁ (S18) The total number of latent cells is the sum of the subset numbers, L = i L i . We only use three types mentioned above (including stem-cell-like and transitional memory cells into naïve and central memory sets respectively). Solutions for the total latent pool when decoupled on ART are thus linear combinations where θ i = α i − δ i − ξ, L i (0) are the initial numbers of each subtype, and because we have assumed there are no couplings between types.
As in Fig. 3 in the main body, simulations assume the same original net clearance rate and activation rates, but different proliferation rates α i and thus different death rates δ i . When the proliferation rate is reduced using an anti-proliferative therapy, the clearance rates are different among types. The initial conditions for each type L i (0) are inferred from Chomont et al. In their models, α L parameter values are chosen between 0.011 and 0.03 per day, consistent with our value of 0.015 [6].
The activation rate ξ is several orders of magnitude smaller than the proliferation rate. Using parameters calculated by Luo et al. from structured treatment interruptions, Hill et al. found that on average, 57 CD4 + T cells per day transition from latency to the activated state [14,8].
The half-life estimate of the latent pool from Siliciano et al.'s quantitative viral outgrowth assay yields the total clearance rate of latent cells, θ L = -5.2 ×10 −4 per day [2]. This result was corroborated closely by Crooks et al. [17]. Thus, the clearance, proliferation, and activations rates are derived from human experiments. However, the death rate is calculated. The total clearance rate of the latent pool is the death and activation rates subtracted from the proliferation rate.
A rearrangement of this equation (using central memory cell values) gives the daily death rate A large body of work has been reviewed in de Boer and Perelson on the variety of experimental techniques employed to measure the turnover of T cells [40]. The modeling and experimental challenges are discussed, and we include the reference here to emphasize how much uncertainty in the parameters indeed exists. The uncertainty and sensitivity analysis in §3 are included to demonstrate our awareness of the uncertainty and provide theoretical ranges for plausible outcomes given the varying measurements of these rates.

Fractional makeup of the reservoir
Chomont et al. determined the percentage of each cellular subset from 31 aviremic individuals.
They found mean contributions of T cm + T tm ≈ 85%, whereas T em ≈ 13% and T n ≈ 2% [19]. Buzon et al. 2014 found higher percentages of naïve and stem-cell-like memory cells, up to 40% [33]. Thus in the model inclusive of diversity we set the initial conditions of each subtype proportional to their fractional makeup, e.g. L i (0) = [85, 13, 2] for the example above.

Susceptible and activated cell dynamics
The production of CD4 + T cells from the bone marrow and thymus is described by α S ; and δ S is the rate of susceptible T cell death. In several of the early HIV modeling papers, a value of 10 per µL-day was estimated for the production rate [39], [45] with δ S estimated at 0.02.
Luo et al. used a Bayesian Markov-Chain Monte-Carlo method to estimate HIV model parameters [14]. They used data from 10 patients who underwent a series of 30-day ART treatment interruptions with viral loads taken three times weekly following interruptions and then weekly following Carlo) and fit their model to data from [46], an AIDS clinical trial comparing dosing regimens for indinavir and ritonavir [47]. This model also incorporated adherence, drug concentrations, and drug susceptibilities [47]. They find α S = 98.1, δ S = 0.08, and δ A = 0.37. Note in Table S3 that the ratio of α S to δ S varies between 500 -1639 with the more recent experiments in relative agreement near 1500. Thus, we chose α S = 300, δ S = 0.2 to reflect this ratio.
The death rate of productively infected cells δ A was initially thought to be 0.24 [39]. Later, Perelson et al. find δ A = 0.5 based on frequent sampling of 5 patients after giving ritonavir monotherapy [7]. Using a fixed viral clearance rate γ = 23 from [48], Markowitz et al. determined δ A = 1 based on potent antiretroviral therapy with lopinavir-ritonavir, tenofovir, lamivudine, and efavirenz in 5 chronically-infected patients [49]. We chose δ A = 1.0 to reflect more recent experimental findings.
We assign α A = 0, as the relative rate of proliferation of actively infected cells likely occurs at neglible rates compared to the death rate of these cells [39].

