Accepted 4 March 2009

JOURNAL OF CLINICAL MICROBIOLOGY, May 2009, p. 1484–1490 0095-1137/09/$08.00⫹0 doi:10.1128/JCM.02289-08 Copyright © 2009, American Society for Microbi...
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JOURNAL OF CLINICAL MICROBIOLOGY, May 2009, p. 1484–1490 0095-1137/09/$08.00⫹0 doi:10.1128/JCM.02289-08 Copyright © 2009, American Society for Microbiology. All Rights Reserved.

Vol. 47, No. 5

Potential of Rapid Diagnosis for Controlling Drug-Susceptible and Drug-Resistant Tuberculosis in Communities Where Mycobacterium tuberculosis Infections Are Highly Prevalent䌤 Pieter W. Uys,1,2* Robin Warren,1 Paul D. van Helden,1 Megan Murray,3 and Thomas C. Victor1 MRC Center for Molecular and Cellular Biology, DST/NRF Centre of Excellence for Biomedical Tuberculosis Research, Faculty of Health Sciences, Stellenbosch University, P.O. Box 19063, Tygerberg 7505, South Africa1; DST/NRF Centre of Excellence for Epidemiological Modelling and Analysis (SACEMA), Stellenbosch University, Western Cape, South Africa2; and PSOH, Harvard University, Cambridge, Massachusetts3 Received 28 November 2008/Returned for modification 13 February 2009/Accepted 4 March 2009

The long-term persistence of Mycobacterium tuberculosis in communities with high tuberculosis prevalence is a serious problem aggravated by the presence of drug-resistant tuberculosis strains. Drug resistance in an individual patient is often discovered only after a long delay, particularly if the diagnosis is based on current culture-based drug sensitivity testing methods. During such delays, the patient may transmit tuberculosis to his or her contacts. Rapid diagnosis of drug resistance would be expected to reduce this transmission and hence to decrease the prevalence of drug-resistant strains. To investigate this quantitatively, a mathematical model was constructed, assuming a homogeneous population structure typical of communities in South Africa where tuberculosis incidence is high. Computer simulations performed with this model showed that current control strategies will not halt the spread of multidrug-resistant tuberculosis in such communities. The simulations showed that the rapid diagnosis of drug resistance can be expected to reduce the incidence of drug-resistant cases provided the additional measure of screening within the community is implemented. Multidrug resistance in Mycobacterium tuberculosis poses a threat to the success of tuberculosis (TB) control programs and creates an enormous financial burden in regions where TB prevalence is high. The current WHO DOTS (directly observed treatment, short course) TB control strategy recommends passive case detection and diagnosis by sputum smear microscopy. TB patients are first detected when they seek help for symptomatic disease, rather than being identified through active screening. Since there is often a long delay between the onset of infectiousness and the time when a patient presents, this means that patients are frequently diagnosed only after they may have transmitted infection to others (3, 16). Once identified, patients suspected of having TB then undergo sputum smear microscopy to detect acid-fast bacilli. Smear positivity correlates well with the bacterial burden in the lungs, as well as with levels of transmissibility, and thus, this approach ensures the identification of the most infectious cases of TB. Nonetheless, recent data demonstrate that smear-negative patients are also able to transmit TB (4), and since they are not rapidly detected, we propose that they may transmit for a longer period than the more infectious smear-positive cases. In settings where TB incidence is high, once a patient is diagnosed with TB by smear microscopy, due to resource limitations, no further microbiological tests are performed, and instead, diag-

nosis is primarily done by microscopy. In South Africa, culture and drug susceptibility testing on solid medium are routinely done only for retreatment cases. These test results are not available for a minimum of 3 weeks and may take as long as 10 weeks in high-throughput laboratories (average, 1,000 samples per week) (11). A full description of the setting in which this investigation was conducted, with an emphasis on the exact way in which TB patients were evaluated and treated, is provided by Verver et al. (21). Several recent operational studies have found that even after diagnosis of drug-resistant TB, further delays are often experienced before patients receive appropriate second-line drug regimens (1). During such delays, further transmission events may take place (6), thereby potentially amplifying or perpetuating the epidemic or ensuring that multidrug-resistant (MDR) TB remains endemic. The potentially serious consequences of a delay in appropriate treatment for MDR TB have been recognized (10, 14). Drug susceptibility testing using molecular techniques can enhance TB diagnosis (11), and various rapid molecular tests for drug resistance are available, but they have not been implemented in settings where the TB burden is high. Currently, the two main diagnostic tests available commercially are the INNO-LiPA TB test (Innogenetics) (12) and the MTBDRplus kit (Hain Lifescience) (15). These assays have recently been approved by the World Health Organization as tools for rapid MDR TB diagnosis (http//www.who.int/tb/dots/laboratory/1pa _policy). Recently, the MTBDRplus kit was tested in a busy routine diagnostic laboratory in Cape Town, South Africa (2). This commercially available molecular line probe assay for rapid

