Averting epidemics of extensively drug-resistant tuberculosis

Abstract

Extensively drug-resistant tuberculosis (XDR TB) has been detected in most provinces of South Africa, particularly in the KwaZulu-Natal province where several hundred cases have been reported since 2004. We analyzed the transmission dynamics of XDR TB in the region using mathematical models, and observed that nosocomial transmission clusters of XDR TB may emerge into community-based epidemics under the public health conditions of many South African communities. The effective reproductive number of XDR TB in KwaZulu-Natal may be around 2. Intensified community-based case finding and therapy appears critical to curtailing transmission. In the setting of delayed disease presentation and high system demand, improved diagnostic approaches may need to be employed in community-based programs rather than exclusively at tertiary hospitals. Using branching process mathematics, we observed that early, community-based drug-susceptibility testing and effective XDR therapy could help curtail ongoing transmission and reduce the probability of XDR TB epidemics in neighboring territories.

Extensively drug-resistant tuberculosis (XDR TB) poses a grave public health threat, particularly in populations with high HIV prevalence and overburdened health care systems. XDR TB is caused by Mycobacterium tuberculosis resistant to the first-line drugs isoniazid and rifampin, and to any fluoroquinolone and to at least 1 of 3 injectable second-line drugs (1). The disease has been identified in all regions of the world (2), including nearly all provinces of South Africa (3, 4). Several hundred cases, including outbreak clusters indicative of extensive primary transmission (5), have been reported in the KwaZulu-Natal province (Fig. 1A).

Fig. 1.

Burden and dynamics of XDR TB. (A) Reported XDR TB cases in South Africa, by province, from January 2004 to March 2007. The KwaZulu-Natal province has reported the highest number of cases to date. (B) Flow diagram of the model. Subscripts “c” and “h” refer to the community and hospital environments, respectively (separated by a dotted vertical line). Non-XDR patients can be admitted to the ward for other TB disease, and risk superinfection/nosocomial infection. Admission and discharge are symbolized by gray arrows. Health states are further stratified by HIV and antiretroviral therapy status, and mortality is subtracted from all compartments. The model equations are detailed in SI Appendix. Persons can be uninfected with XDR TB (S), latently-infected (L; long latency; E, brief latency), actively infected and not yet detected (U, undetected for TB generally; I, detected for TB but placed on empirical first-line therapy before drug-susceptibility testing; X, queueing for drug-susceptibility testing as a result of having suspected XDR; D, defaulted from the testing and treatment system), identified as XDR and initiated on XDR therapy (T_h reflecting inpatient XDR therapy; T_c reflecting potential community-based XDR therapy programs), or recovered after therapy (R, among a proportion for whom XDR therapy is potentially effective). Proportion z of the population is also infected with non-XDR TB strains. Dashed lines indicate a strategy of early outpatient screening among detected TB cases for XDR TB and potential community-based XDR therapy.

Weeks or months are required to detect mycobacterial drug resistance through the culture and drug-susceptibility testing methods commonly available in the region (6). The capacity for safe airborne isolation also does not exist in South African hospitals (7, 8). Therefore, most XDR TB patients are not detected as having XDR for months, and are often hospitalized in communal wards together with other patients. Nosocomial transmission of the disease has been suspected, because such transmission has been a predominant feature of prior drug-resistant TB outbreaks, particularly among HIV-infected individuals (9). At the district hospital in Tugela Ferry, a region in the KwaZulu-Natal province where XDR TB was first reported, transmission of XDR TB was observed among mostly HIV-infected patients with recent histories of hospitalization (10). Subsequently, a common new strain (F15/LAM4/KZN) was identified as the principle circulating pathogen generating XDR TB disease in the province (11).