Viral dynamics
The viral 'burst rate' π is the amount of virus a single actively infected cell emits in a day (roughly its lifetime). Rong  . Both of these estimates were obtained experimentally rather than derived from models; however, they do not necessarily reflect the number of copies produced in a day per cell or in a cell's lifetime, rather the amount that was being produced at the instant the experiments were performed. In any case, the value of π appears to be in the neighborhood of 10 3 , which is the value we will adopt. For the viral clearance rate γ, we use  Table S5: Viral burst and clearance rates as estimated in prior experiments and mathematical models. Note: [14] used the data from [48] but used the geometric mean rather than the arithmetic mean, i.e. 18.8 rather than 23.
The 'latent cell fraction' τ is the rate at which newly infected cells join the latent cell pool, whereas 1 − τ is the rate at which they join the actively infected pool. There are few estimates available for this parameter, and the estimates vary across a wide range. Despite this, the choice of τ within the given range does not affect our cure estimates or our estimate of the critical epsilon ( c ). Because their estimates are based on modern measurements of the reservoir, we chose Conway and Perelson's value τ = 10 −4 , which is the upper bound on their estimates ranging from 10 −7 to 10 −4 [10].
Parameter Callaway [45] Jones [52] Conway [10] Units τ 10 −6 10 −3 10 −4 unitless Table S6: Latency fraction as estimated in prior experiments and mathematical models. Working within the intervals given in Table S7 we first conduct a local uncertainty analysis [53] that shows how anti-proliferative therapy coupled with ART depletes the latent reservoir. First, we fix each parameters to its typical value and vary one parameter at a time within its bounds as given in Table S7 and with the constraint that R ART 0 < 1. The results are shown in Fig. S6. We see that the variables affecting the basic reproductive number (here for example τ , ART ) have no effect on the time to cure. As we have shown, if R ART 0 < 1, the dynamics of the latent reservoir effectively decouple from the viral dynamics. The original number of cells in the reservoir has a large impact on the time to cure. Additionally, as discussed in the main body, the makeup of the reservoir impacts the clearance: a higher percentage of naïve cells slows decay and extends the time to cure. Also, the value of the clearance rate changes time to cure as expected by the compound interest framework. However, the expected range of this value is not large, so our results are not drastically changed by this variation.
We also complete a global uncertainty and sensitivity analysis in which all variables for the model are drawn from uniform distributions based on the ranges in Table S7 and sampled using Latin Hypercube sampling [53]. Any variable whose range spans more than one order of magnitude is  Table S7: A summary of all parameters used in our simulations. * δ L is back-calculated from known α L , θ L , and ξ. 95% confidence intervals are given in parentheses () where applicable from experimental data. Otherwise, the range is taken from our literature search or from the ranges given in the cited works not assumed to be normally distributed; these values are given in square brackets [].
sampled to achieve even sampling on a logarithmic scale. The simulations are carried out in Matlab using lhsdesign and ode23s. We note that a stochastic model is not necessary because we are studying decay dynamics with large numbers of cells as opposed to rebound dynamics.
The simulations from the global uncertainty analysis are used in a sensitivity analysis to correlate each parameter with time to cure on ART and anti-proliferative therapy. Time to cure is defined by the number of latent cells as compared to the thresholds described in the main body. We calculate correlation with the the Pearson correlation coefficient ρ T,p , the ratio of the covariance of the time to cure T with the parameter of interest p normalized by each's standard deviation: where · indicates the expectation value. The results of the simulations are shown in Fig. S7.  Figure S6: Local uncertainty analysis of the complete model, Eq. S1 where ART is coupled with anti-proliferative therapy (potency AP = 5) to analyze the variability of solutions and time to cure. Note that the model includes varying phenotypes of T cells with varying proliferation rates as in Table S7. Each parameter that is varied is listed above the panel with the range of variability shown in the panel legend, and all other values are constant as in the Value column of Table S7.
One thousand simulations are carried out and only the parameter ranges that obey R ART We suspect that the anti-proliferative drugs are not successful in all patients, so that the 3 patients who had no reservoir reduction may not have responded well to MMF. The suspicion is corroborated in the next subsection by the fact that the authors of a separate MMF study had to rearrange their clinical cohorts to deal with participants who received MMF but showed no anti-proliferative effect.