* Corresponding author. Mailing address: Division of Molecular Biology and Human Genetics, Faculty of Health Sciences, Stellenbosch University, P.O. Box 19063, Tygerberg, 7505, South Africa. Phone: 27-83-5571151. Fax: 27-86-5140309. E-mail: pieter@edserve .co.za. 䌤 Published ahead of print on 18 March 2009. 1484

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detection of rifampin (rifampicin) and isoniazid resistance was assessed and provided the following measurements. Overall, 97% of smear-positive specimens gave interpretable results within 1 to 2 days using the molecular assay. The sensitivity, specificity, and positive and negative predictive values were 98.9, 99.4, 97.9, and 99.7%, respectively, for detection of rifampin resistance; 94.2, 99.7, 99.1, and 97.9%, respectively, for detection of isoniazid resistance; and 98.8, 100, 100, and 99.7%, respectively, for detection of multidrug resistance compared with conventional results (2). The results show that this molecular assay is an accurate screening tool for MDR TB and that it has the potential to reduce diagnostic delay. In order to examine the potential benefits of rapid diagnosis of MDR TB, we developed a mathematical model of TB to simulate the trajectories of the course of the epidemic under continued application of current strategies and under the application of rapid diagnostic tools. To place this analysis in a realistic context, we calibrated the model to epidemiologic data reported for a region in South Africa where the incidence of TB is high. MATERIALS AND METHODS In a community where TB is prevalent, at any given time, an individual could be in any one of a number of different states with regard to TB (Fig. 1). These states include (i) actively diseased and infectious, (ii) diseased and undergoing therapy, (iii) recovered after undergoing therapy, (iv) developing resistance while undergoing therapy, (v) latency, (vi) reactivation, (vii) reinfection, and (viii) immunity. Either susceptible disease treatable by first-line drugs or resistant disease involving drug-resistant bacteria is possible. In order to properly analyze the effect of time delays in treating TB cases, all of the possible sequences of these various states must be taken into account. This must be done both for first-line drug-susceptible and -resistant TB cases. Some of the simpler scenarios are listed in Table 1. A schematic flow diagram depicting the various states and transitions among them is shown in Fig. 1. Note that a patient infected with a first-line drug-resistant strain of TB would (under current practice) be started on regular therapy as for a susceptible TB case. Only later would it be determined that resistant TB was involved so that appropriate therapy for drug-resistant TB could be started. Similarly, patients who develop resistance during regular therapy would experience a delay before their new condition would be diagnosed. Delays incurred before diagnosis of an actively diseased, and hence infectious, patient allow transmission events to take place. The number of such events could be reduced if rapid diagnostic strategies were instituted. These delays, therefore, form an important integral part of the dynamics of the epidemiological process. There are several other delays that also affect this process. A patient with MDR TB who is receiving therapy for drug-susceptible disease remains infectious and could infect others. The delay prior to diagnosis of the true condition therefore affects the dynamics of TB in the community. The same applies to patients acquiring resistance while undergoing a course of therapy for drugsusceptible TB. Detailed descriptions of the epidemiological principles incorporated in the mathematical model used can be found elsewhere (13). Only some of the main considerations are repeated here. (i) The essential mechanism underpinning many of the dynamic processes is the principle of mass action, in which the rate of infection is proportional to the number of infectious cases and also to the number of nonimmune persons. (ii) It is assumed that when therapy is completed, the status of a newly cured patient returns to that of being disease free. Such patients may experience reinfection followed by a further episode of active disease, termed exogenous disease. This type of event was evident among the case histories observed in our data set. Indeed, it is not uncommon for persons to experience several disease episodes, each time with a different strain of bacteria (17). Instead of active disease resulting from an infection or reinfection event, it is possible that a person may experience a relapse of a previous disease episode. Such cases are not uncommon. Finally, it is also possible that a patient may suffer a disease episode as the result of reactivation of a long-standing latent infection. (iii) Like Vynnycky and Fine (25), we have assumed that the infection and reinfection rates are the same. We have, however, also assumed that infec-

