Predicting the Long-Term Impact of Antiretroviral Therapy Scale-Up on Population Incidence of Tuberculosis

Objective To investigate the impact of antiretroviral therapy (ART) on long-term population-level tuberculosis disease (TB) incidence in sub-Saharan Africa. Methods We used a mathematical model to consider the effect of different assumptions about life expectancy and TB risk during long-term ART under alternative scenarios for trends in population HIV incidence and ART coverage. Results All the scenarios we explored predicted that the widespread introduction of ART would initially reduce population-level TB incidence. However, many modelled scenarios projected a rebound in population-level TB incidence after around 20 years. This rebound was predicted to exceed the TB incidence present before ART scale-up if decreases in HIV incidence during the same period were not sufficiently rapid or if the protective effect of ART on TB was not sustained. Nevertheless, most scenarios predicted a reduction in the cumulative TB incidence when accompanied by a relative decline in HIV incidence of more than 10% each year. Conclusions Despite short-term benefits of ART scale-up on population TB incidence in sub-Saharan Africa, longer-term projections raise the possibility of a rebound in TB incidence. This highlights the importance of sustaining good adherence and immunologic response to ART and, crucially, the need for effective HIV preventive interventions, including early widespread implementation of ART.


Model of life-expectancy
Mortality on ART has been shown to depend strongly on age and CD4 cell count at initiation [1,4]. We construct a life-expectancy at ART initiation that follows the general patterns described in [1], while allowing flexible variation. We take life-expectancy on starting ART to be where [X] + = max(X, 0), and where CD4 A and CD4 1 are points in CD4count schedule described in Section 2 and Figure 1B in the main text. This means that those who state ART later, for a given age, live less long than those who start early. More life-years are lost to later starts as γ increases. For γ < 1, differences in CD4 counts for ART initiation have a greater impact on survival at low CD4 counts than at high CD4 counts.

Model of TB risk and CD4 positive cell count decline
We assume HIV-uninfected individuals have a CD4 cell count of 1000 cells/µL and that, as in [3], this declines immediately by 25% upon HIV infection. Their CD4 cell count is then assumed to decline linearly to death at τ m . Upon starting ART, an individual's CD4 cell count is assumed to be boosted by an amount ∆. In the main paper ∆ is chosen in conjunction with the risk model of Section 4 to produce the required rate ratio for TB on ART. Given our interest in the long-term, we caricature an increase that in  reality occurs over a few years [2] after starting ART as instantaneous. The CD4 level achieved and the associated protection from TB should therefore be characteristic of a patient on ART once this initial rise has taken place. A linear decline over the individual's lifetime on ART is assumed to result in the loss of a fraction f of this new CD4 count at time of death τ m , due to immunologic failure, imperfect adherence, or other reasons. f = 1 corresponds to our 'pessimistic' scenario, wherein an individual's CD4 count declines to zero, although at a slower rate than untreated individuals. f = 0 corresponds to our 'optimistic' scenario, wherein the level of CD4 reconstitution and protection from TB are maintained throughout an individual's life, until death at τ m .
This implies: 3 HIV/ART model Figure 1A in the main text summarizes the logic of the model. Individuals are either in class I -infected with HIV, not on ART; or class A -infected with HIV, on ART. New individuals enter the model on the left at a rate that may vary with calendar time, and propagate rightwards. Those who are not started on ART die at point τ m , whereas those who are started on ART transit to class A at point τ A and die at a later point τ m . The mathematical details are described in the sequel.

HIV only
Let I(t, a, τ ; τ m ) be the number of individuals at time t that are aged a, where infected with HIV time τ ago, are not on ART and who will die a time τ m after their initial infection in the absence of treatment. This quantity satisfies the partial differential equation where the mortality is defined in terms of a Dirac delta function enforcing death τ m years after infection: and with boundary condition Here, i(t) is the HIV incidence at calendar time t, φ(a) represents the distribution of ages at HIV infection, and P (τ m ) the distribution of life-expectancy at HIV-infection in the absence of treatment. The distributions P and φ are modelled as Weibull distributions (as described in the main text), and averages over these distributions performed as described later in this document. Defining Equations 6 & 8 can be solved by the method of characteristics to give andĨ(0, a, τ ; τ m ) determined by the initial condition.