We studied hospital-based XDR TB infection control measures to reduce transmission of the disease at the initial epidemic reporting site (7, 8). Continuing concerns about nosocomial transmission have resulted in considerable efforts to improve hospital conditions. However, transmission of XDR TB in the community has not been as widely addressed. Many patients submit sputum samples to outpatient clinics to diagnose TB and test for drug resistance, but those who have cultures indicative of drug-resistance are often lost to follow-up when test results are received, weeks after sputum collection. In addition, we found that at least 1,790 (72%) of 2,476 patients with multidrug resistant TB (cases resistant to at least isoniazid and rifampin, including XDR TB cases) diagnosed in 2006 in the KwaZulu-Natal province were not known to have received therapy (3). These patients are believed to be either in the community or to have died, having been lost from the care system. XDR TB has been recently observed among individuals without histories of hospitalization (suggesting community transmission), among those without HIV coinfection (suggesting evolution of the epidemic beyond those at highest risk), and in areas without second-line TB drugs (suggesting migration of cases between regions) (3, 12, 13). Therefore, a critical question is whether XDR TB transmission is likely to be restricted to hospital wards and small sectors of the population, or emerge into a community-based epidemic.

Here, we evaluated the determinants of XDR TB epidemics. We calculated the effective reproductive number (R) for the disease, which is the number of secondary cases generated by each primary case in the context of ongoing disease control efforts and partial immunity from other TB infections (14). A mathematical model was used for this estimate (Fig. 1B). We simulated interventions to drive R below a critical threshold of 1, which is required to curtail epidemic transmission. The effectiveness of key measures debated in the control effort against XDR TB were compared, including intensified case finding, infection control, and community-based XDR therapy. Branching process mathematics were also used to evaluate the impact of these interventions on the probability of XDR TB epidemics in different communities.

Results

Epidemic Trajectory.

Stochastic simulations of our model (Fig. 1B) revealed that XDR TB outbreaks initiated by a single hospital-based case typically developed into community-based epidemics (Fig. 2A). What factors contributed to this epidemic course? We observed that delays in identifying and treating XDR patients, and losses from the queue as patients awaited inpatient therapy, were critical to establishing community-based transmission. Even when patients were nosocomially-infected with XDR TB, they often were discharged from the hospital with latent XDR TB. When patients presented to outpatient health clinics after latent XDR disease became active symptomatic disease, the waiting times for testing and therapy were often several months long (Fig. 2B). Long delays in obtaining drug-susceptibility test results, and shortages in the number of beds available to admit patients for inpatient XDR treatment, maintained infectious individuals in the community to produce community-based transmission.

Fig. 2.

Epidemic dynamics. (A) Epidemic trajectory. Results of 10,000 stochastic simulations displaying the cumulative nosocomial (red) and community-based (blue) XDR TB infection rates in a population of 100,000 served by a 60 bed TB hospital facility. Note epidemic die-out occurs in some simulations (bottom of error bars). (B) Queuing for care. Waiting times for therapy under the current system (blue), with rapid drug-susceptibility testing of 1 week turnaround (green), or with community-based XDR TB screening and therapy with capacity of 20 treatment slots at a time per 100,000 population (red; 20 slots per 100,000 have been provided in the Tugela Ferry pilot program). Cumulative XDR TB mortality among those patients waiting for therapy in each system (dashed lines) is also displayed. The y axis scale is both the waiting time to obtain therapy (in days) and deaths among those waiting (per 100,000 population).

System Delays.

We observed that rapid drug-susceptibility testing to obtain XDR results in 1 week, instead of the current delay of 6 weeks, reduced the typical waiting time for XDR detection and therapy from 220 days to 40 days (Fig. 2B) (the waiting time is from the point of sputum submission for drug-susceptibility testing among XDR suspects to the point of XDR therapy). Cumulative 5-year mortality among this population, however, reduced only from 230 to 215 deaths per 100,000 population under this rapid testing intervention. An alternative strategy involved screening all newly-diagnosed TB cases for XDR TB (drug-susceptibility testing) in outpatient health posts, and providing XDR cases with community-based XDR therapy. This early screening and therapy strategy involved testing all new TB patients for drug-resistance at the time of TB diagnosis (to detect primary XDR transmission), rather than waiting for 2 months of nonresponse to empiric first-line therapy before persons become XDR TB “suspects” and undergo drug-susceptibility tests (the current approach) (15). Of course, early screening does not apply to those patients who acquire XDR resistance while currently on treatment for non-XDR TB (hence, early screening applies to primary transmission rather than acquired resistance) (Fig. 1B). The early screening and treatment strategy also involves outpatient therapy rather than exclusive inpatient therapy. The strategy reduced delays from 220 days to zero days near the beginning of the epidemic, and from 220 days to 110 days by year 5 of the epidemic as the community-based therapy queue filled (Fig. 2B). Early screening and treatment was more successful than rapid testing technology at averting mortality, preventing just >50 deaths per 100,000 population over a 5-year epidemic course. This community-based XDR screening and outpatient treatment process was able to identify cases of XDR TB earlier in the infectious period, reducing transmission and mortality (Fig. 2B), even though the risk of community-based transmission and further amplification of resistance was included in the model.