FIG. 1. Schematic flow diagram depicting the various disease states and transitions among them. The boxes represent cohorts of persons in a particular state. Single arrows indicate flows of people from one state to the next according to some probability. Double arrows indicate flows, involving time lags, of people from one state to the next. Broken arrows indicate removal of persons from the indicated state due to death. The subscripts S and R refer to susceptible and resistant strains, respectively, of TB. Endogenous refers to reactivation of an earlier infection to produce an actual episode of disease. Double borders signify a state involving resistant TB. The large block arrows indicate a flow of persons at a rate dependent on the time since infection.

tion rates are not age dependent (13), since the focus of this investigation was to determine the benefits of rapid diagnosis. (iv) Whether a person who has been infected develops active disease shortly thereafter or remains latent with the potential to develop disease much later can, for our present purposes, be regarded as largely a matter of chance. For a population, such random factors average out to probabilities. Thus, we have a probability of infection, a probability of reactivation, or a probability of developing resistance. These probabilities can be treated as rates. The probability of an infection progressing to disease varies according to the time elapsed since the infection event (25). (v) We assume the presence of MDR strains that have the same level of fitness and present the same annual risk of infection as the average for susceptible strains (13). (vi) A core capability of the model is the way it represents the acquisition of drug resistance by patients undergoing treatment for susceptible TB. A model for TB has to account for all the different states that an individual could be in, as well as the transitions of persons between states. Several of these transitions are predominantly deterministic and involve a specific time lag. An example is regular therapy provided to a patient with susceptible disease for a period of 6 months with a successful outcome. The patient would be in the therapy state for precisely that period at the conclusion of which the status of the patient was disease free. It should be noted that during the period of treatment, and from within a week or so of the commencement of such treatment, the patient is not infectious. Other transitions are probabilistic. Two examples are the probability of death and the probability of the transition from the state of being infected to that of being actively diseased. Within the context of the community, the latter transition is also density dependent. Thus, for an individual, the transition from merely infected to actively diseased is effectively a random event with a certain probability, while for the cohort of persons who are

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J. CLIN. MICROBIOL. TABLE 1. Examples of partial sequences of TB states Sequence of states

Scenario

1 2 3 4 5 6

1

2

3

4

Naı¨ve Exposure to infection Primary infection Therapy for susceptible TB Latent Latent

Exposure to infection Primary infection Active disease episode Development of resistance Reactivation (endogenous disease) Reinfection (exogenous disease)

Immune Latent Therapy for susceptible TB Resistance diagnosed Diagnosis Diagnosis

Death Death Disease free Therapy for resistant TB Therapy Therapy

infected, the number who becomed diseased is that same probability multiplied by the number of infected people. No single specific time lag or delay can be associated with this kind of transition event. In mathematical terms, a model of this nature can be expressed as a system of simultaneous differential equations governing the number of people in each state. These equations are presented in the appendix. Straightforward standard numerical methods are available for the solution of such systems. These methods enable simulations mimicking the course of an epidemic and can be implemented quite readily on a spreadsheet, such as Microsoft Excel. Excel also has a feature (indexing of arrays) that makes it possible to incorporate the modeling of time delays. These time delays may differ from one differential equation to another, and Excel can accommodate this, as well. Moreover, the indexing feature in Excel allows the user to specify particular values for such delays before each simulation is performed. In this way, the effect, for instance, of a reduction in the time from onset of illness to the correct diagnosis of susceptible or resistant TB, as the case may be, can be ascertained by performing suitable simulations with appropriate values set for the time delay parameters. The model was calibrated so as to yield simulations matching as closely as possible the conditions historically observed in a study area in the Western Cape Province of the Republic of South Africa, where the population is low income and the incidence of TB exceeds 700/100,000 per annum (5, 20, 21). A recent survey of 366 new adult smear-positive TB cases (2000 to 2002) at this epidemiological field site showed that 10% of the TB cases were human immunodeficiency virus (HIV) positive (2). The HIV prevalence in this community thus has been and still is relatively low, so modeling the epidemiology of TB here is free of the complications of having to consider the effects of HIV. This model, therefore, does not include any HIV effects and can only be applied to communities with low HIV prevalence. This community has been the subject of careful local and international investigations over the past decade (17–24, 26), with the result that considerable data have been accumulated. A relatively good standard of health care delivery to TB patients has been in place over the same period, and this has been carefully monitored. Thus, the community presents an ideal case study for the application of a mathematical model. The community has a population of about 35,000, but all