ART population
We imagine that a proportion V (t) at time t of those reaching a threshold time-since-infection τ = τ A (τ m ) commence ART. The dependence of τ A on τ m allows this threshold to be specified in terms of a CD4 cell count threshold, given a model that determines CD4 cell count in terms of τ and τ m , like Equation 5. We assume that a new time of death τ m is determined by the age, a, time of ART commencement, τ A , and original time of death τ m . Given our dependence of τ A on τ m , we can treat the new time of death as a function τ m (a, τ m ). Analogously to Equation 6, A(t, a, τ ; τ m ), the number of individuals on ART at time t with age a, who were HIV-infected a time τ ago, and with time of death without ART τ m , satisfies with and with the right hand side of Equation 15 understood to be evaluated with τ just less than τ A and imposing A(t, a, τ ; τ m ) = 0 for τ < τ A .

Combined solution
If at t = 0 no-one is on ART, the overall solution for A and I is specified by and with the boundary condition forĨ specified by Equation 8.

Starting from equilibrium
If the system begins at equilibrium at t = 0 with HIV incidence equal to i 0 , a constant birth-rate, and no-one on ART, and thereafter the HIV incidence follows Equation 8, then this solution becomes

TB incidence
We will take the hazard ratio for developing incident TB to be different according to whether on or off ART, and functions of τ and τ m (τ m for those on ART), denoted: h H (τ, τ m ) and h A (τ, τ m ) respectively. Let H denote the respective cumulative hazards. The total TB incidence in those infected with HIV,Ḋ(t) is given bẏ

Equilibrium incidence
We calculate the new equilibrium reached after the introduction of ART, when HIV incidence is constant. Writing i(0) = i 0 , andV = (1 − V ) for the proportion not starting ART, and using Equations 21 & 22, d H becomes where we have used the substitution x = a − τ , and used the shorthands τ m = τ m (x + τ A , τ m ) and a ∧ b = min(a, b).
with τ m = τ m (x + τ A , τ m ). This is easily interpreted in terms of areas under portions of the hazard curves in Figure 3: the first term is the area under the curve up to τ = τ A ; the second term is (1−V ) times the area under the curve from τ A to τ m ; the last term is V times an age-average of the area under the ART hazard curve from τ A to τ m . It is thus clear that the equilibrium TB incidence will be higher after ART roll-out when the population mean of the forward-hatched area exceeds the mean of the backward-hatched area.

Specific solution
Following [3], we take TB risk to depend exponentially on the decline in CD4 cell counts: (32) withρ = 0.36 per 100 CD4 cells/µL lost. The TB incidences in the model depend directly only on the model of hazard ratios through time (i.e. Figure 3). We have arrived at these via a model of CD4-decline and the an assumption that CD4 cell count exponentially influences the rate of TB in the same way for HIV-infected individuals regardless of whether they are on ART. For the model of CD4 cell count decline We will also take V (t) to be: illustrated for = 5 years in Figure 4, and HIV incidence to be declining at a rate r for t > 0: Then, from Equation 27 where it is understood that the integrals and square brackets evaluate to zero when the top limit is smaller than the bottom, i.e.
and where a ∨ b = max(a, b). The contribution to TB incidence from those on ART is similarly given by where with τ m = τ m (x + τ A , τ m ) and y = τ m ∧ t and With the model of CD4 cell count decline described in Section 2, we have where Comparing with Equation 34, we have The unevaluated integrals in Equations 25, 26 & 39 can be calculated in a Monte Carlo fashion by sampling τ m and x from the distributions P and φ respectively and taking the mean.

Supplementary results
In Figure 5 we display contour plots identical to Figure 1(d) in the main paper, but including different ART initiation thresholds. The IRR, α, marked with the horizontal dashed line corresponds to a protection from ART that brings the TB rate just after ART initiation down to 4.4 times the general population. The uncertainty envelopes arising from varying life-expectancy become broader at low CD4 count since a fixed range of values for γ (see Section 1) results in a larger percentage variation in life-expectancy for lower ART initiation CD4 thresholds, i.e. the model of life-expectancy is more sensitive to the parameter γ for late ART starts. Figure 6 we display contour plots identical to Figure 1(d) in the main paper, but mapping the regions where any subsequent peak in TB incidence among people living with HIV is higher than the initial level.

Comparison of notation
In the main text γ is denoted p, α is denoted a, and l is used to represent (τ A /τ m ).  Figure 5: Contours of regions where the cumulative TB incidence over 50 years is equal to the scenario without ART for different ART initiation CD4 thresholds. Other parameters and interpretation as for Figure 1(d) in the main paper. Below and right of the contours, cumulative incidence is lower than the initial level; above and left of the contours, the peak is higher. See Section 5 for discussion of α.  Figure 6: Contours of regions where the peak in subsequent TB incidence is equal to the initial level for different ART initiation CD4 thresholds. Other parameters and interpretation as for Figure 1(d) in the main paper. Below and right of the contours, any subsequent peak is lower than the initial level; above and left of the contours, the peak is higher. See Section 5 for discussion of α.