Comparison with time to rebound from García et al. 2004
In 2004, García et al. also assessed the effect of MMF in treatment interruption studies of ART [28]. In the study, seventeen HIV patients received ART for a year. They were then randomized into two groups, the control group that remained on ART only, and the experimental group that also received MMF for 17 weeks. After this time, ART was interrupted in both groups and viral rebound was measured. The benchmark to compare the two groups was to be the number of individuals who maintained a viral load set-point below 200 copies/mL. 5/9 in the experimental group and 1/6 in the control group met this standard, though this was not statistically significant.
The MMF recipients were subsequently reclassified into a group of the 6 that successfully inhibited proliferation (as measured by an in vitro assay) and a group of 3 with no clear inhibition. The viral load set-point measurement was indeed significantly different between these two groups, as was the time to rebound. The authors provide two mechanisms by which they expect MMF to control HIV infection: 1) antiviral in that MMF may deplete a substrate necessary for reverse transcription, and 2) immunologic in that the number of target cells is reduced.
Previous studies allow us to estimate the possible decrease in reservoir size due to MMF or similarly potent anti-proliferative therapy in general. Examining the patient data of García et al. (Fig. 1 of Ref. [28]), we see that the times to achieving a viral load above the typical detectable limit (10-50 copies per mL) is 1-4 weeks for the control group and 6-12 weeks for the MMF responder group.
We can compare this experimental data with several theoretical predictions for the fold decrease in the latent reservoir. Using the results of Pinkevych et al., a median time-to-detection of roughly 7 weeks (see Fig. 5B of Ref. [21]) corresponds to a 7-fold decrease in the latent reservoir. Using the results of Hill et al. the same median time-to-detection of roughly 7 weeks (see Fig. 4 of Ref. [8]) corresponds to a 50-fold reduction in the latent reservoir. The result of Hill matches well with our calculation. Because MMF was given to patients in the study for 17 weeks, and we assume the potency of MMF is above a factor of 2 ( AP > 2) [29], our simulations suggest that the size of the latent reservoir would be reduced by approximately a factor of 40 (visualized in Fig. 1D in the main body).

Bone marrow/thymic production of T cells and the safety of MMF as an anti-proliferative therapy
There is some debate about how the susceptible cells are sustained [54,55,56]. In our model we assumed that bone marrow or thymic production is the mechanism that produces T cells. However, density-dependent growth could also contribute to the production of T cells. To account for this possibility, we and other modelers consider this effect to be grouped into the density-dependent death rate δ S . Thus, assuming that anti-proliferative drugs have no impact on thymic/bone marrow production, the total number of T cells should stay roughly constant. To be precise, the equilibrium solution for the susceptible cells S * = α S /δ S depends only on bone marrow/thymic production and death rate. Perhaps more important than our theoretical perspective, experimental results from studies discussed in the previous 2 subsections show that the total number of CD4 + and CD8 + T cells stayed roughly constant even as the number of proliferating cells decreased on MMF [29,28].
Beyond the stability of CD4 + T cell counts, the safety of MMF should be vetted in effective dosages in HIV patients. Again, in the Chapuis and Garciái studies, no opportunistic infections or adverse events were observed, but these were performed in small populations for relatively limited durations [29,28]. On the other hand, MMF has been given in many patients at higher dosages and for much longer durations in autoimmune disease and solid organ transplantation though much of this data is difficult to interpret given the usual, concurrent administration of steroids, calcineurin-inhibitors, and/or cyclosporine [35]. However, Kriss et al. report retrospectively on 23 liver transplant patients who were transitioned to MMF monotherapy after a median of 2 years on calcineurin-inhibitors +/-MMF. Following conversion to MMF monotherapy, patients were followed for an average of 4 years. During that time, one patient discontinued MMF due to persistent diarrhea; one required dose-reduction due to anemia; one developed zoster with resolution on acyclovir; and one developed post-transplant lymphoproliferative disorder (PTLD) but achieved complete remission with MMF dose-reduction and R-CHOP chemotherapy. Thus, the incidence of adverse events over 4 years was small; but if the case of PTLD was caused by the MMF, the consequences are too severe to merit treatment in HIV patients controlled on ART. The median dose in this trial was 2g/d, the same dose given in [29] and twice the dose given in [28]. Questions of causation and relatedness to dosage remain to be answered, but we propose worthy to be attempted with low doses and careful monitoring.
As a final note, it is also possible that the number of divisions resulting in cell proliferation is finite.
This upper bound on rounds of proliferation is called the "Hayflick limit." This limit is unlikely to apply to our model, as destructive telomere shortening is not as relevant for T cells as other cell types. See Ref. [57] for further discussion and references.