the parameter values have been normalized to correspond to a population of 100,000. In addition, since the population has remained fairly constant over a decade and there is minimal immigration or emigration, we may assume that the birth rate and the average death rate are approximately equal to each other. The data used for calibrating the model, that is, to determine the various parameters used in the model, are listed in Table 2. The aim of this study was to ascertain the extent of the reduction of the epidemic of drug resistance that could be achieved by earlier diagnosis of drug resistance. The potential benefits of screening contacts were also investigated. A time frame of 20 years was used (8). A longer period would imply many assumptions about future conditions that could render the predictions highly speculative. Several simulations were performed in order to compare various strategies for the control of the TB epidemic. The first simulation assumed that the present strategy continued unchanged. The second simulation assumed rapid detection of resistance was in place. A third simulation assumed a situation of proactive case finding in addition to rapid diagnosis. Variations of these basic simulations were performed to ascertain the sensitivity of the predictions to the parameters being varied. This helps identify those factors that play the major roles in any effort to control a TB epidemic. The total treatment costs for drugs only were calculated during the running of each simulation.

RESULTS The main results obtained from the simulations are set out in Table 3. The model predicts that if the present control strategies remain unchanged, then by the end of a 20-year period, the incidence of MDR TB cases will have increased from 2 cases per month to 11 cases per month (per 100,000). With rapid diagnosis (Table 2) in place, the increase will still be high at 9.6 cases per month, even if the rapid-diagnosis method has 90% sensitivity, i.e., 90% of positive cases are identified. If

TABLE 2. Data used to set parameter values used in simulationsa Parameter

Value

Population at beginning of 20-yr period .................................................................................................................................................................100,000 Birth rate (no. of live births per 100,000 population per annum)....................................................................................................................... 1,500 Avg death rate (all causes, per 100,000 population per annum)......................................................................................................................... 1,500 63 Initial monthly incidence of drug-susceptible cases (per 100,000 population)b ................................................................................................. Initial monthly incidence of MDR cases (per 100,000 population)b ................................................................................................................... 2 Initial annual risk of infection (%)c ........................................................................................................................................................................ 3.5 Risk of conversion during the treatment period from susceptible to resistant diseased .................................................................................. 0.1 Risk of progression to disease for an infected person within 2 yr of infectione ............................................................................................... 0.05 Mean no. of days for executing diagnosis (rapid method) ................................................................................................................................... 2 Mean no. of days for executing diagnosis (culture method) ................................................................................................................................ 40 Mean no. of wk for standard treatment.................................................................................................................................................................. 26 Mean no. of wk for treatment of a resistant case ................................................................................................................................................. 78 a

Reported historical data for the study population were used. The rate of reinfection was assumed to be the same as the rate of infection, and such rates were assumed to be the same for susceptible and drug-resistant TB (21, 25). c This value is more conservative than the 5% used by Resch et al. (13). d The value was determined by making adjustments in order that simulations yielded outputs corresponding to historical data. This value is less than the value of 0.4 found by Resch et al. (13) for data pertaining to Peru. e This is less than the 8% used by Resch et al. b

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TABLE 3. Predicted epidemiological outcomes over a 20-year period

Strategy

a

Avg no. of cases diagnosed in 1 mo at the end of the 20-yr period Susceptible

Resistant

60 57

11.2 9.6

57

2.4

57

1.6

57

2

Current Rapid diagnosis of resistance (90% sensitivity) Rapid diagnosis of resistance (97% sensitivity) Screening of contacts plus rapid diagnosis of resistance Screening of contacts with 4-wk delay in detecting development of resistance

a At the commencement of each simulation, the average monthly incidence of susceptible and resistant TB cases was set at 63 and 2 respectively. This corresponds to a caseload during the early stages of the simulation of the order of 750 and 36 respectively. The point estimates shown need to be qualified by taking into account their sensitivity to parameter values given in Table 4.