Control Efforts.

We performed sensitivity analyses to examine how addressing waiting times and other key health system challenges could alter R and affect epidemic transmission of XDR TB (Fig. 3). Using demographic and treatment parameters from Tugela Ferry, where XDR TB was first identified (see Methods and SI Appendix), our model produced an estimate of 1.62 for R. However, when we performed uncertainty analyses around the key demographic and system parameters from South African communities generally, incorporating delays and system limitations from all provinces (SI Appendix), the average value of R increased to 1.97 (Fig. 4) (range 0.7–4.6).

Fig. 3.

Sensitivity analysis of the effective reproductive number. Contour plots of R (right-side color bar) are displayed under different transmission conditions and public health measures. See Table 1 for parameter definitions. (A) Varying the efficacy of infection control on the TB ward (y axis) and the speed of drug-susceptibility test results (x axis). (B) Varying the case detection rate of incident TB cases (y axis) and the proportion of cases lost to follow-up between drug-susceptibility testing and the initiation of XDR therapy (x axis). (C) Varying the efficacy of XDR therapy (y axis) and access to outpatient community-based drug-susceptibility testing (DST) and XDR therapy (x axis).

Fig. 4.

Distributions of R and the probability of an XDR TB epidemic. Baseline, existing health system; default, reduce the rate of default from therapy to 5%; detection, increase case detection to 70%; DST loss, prevent loss to follow-up during the drug-susceptibility testing process and accelerate turnaround of results to 1 week; nosocomial, institute fully effective infection control program to avert nosocomial transmission; CBST, institute community-based outpatient DST screening and XDR therapy; combined, combine all of the preceding measures. (A) Box plots of R displaying 10,000 iterations of Latin Hypercube Sampling from the probability distributions of the input parameters. The horizontal mark is the median, the box designates the interquartile range, whiskers extend to 1.5 times the interquartile range, and outliers are plotted individually. (B) Probabilities of XDR TB epidemics under each of the simulated public health control measures. (C) Variation in the probability of XDR TB epidemics in communities with different adult HIV prevalences and congregate TB hospital ward occupancy (per 100,000 persons).

Surprisingly, although hospital-based infection control measures and rapid drug-susceptibility tests in hospitals were helpful to reducing the reproductive number, the value of R was not typically reduced to <1 by these measures alone (Fig. 3A). A system that implemented perfect infection control measures, after previously having no infection control measures [as is the case with most South African district hospitals (6)], would reduce R by at most 0.3. Hence, infection control alone did not reduce R sufficiently to curtail long-term epidemic transmission. Accelerating drug-susceptibility test results from 6 weeks to <1 week reduced R by at most 0.1. This strategy alone did not fully relieve the queue for therapy given limited hospital treatment beds, and did not change the delay caused by waiting to screen many patients for drug-resistance after 2 months of empiric first-line treatment, which is the current South African standard for newly-detected TB cases before they are suspected of having drug-resistance and tested for MDR or XDR TB.

We found that the estimate of R was more sensitive to community-based interventions than to interventions exclusively focused on hospitals (Fig. 3 B and C). Access to early XDR screening by drug-susceptibility testing in outpatient health clinics combined with follow-up community-based XDR therapy (15), was able to reduce R by as much as 0.4 (Fig. 3C). Approximately 2/3 of this impact was from the screening aspect alone, by virtue of earlier identification and earlier queuing for suppressive therapy (reducing transmission). R was reduced by community-based XDR therapy even though we incorporated the possibility of sustained community-based transmission and amplified resistance as potential risks of the community-based XDR therapy strategy. R was also sensitive to the proportion of patients lost to follow-up while waiting in the community for drug-susceptibility tests results (Fig. 3B). A 20% drop in the proportion of patients lost to follow-up could reduce R by 0.2.

