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Journal of Clinical Microbiology, May 2009, p. 1484-1490, Vol. 47, No. 5
0095-1137/09/$08.00+0 doi:10.1128/JCM.02289-08
Copyright © 2009, American Society for Microbiology. All Rights Reserved.

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 Africa,1 DST/NRF Centre of Excellence for Epidemiological Modelling and Analysis (SACEMA), Stellenbosch University, Western Cape, South Africa,2 PSOH, Harvard University, Cambridge, Massachusetts3
Received 28 November 2008/ Returned for modification 13 February 2009/ Accepted 4 March 2009
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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 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.
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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.
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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.
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TABLE 1. Examples of partial sequences of TB states
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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 infection 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 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.
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TABLE 2. Data used to set parameter values used in simulationsa
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TABLE 3. Predicted epidemiological outcomes over a 20-year period
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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 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.
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TABLE 4. Model sensitivity analysis
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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.
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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.
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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.
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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.
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(A
B) represents the probability during 1 week of a transition from state A to state B. Depending on the particularities of the transition, the actual numbers of individuals involved in the transition may equal
(A
B) x A or may additionally entail a further density-dependent factor. The expression
(A
B) 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.
(A
B) 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 µ(A) represents the per capita weekly mortality rate for the cohort in state A. |
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TABLE A1. The model equations
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TABLE A2. Parameter values used in simulations
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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.
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TABLE A3. Symbols useda
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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 strategy of diagnosis and treatment can be estimated and compared with the costs of other strategies.
Published ahead of print on 18 March 2009. ![]()
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