the diagnosis method has a sensitivity rate of 97%, then the model predicts that the incidence will still increase, but from 2 cases per month only to 2.4 cases per month. The projected incidence of 11 cases per month would thus have been reduced to only 2.4 cases per month. This shows that it is important that the diagnostic technique used be not only rapid, but also highly reliable and sensitive. In this analysis, it was assumed that cases were diagnosed within 2 weeks of having attained such an infectious stage. In the community studied, it is unlikely that patients would visit a clinic at so early a stage. Thus, screening of the community would have to be in place to locate such cases. However, the model predicts that with the addition of the implementation of screening, together with prophylactic treatment of contacts at only a 10% level of success, the incidence of MDR TB will be reduced to almost zero. This demonstrates the critical importance of rigorous and proactive implementation of any rapid diagnostic technology. The main results inferred from simulations performed using the mathematical model are summarized in Table 3. The point estimates given need to be qualified by considering the sensitivity to parameter values, as shown in Table 4. The comparative cost implications are shown in Fig. 2. These costs were computed during the corresponding simulations by adding, at

FIG. 2. Annual costs of treating susceptible cases and MDR cases shown as a multiple of the present-day annual cost of treating susceptible cases. Although the actual case load for MDR TB is considerably less than that for susceptible TB, the costs are far greater, since the cost of treating a patient with MDR TB is 2 orders of magnitude greater than that for susceptible TB.

each time step, only the cost of the drugs used during that time step. Again, these results must be qualified by referring to Table 4. The relative importance of time delays in the control of TB, and especially MDR TB, is illustrated in the conceptual diagram in Fig. 3.

TABLE 4. Model sensitivity analysis % Increase in incidencea Rate

Susceptible disease

Resistant disease

Mortality Progress to infectious disease Infection Reinfection Reactivation Conversion to resistant disease

⫺2.22 2.39 6.00 ⫺5.77 1.45 8.93

⫺3.32 3.43 9.95 ⫺9.05 10.43 2.21

a The percentage increase in incidence for susceptible and resistant disease is shown for separate 10% increases in the stated parameters. The sensitivities to the various mortality rates were investigated, and only the maximum sensitivities are shown. These data show that the model predictions are not unduly sensitive to parameter value uncertainties.

FIG. 3. Conceptual chart showing relative numbers (represented approximately by the areas of the shapes and not to accurate scale) of people in different categories and relative durations (represented by the lengths of the boxes) of the illness and treatment phases. Note that patients with MDR TB may incorrectly receive treatment for susceptible TB for a period (dark shaded box). The situation of resistant disease compared to susceptible disease is asymmetrical by virtue of the one-way flow depicted by the double arrow from the box representing patients undergoing therapy for susceptible TB who develop resistance. The time a TB patient is ill before diagnosis and the start of treatment is indicated by the lengths of the hatched boxes. All actively diseased people contribute to further infection events until their treatment starts. All flows are directly proportional not only to the numbers in the source categories, but also to the length of time before commencement of treatment or the detection of conversion to resistant disease.

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FIG. 4. Incidence of susceptible disease for various diagnostic sensitivities as a multiple of the base year incidence with 2- and 28-week diagnostic delays.

FIG. 5. Incidence of resistant disease for various diagnostic sensitivities as a multiple of the base year incidence with 2- and 28-week diagnostic delays.

The effects of different diagnostic sensitivities and delays were investigated. The results are summarized in Fig. 4 and 5. These results seem to indicate that the delay until diagnosis has a relatively small effect on incidence. This may be so for susceptible disease and low diagnostic sensitivity, but for resistant disease, diagnostic delay has a major effect on incidence (Fig. 5). In a sense, it could be said that low diagnostic sensitivity effectively produces a long delay until diagnosis. Low sensitivity results in patients not being diagnosed at their first visit and remaining in the infectious cohort. Thus, although the delay until the first visit could be short, actual successful diagnosis of a TB-positive case may take much longer. Conversely, at high diagnostic sensitivities, the chances are good that the patient will be diagnosed at the first visit, and the time to diagnosis is then reduced to the time from onset of symptoms to the actual visit. The model simulations reflect that situation. It should be noted that even with effective control strategies in place, one can expect the incidence of disease to increase for several years before actual declines can be observed (Fig. 4). This is because the large cohort of latently infected people needs several years to shrink through mortality and reactivations, even when the recruitment rate has decreased. A modeling-sensitivity analysis was also performed to ascertain how sensitive the predictions of the model are to parameter values (Table 4). The model predictions were not found to be unduly sensitive to uncertainties in any of the model parameters.