Taken together, the estimated values of R reduced from 1.97 (range 0.7–4.6) to 1.23 (0.4–3.1) when community-based screening and therapy was combined with the measures included in the new South African strategic plan to control TB (Fig. 4A); this involves averting nosocomial transmission, improving case detection from 55% to 70%, eliminating loss to follow-up, reducing treatment default <5%, and improving drug-susceptibility testing speeds to 1 week turnaround (4). Without community-based screening and therapy, R was reduced from 1.97 to 1.38 (range 0.6–3.3) by the South African strategic plan alone. Uncertainty analyses around these estimates revealed positively-skewed distributions in which some combinations of parameters produced values of R > 4 (Fig. 4A; outliers are plotted individually). Less than 20% of simulations produced R < 1 in the context of existing control efforts, whereas >40% produced R < 1 when all of the simulated interventions where combined, including those in the strategic plan and the community-based screening and treatment approach.

Probability of XDR TB Epidemics.

Using a branching process model to estimate the probability of an epidemic from the distributions of R in the population, we estimated that the probability of an XDR TB epidemic reduced from 12% in the context of current control efforts to just >8% with intensified community-based screening and treatment alone (Fig. 4B). With a combination of this screening and treatment measure, and the other measures included in the strategic plan, the probability of an epidemic reduced to 2.5% in this model (Fig. 4B). Because these point values are estimates based on the distributions of reproductive numbers in the population, they will inherent vary with heterogeneity in the population.

The probability of an epidemic may also vary with community size, because smaller communities may experience rapid die-out. Within a plausible range of South African community population sizes, which we varied from 50,000 to 500,000, we found that our estimates of the probability of an epidemic changed by <1%. Upon further sensitivity analysis, however, we observed that the probability was more sensitive to the HIV prevalence and the size of congregate TB wards. Both a high HIV prevalence and large congregate hospital wards increased the probability of an epidemic to as high as 15% among South African communities with extensive adult HIV prevalence rates and large TB ward occupancy; in contrast, the probability lowered to 4% without congregate TB wards or HIV (Fig. 4C).

Discussion

XDR TB transmission is a significant concern for TB control efforts in regions with high HIV prevalence and strained health systems. Detection of XDR TB appears markedly delayed in South Africa. An analysis of the TB detection and treatment system in the country reveals that most individuals who acquire XDR TB through primary transmission must fail empirical first-line therapy and wait for weeks or months before being detected as nonrespondents to therapy and subsequently undergo a drug-susceptibility test. These individuals are then subjected to further delays as limited hospital capacity to provide inpatient treatment results in long queues for therapy, maintaining infectious individuals in the community and often losing them from follow-up treatment. Indeed, XDR TB treatment queues at Durban's central hospital are typically longer than 70 patients (16). We observe XDR TB patients to be at end-stage disease when they are ultimately admitted for inpatient therapy (10), surviving for only a few weeks on average. In our model, epidemics of XDR TB in South African hospitals typically emerged into community-based epidemics during the first few years of transmission. Individuals who are infected in hospital wards can manifest active XDR TB in the community, and delays in identification and therapy allow the majority of infected persons' infectious periods to be spent in the community before detection and treatment.

The two most common recommendations for averting transmission and reducing R in South Africa are to introduce infection control measures and rapid drug-susceptibility tests in hospitals (17). Surprisingly, we observed that these hospital-based interventions may not be sufficient for epidemic control when accounting for other health system features. Reducing nosocomial transmission or the speed of detection among patients who survive long enough to be admitted to the hospital did not alleviate community-based delays in identification. XDR TB patients are delayed in being identified as suspects, and limited system capacity to treat XDR TB patients was more effectively addressed by community-based screening through drug-susceptibility testing when patients first present to outpatient clinics with TB symptoms. XDR TB treatment in the community was also found to have a net benefit despite the potential for community-based transmission and amplified resistance, which were risks of the strategy included in this model. The withdrawal of patients from the long queue for inpatient treatment and suppression of their infectiousness was beneficial for epidemic control. Even if XDR TB therapy is unavailable or ineffective, early community-based screening through wider access of drug-susceptibility tests to local health clinics was substantially beneficial in reducing transmission by allowing persons to more quickly enter treatment queues for hospital therapy rather than undergoing empiric first-line therapy before becoming XDR TB suspects.