situation in which drug-resistant TB has already become prevalent, we found that an intensively proactive approach was needed to reduce the prevalence of drug resistance. Rapid diagnostic techniques with a high level of accuracy are necessary. Such methods are now becoming available and are practical, highly sensitive, and similar in cost to standard culture methods. In addition, it was found that rapid diagnosis must be supplemented by screening of contacts and that infected persons should be given prophylactic therapy to reduce the burden of disease. Only then is a substantial reduction in incidence rates achieved. It was found that such reductions in incidence would occur even if only 10% of the infected contacts of an infectious person with drug-resistant disease are located and treated. However, any delays in diagnosis or failures to locate at least a significant number of the contacts of patients with drug-resistant disease allow the incidence of resistant TB cases to increase. It should be noted that simply screening traditional or household contacts may not constitute an effective screening protocol. Regular screening of the entire community may be necessary in order to locate the many possible casual contacts (13). These conclusions are supported by investigations using clinical data in various regions of the world (7, 18). It should be noted that the conclusions that we have drawn using this model are subject to the conditions described above and, in particular, apply only to regions where the incidence of TB is high and that of HIV is relatively low. The necessary measures are technically achievable with current rapid diagnostic methods (2). The benefits for individual patients with MDR TB, and also for the community at large, are substantial and important. Moreover, enormous savings in therapy costs would be realized compared to the situation that would otherwise develop should current control strategies continue to be used. The greatest challenge will be the institution of a cheap and effective community-wide screening system. Thus, cheap mass screening technologies need to be developed. Failure to do this will lead ultimately to a daunting problem in the not-too-distant future.

DISCUSSION Dye and Williams (9) found that merely to prevent outbreaks of drug-resistant TB, high rates (70%) of detection of active cases, combined with high cure rates (80%), are needed. It was noted that these factors act synergistically. When either one is low, the other cannot succeed alone. Recently, Dowdy et al. (8) found that expanded testing of people suspected of having TB with a diagnostic test with 100% sensitivity and no diagnostic delay could reduce the cumulative incidence of MDR TB over a 20-year period by 19.9%. Using our model, under optimal conditions, the projected incidence of 11 cases per month is reduced to only 2.4 cases per month. Unfortunately, this cumulative figure cannot be directly compared to a monthly incidence figure, but both results demonstrate the need for rapid diagnosis with a high-sensitivity test. Given a

APPENDIX Notational conventions. In order to aid the presentation of the model equations (Table A1), the following notational conventions have been applied for the parameters used. Table A2 lists all the parameters used and their values. The expression ␲(A3B) represents the probability during 1 week of a transition from state A to state B.

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TABLE A1. The model equations dU/dt ⫽ Recruitment rate ⫺ {␮(U) ⫹ ␲(U 3 Ir) * Er ⫹ ␲(U3 Is) * Es} * U ⫹ dC/dt dC/dt ⫽ ␭(STs 3 C) ⫹ ␭(Tr 3 C) Ek ⫽ Pk ⫹ EXrk ⫹ EXsk (k ⫽ s, r) dIk/dt ⫽ ␲(U 3 Ik) * U * Ek ⫺ {兰␲(Ik 3 Pk) ⫹ ␮(Ik) ⫹ ␲(Ik 3 Lk)} * Ik (k ⫽ s, r) dPk/dt ⫽ 兰 ␲(Ik 3 Pk) * Ik ⫺ ␮(Pk) * Pk ⫺ ␭(Pk 3 STk) (k ⫽ s, r) dSTs/dt ⫽ ␭(Ps 3 STs) ⫺ ␮(Ts) * STs ⫺ ␭(STs 3 C) ⫹ ␭(Ns 3 STs) ⫹ ␭(EXss 3 STs) ⫹ ␭(EXrs 3 STs) dSTr/dt ⫽ ␭(Pr 3 STr) ⫺ ␮(STr) * STr ⫺ ␭(STr 3 Tr) ⫹ ␭(Nr 3 STr) ⫹ ␭(EXsr 3 STr) ⫹ ␭(EXrr 3 STr) dRTs/dt ⫽ ␭(STr 3 RT) ⫹ ␭(Nr 3 RTs) ⫹ ␭(Pr 3 RTs) ⫹ ␭(Xrr 3 RTs) ⫹ ␭(Xsr 3 RTs) ⫺ ␮(RTs) * RTs ⫺ ␭(RTs 3 Tr) dLs/dt ⫽ ␲(Is 3 Ls) * Is ⫹ ␲(Xss 3 Ls) * Xss ⫺ {␲(Ls 3 Xsr) ⫹ ␲(Ls 3 Ns) ⫹ ␮(Ls)} * Ls dLr/dt ⫽ ␲(Ir 3 Lr) * Ir ⫹ ␲(Xrr 3 Lr) * Xrr ⫹ ␲(Xsr 3 Lr) * Xsr ⫹ ␲(Xrs 3 Lr) * Xrs ⫺ {␲(Lr 3 Xrs) ⫹ ␲(Lr 3 Xrr) ⫹ ␲(Lr 3 Nr) ⫹ ␮(Lr)} * Lr dNk/dt ⫽ 兰 ␲(Lk 3 Nk) * Lk ⫺ {␲(Nk 3 Tr) ⫹ ␮(Nk)} * Nk (k ⫽ s, r) dXkl/dt ⫽ ␲(Lk 3 Xkl) * Lk * (Pk ⫹ EXsl ⫹ EXrl) ⫺ ␮(Xkl) * Xkl ⫺ ␲(Xkl 3 Lk) * Xkl (k, l ⫽ s, r) dEXkl/dt ⫽ 兰 ␲(Xkl 3 EXkl)*Xkl ⫺ ␮(EXkl) * EXkl ⫺ ␲(EXkl 3 Ts) * EXks (k, l ⫽ s, r) dEXkr/dt ⫽ 兰 ␲(Xkr 3 EXkr) * Xkr ⫺ ␮(EXkr) * EXkr ⫺ ␲(EXkr 3 Trs) * EXkr (k ⫽ s, r) dTr/dt ⫽ ␭(RTs 3 Tr) ⫺ ␮(Tr) * Tr ⫺ ␭(Tr 3 C)

Depending on the particularities of the transition, the actual numbers of individuals involved in the transition may equal ␲(A3B) ⫻ A or may additionally entail a further density-dependent factor. The expression ␭(A3B) represents the number of persons transferring during 1 week from state A to state B and entailing a specific time lag, depending on the pair A and B. ␭(A3B) requires the simulation program to view the number of persons in cohort A at an earlier time given by the current time minus the time lag, ␭t. The number of persons transferring to cohort B is a fraction, ␭f, of the number of persons in cohort A at that earlier time. For example, if A is the cohort of patients starting treatment at time t0 and B is the cohort of cured patients, then the time lag is the duration of the treatment and ␭f is the cure rate. The expression ␮(A) represents the per capita weekly mortality rate for the cohort in state A. A distinction was maintained between people infected with susceptible and resistant disease, and each state is therefore, where applicable, represented by a state comprising people with susceptible disease or a state comprising resistant disease and is indexed with a subscript s or r. Throughout, boldface uppercase letters refer to the number of people in the relevant state. In view of the large number of variables required, it was deemed helpful to resort to the use of short acronyms. The various symbols used for this purpose (representing numbers of persons in the designated cohorts), in the order of appearance, and their meanings are listed in Table A3. As described above, the parameters used in this system of differen-

tial equations are listed, together with their values, in their order of appearance in Table A2. The various parameter values were determined by running computer simulations and varying parameter values until the simulation output yielded results that corresponded to the data in Table 2. The solution method. The system of differential equations constructed for this model is nonlinear and cannot be solved by analytical means. Instead, numerical methods have to be employed. A standard algorithm for doing this is the Euler method, with a time step size of 1 week being suitable. This can readily be done on a spreadsheet, such as Microsoft Excel. Excel also has a feature (indexing of arrays) that makes it possible to incorporate the modeling of time delays. Thus, for example, the number of people successfully completing a course of regular therapy during a particular week is not simply related to the total number of people undergoing such therapy at that time. Rather, it is found by referring to the number of persons who commenced such treatment 24 weeks earlier (24 weeks being the duration of standard therapy) less the number of those same people who died or developed resistance during the same period. Other time delays can be treated in the same way. Simulations representing time periods of several decades were performed. The computer solution comprises values of the various state variables over a sequence of time steps and represents a computer simulation of the progress of the epidemic being modeled. Simultaneously, the cost over the simulation period associated with a specific