If regions neighboring XDR TB hot zones were to undertake the interventions we simulated, how much protection might they afford? How likely is the spread of XDR TB in neighboring provinces? The emergence of epidemics is a stochastic process, in which stochastic die-out of a transmission chain may occur, averting an epidemic (lower bounds of Fig. 2A). At values of R <5, this probability of die-out rather than an epidemic may be significant, and critically determined by local health system capacity and response (18). We constructed a branching process model to estimate what factors critically alter the probability of an XDR TB epidemic. We estimated that the probability of XDR TB epidemics was reduced from 12% on average in the context of current practices in South Africa to just >8% with community-based XDR screening and treatment at outpatient health clinics, and to 2.5% when additionally implementing the rapid detection, infection control, and follow-up protocols specified in the South African government's TB control plan (4). The amplification of an initially nosocomial epidemic by poor infection control in TB wards, and high adult HIV prevalence, could notably increase the epidemic probability. These findings can be taken in the broader context of budget outlays and plans for addressing South Africa's XDR epidemic. Although there is insufficient incidence data, cost data or price stability in the simulated interventions to accurately estimate the potential cost-effectiveness of different XDR control strategies at present, the general trends observed in this reproductive number analysis are consistent with analyses suggesting the potency and cost-effectiveness of community-based detection and treatment strategies for drug-resistant TB (19, 20).

As with all mathematical models, our estimates are subject to demographic, pathogenic, and transmission-related assumptions and uncertainties. Our modeling of the XDR TB epidemic involved sampling from wide distributions of parameter estimates, to incorporate variable possibilities in natural history and transmission. Yet differences in the structure of models themselves will also affect model conclusions. For example, we separate rapidly progressing and slowly progressing infected individuals, whereas some authors assume that primary progressive disease involves essentially no latent period (21); our estimates of epidemic emergence may therefore be slower than among these other models. Assumptions also differ about whether and how to represent exogenous reinfection; we included this phenomenon, given its potential importance in nosocomial environments in South Africa (22), yet the death of new XDR strains will be more probable outside of close-contact settings. Because of these unavoidable uncertainties around TB pathogenesis and natural history, it would be imprudent to set an exact target for detection or treatment levels that would definitively achieve R < 1. Rather, we used stochastic sampling from wide ranges of parameters to detect robust trends in emergence and intervention.

Our results consistently revealed that infection control and rapid diagnostic tests within hospitals, the current international focus for XDR TB control, do not sufficiently account for the critical impact of health system delays that facilitate community-based transmission. Hospital-based interventions are necessary, but possibly insufficient, to control the transmission of XDR TB in South Africa. Intensified community-based screening and treatment for XDR TB at outpatient health posts could critically lower the probability of XDR TB epidemics. Enhancing the identification of patients before they arrive at hospitals with late-stage disease may substantially reduce primary transmission of XDR TB in South Africa and avert future epidemics.

Methods

Model Structure.

To simulate XDR TB epidemics and calculate R, we constructed a model of tuberculosis transmission that incorporated the existing XDR detection and treatment system in South Africa (Fig. 1B). The model simulates XDR TB transmission and pathogenesis in both community and hospital environments, and incorporates both HIV-negative and HIV-positive patients, including the potential impact of antiretroviral therapy. The model's parameters and equations are extensively detailed in SI Appendix. Here, we review the model's base structure and critical assumptions.

The model uses a natural history description updated from prior epidemiological models of TB to reflect a modern understanding of disease pathogenesis (21, 23, 24). Specifically, we describe the process of infection as leading to brief or prolonged latent states of infection, from which active disease may occur through reactivation or reinfection. Persons infected with other circulating TB strains may harbor partial immunity to reinfection. XDR TB cases may be generated through primary transmission (in which case persons must undergo empiric therapy before becoming suspects), or acquire resistance during therapy for other TB disease (leading them directly to the suspect queue). Throughout the process, mortality from TB and default from the system are incorporated, including the potential for representation for diagnosis, drug-susceptibility testing, and therapy. Currently, therapy for XDR TB in South Africa is only available at hospitals, rather than outpatient health clinics. Therefore, those with confirmed XDR TB after drug-susceptibility testing must remain in the queue until hospital space is available. Other individuals may also be admitted for non-XDR TB disease, and are subject to potential nosocomial transmission of the disease, or acquired resistance generating new XDR TB. Those persons effectively treated for XDR TB are discharged. Because there is great uncertainty as to how effective therapy for XDR TB may be in this environment, particularly among HIV-infected individuals, we varied the efficacy of therapy around a wide range of estimates from 0 to 83% (Table 1) (25–28).