TABLE A2. Parameter values used in simulations Parameter(s)

Value

␮(U), ␮(Ik), ␮(Pk), ␮(Tk), ␮(STr), ␮(RTs), ␮(Lk), ␮(Nk), ␮(Xkl), ␮(EXkl)..............................................0.000288462/wka ␲(U3Ik), ␲(Xkl3EXkl) ...................................................................................................................................0.031154/wkb ␭(STs3C) ..........................................................................................................................................................␭t ⫽ 24 wkc, ␭f ⫽ 0.99d ␭(Tr3C).............................................................................................................................................................␭t ⫽ 48 wkc, ␭f ⫽ 0.99d ␲(Ik3Lk), ␲(Xkl3Lk) ......................................................................................................................................0.019230769/wke Time lag from onset of disease to diagnosis ␭(Pk3STk), ␭(Nk3STk), ␭(EXss3STs), ␭(EXrs3STs), ␭(EXsr3STr), ␭(EXrr3STr), ␭(STr3Tr), ␭(STr3RT), ␭(Nr3RTs), ␭(Pr3RTs), (wk) and detection rate; both varied in ␭(Xrr3RTs), ␭(Xsr3RTs), ␭(RTs3Tr) ...................................................................................................... simulations.f ␲(Lk3Xkl) .........................................................................................................................................................0.00147929/wkg ␲(Lk3Nk)..........................................................................................................................................................0.000106509/wkh ␲(Nk3Tr), ␲(EXkl3Ts), ␲(EXkr3Trs)..........................................................................................................0.9i a

The simplifying assumption was made that mortality rates were the same for all cohorts. No data are available for most types of cohorts. The infection and reinfection rates, respectively; they were assumed to be equal. The value was determined by simulations using the known prevalence as the goal. The usual therapy duration times. d The rates of cure; they were varied in simulations to explore the minimum cure rates needed to control an epidemic. e Most infection and reinfection events do not lead to disease, but rather latency. f The time lag from onset of disease to diagnosis was assumed to be the same irrespective of the manner in which the active disease originated. Similarly, detection rates were assumed to be the same. Various values for the time lag and the detection rate were explored in simulations. This aspect is the actual focus of this work. g The reinfection rate for the latent cohort, which was varied in simulations. h We assumed that over a lifetime of 70 years, the probability of latent disease reactivating is 0.1. i The sensitivity of the test for the diagnosis of disease; simulations can be done with various values of this paramenter. b c

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TABLE A3. Symbols useda Symbol

Meaning

U...............Uninfected C...............Cured Ik...............Infected Ek .............Total cohort of active disease episode Pk ..............Primary disease episode EXrk .........Secondary disease episode; latent resistant and reinfected with susceptible or resistant according to the index k EXsk .........Secondary disease episode; latent susceptible and reinfected with susceptible or resistant according to the index k STs ............Susceptible disease treated with therapy for susceptible disease STr ............Acquired resistant disease treated with therapy for susceptible disease Lk .............Latent Nk .............Endogenous (reactivation) RT............Acquired resistance Xsk ............Exogenous (secondary) reinfection; latent susceptible and reinfected with susceptible or resistant according to the index k Xrk ............Exogenous (secondary) reinfection; latent resistant and reinfected with susceptible or resistant according to the index k Tr ..............Resistant disease on therapy for resistant disease RTs ...........Resistant disease but commenced therapy for susceptible disease

5.

6.

7.

8.

9. 10.

11. 12.

13. 14.

15.

16.

a

The symbols stand for the number of persons in the cohort. Throughout, the index k may take the value s or r to indicate susceptible or resistant disease, respectively.

17. 18.

strategy of diagnosis and treatment can be estimated and compared with the costs of other strategies.

19.

ACKNOWLEDGMENTS

20.

We thank the South African National Research Foundation (IFR 2008042900006), the Harry Crossly Foundation, and the IAEA (SAF 6008 and SAF2007016 grants) for support.

21.

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