Table 1.

Ranges of key parameters varied in simulations

Parameter	Initial value (range over which parameter is varied)
Efficacy of infection control strategy; proportion of potential nosocomial transmissions averted (χ)	0% (0–100%)
Speed of drug-susceptibility testing results; time from resistance testing to receipt of results and initiation of therapy (1/δ) (6, 7, 45, 46)	6 weeks (1 day–10 weeks)
Case detection ″rate″ of all TB cases; proportion of incident cases detected in community health posts (d) (4, 47, 48)	55% (40–100%)
Proportion of patients lost to follow-up while waiting for drug-susceptibility testing (q) (3, 13, 49)	25% (0–72%)
Outpatient screening access; proportion of detected cases presenting to outpatient health posts whose sputa are submitted for drug-resistance testing (π)	0% (0–100%)
Proportion of detected patients with access to community-based XDR TB therapy (c)	0% (0–100%)
Efficacy of XDR therapy; proportion of treated patients with potential for recovery from therapy tailored by drug-susceptibility testing results (j) (22–25)	50% (0–83%)

Model Assumptions.

As with all mathematical models, estimates of epidemic transmission from this model are subject to assumptions. We adopted the Wells–Riley system to describe airborne nosocomial transmission in hospital wards (29, 30); this model assumes relatively even mixing of air, and thus does not account for architectural specifics that may vary among facilities. The infectiousness of active TB patients was varied across a distribution observed in in vivo air sampling studies (31–33). We varied the infectiousness of active TB patients across this broad range, rather than discretely categorizing active TB patients into infectious or noninfectious categories. The classification of active TB patients into infectious and noninfectious categories was previously based on the assumption that sputum-negative individuals were noninfectious (23); we now know this is not necessarily true, and that active TB patients may be widely varying in their individual infectiousness (34, 31–33). As a result, we varied the rate of per-person transmission in the community across a broad range from a very conservative value of 3 per person-year to a more typical value of 10 per person-year used in previous drug-resistant TB modeling studies (24). We assumed homogeneous mixing in the community, in the absence of more detailed contact-network information. The low-end of the transmission rate range (3 per person-year) was chosen to reflect the most conservative observed rate of transmission of drug-resistant TB transmission cluster studies, and results of previous model calibrations against a lower-bound estimate of extensively drug-resistant TB incidence in South Africa (35, 19). This is likely to result in conservative estimates of R under the assumption that drug-resistance confers significant transmission fitness costs; empirical studies suggest lower or minimal costs among clinically-observed drug-resistant strains, possibly as a result of compensatory mutations (36, 37). Similarly, we varied the per-person infectiousness of inpatient TB cases from a low estimate of 1.3 quanta per person to a high value of 13 quanta per person per hour on average (30, 38). The variance around these average was also input into the stochastic branching process model to incorporate heterogeneity in infectious potential among patients observed in vivo air sampling studies (see below).

We followed common TB models in simulating the adult (>15 years old) population, given that the epidemic impact of pediatric TB cases is believed to be limited, and the parameters describing adult TB pathogenesis are better understood (24, 35, 7, 39). Because TB pathogenic parameter values remain imprecisely defined, we varied the key parameters defining TB pathogenesis, nosocomial and community-based transmission risks across broad ranges; we also varied the parameters describing community demography and public health infrastructure to reflect the range of observed parameters among South African communities (Table 1 and SI Appendix, Tables S1 and S2). Gillespie's τ-leap algorithm and Latin Hypercube Sampling were used to iterate the model 10,000 times to explore the impact of stochasticity and parameter uncertainty on the model results (40, 41). The error bars in our figures reflect the contribution of this uncertainty on model output.

Queuing for Testing and Therapy.

At present, patients who become XDR TB suspects are put in a queue to test for drug-resistance. If they have XDR TB, they must wait for an inpatient bed to become available to obtain XDR therapy (7). We derived a queueing system to simulate the logistics of this circumstance, and to estimate the waiting time for those queuing for detection and treatment, which is derived in detail in SI Appendix. The generalized form of the queueing system is as follows, based on insights from queuing systems applied to housing shortages and vaccination programs (42, 43). The proportion of persons who enter a treatment program will be the rate of entry ξ multiplied by the minimum of either the entire demanding population (the population eligible to enter at any given time) or the fraction of the demanding population for which there is program capacity. If a total of Y persons are available to enter treatment at a given time, then the number admitted will be ξmin(Y, b − N), where b is the number of program spaces (e.g., beds in a hospital ward), and N is the number of current occupants.

To determine waiting times among queuing patients, we observe that for a queue of Q patients who dropout from the queue (due to mortality or default from the system) at exponential rate δ, the proportion dropping out rather than receiving therapy will be 1 − e^−δW, where W is the waiting time in the queue. Hence e^−δW are the fraction of patients who receive therapy. If ξY persons per unit time enter the queue and the rate of exit from the queue is σ, then the fraction of patients who receive therapy are σQ/ξY. Hence e^−δW = σQ/ξY, and the waiting time W = (1/δ)ln(ξY/σQ). The specific form of this generalized system that applies to our XDR TB model is detailed in SI Appendix.

Reproductive Number.

We adopted the next generation approach to estimate the reproductive number of XDR TB (44). Equations describing nosocomial and community-based transmission (see SI Appendix) were categorized into vector F describing new infections, whereas movement among infected states was described in vector V of transported infections. We numerically computed the corresponding Jacobian matrices F and V at the disease-free equilibrium, and solved for R as the dominant eigenvalue of F(V⁻¹). R was computed by Latin Hypercube Sampling 10,000 times from demographic, system, and pathologic parameter distributions given in SI Appendix, Tables S1 and S2 (41).

Epidemic Probability.

We calculated the probability of XDR TB epidemics in the context of different South African communities (SI Appendix), and the public health interventions included in a current South African government TB strategic plan (4). We define the probability of an epidemic as the likelihood that a primary case will infect secondary cases before death or recovery, and that secondary cases will infect tertiary cases, and so on, to sustain chains of transmission (45). The probability is calculated from the estimated value of R in the population, including the variance in the infectiousness among patients from in vivo air sampling studies to incorporate heterogeneity among individuals (31–33). The derivation of the branching process is provided in SI Appendix. A simple branching process can be defined as follows. The probability of emergence is 1 minus the probability of extinction (P_loss). The probability of extinction is the probability that j secondary cases are produced by a primary XDR TB case, multiplied by the probability of ultimate extinction given that j secondary cases are produced, summed across all j: P_loss = Σ_{j = 0,∞}P(ultimate extinction|j secondary cases produced) × P(j secondary cases produced). If the number of secondary cases that are produced are Poisson distributed, the probability of extinction simplifies to P_loss = Σ_{j = 0,∞}P _loss^je^−RR^j/j! = e^−RΣ_{j = 0,∞}(P _loss^jR)^j/j!. Because Σ_{j = 0,∞}x^j/j! = e^x, it follows that Σ_{j = 0,∞}(P _loss^jR)^j/j! = e^P_lossR (46). Hence, P_loss = e^−Re^P_lossR=e^{−(1 − P_loss)R}. This equation is solved numerically for P_loss given R, and the probability of emergence is P_emergence = 1 − P_loss = 1 − e^{−(1 − Ploss)R}.

To deviate from the homogeneity assumption implicit in this derivation—that is, to incorporate the possibility that individuals in a population have different transmission potential—we extended this branching process, using moment-generating functions to create a mixed model. The mixed model is captured in the equation P_loss = [1 − b(P_loss − 1)]^−c, where bc = R, and the ratio of the variance to the mean of infectiousness among individuals will be b + 1 (see SI Appendix for derivation). We take the variance-to-mean ratio from the distribution of infectiousness among patients from in vivo air-sampling studies (31–33), which is the most dispersed data on infectious potential among active TB patients, and thus produces conservative epidemic probability estimates.