Multivariate Markovian modeling of tuberculosis: forecast for the United State.

We have developed a computer-implemented, multivariate Markov chain model to project tuberculosis (TB) incidence in the United States from 1980 to 2010 in disaggregated demographic groups. Uncertainty in model parameters and in the projections is represented by fuzzy numbers. Projections are made under the assumption that current TB control measures will remain unchanged for the projection period. The projections of the model demonstrate an intermediate increase in national TB incidence (similar to that which actually occurred) followed by continuing decline. The rate of decline depends strongly on geographic, racial, and ethnic characteristics. The model predicts that the rate of decline in the number of cases among Hispanics will be slower than among white non-Hispanics and black non-Hispanics-a prediction supported by the most recent data.

After many years of decline, tuberculosis (TB) cases began to resurge in the United States in 1984, and continued to climb for a decade before declining again (1-4). Reemergent TB did not affect all segments of the population equally. While incidence continued to decline in some geographic areas, it increased in others, notably in the greater New York City area, southern Florida, the border areas of Texas, and urban California. Disadvantaged populations were disproportionately affected, with peak incidences in blacks and Hispanics now seen in young adults. TB in foreign-born persons accounts for nearly 40% of the total annual cases in the United States (5). Finally, HIV infection has played a large part in the recent increase in TB incidence. Effective TB control requires reliable estimates of future secular trends. We have constructed a population dynamics, mechanistic Markov chain model for the epidemiology of TB in the United States by separate modeling for disaggregated groups defined by race, ethnicity, age, and geography. With this model, we have projected TB incidence for the U.S. population at large, as well as for racial and ethnic and geographically defined groups within the population.

The Study

In support of this work, we obtained datasets from the U.S. Bureau of the Census and the Centers for Disease Control and Prevention (6-19).

The epidemiology of TB in the United States has an underlying Markovian nature (20-22). Except in unknown or difficult-to-predict influxes of immigrants, future infections and cases of TB may be probabilistically estimated by knowing sufficiently well the present situation of susceptible, infected, and ill persons; rates of transition from healthy to infected and infected to ill; patterns of contact; and rates of fertility and death. Indeed, a Markov chain is completely specified by an initial state and transition probabilities. To model the future course of TB in the United States, we used the structure depicted in Figure 1.

[Figure 1 ILLUSTRATION OMITTED]

The U.S. population was disaggregated into states multivariately defined by sociodemographic, health, and disease descriptors: age (single-year age groups); sex; race (white, black, Asian or Pacific islander, Native American or Alaskan native, or other race); ethnicity (Hispanic or non-Hispanic); TB infection or disease status; and location. One hundred sixty locations were defined: 60 corresponded to the largest standard metropolitan statistical areas (SMSAs) in the United States, 50 to groupings of smaller SMSAs in the 50 states, and 50 to groupings of non-SMSA counties in the 50 states. The impact of the HIV epidemic on the TB epidemic was addressed by dealing with certain location-specific populations in a way that recognizes that a fraction of them are dually affected. A state in the Markov chain was completely defined by the joint specification of all these descriptors.

In the absence of longitudinal data, the population database for the model was constructed by assembling population profiles for each of the individual years from historical surveys and population projections (5-17,23). These profiles consisted of the sizes of population groups defined multivariately by age, race, sex, ethnicity, and geographic location. The population projections of the Bureau of the Census incorporate expected immigration patterns.

Transition probabilities for the model were based on rates cited in the literature to describe how these transitions occur for individuals. The probability of transition from healthy to infected has been termed the "annual risk for infection." To estimate this probability from tuberculin skin test surveys, we used the catalytic method of Muench (19,24,25). This method relates incidence to prevalence, given several data points of the latter, under the assumption of an exponential model, the exponential curve being called "catalytic" at the time of Muench, who used graphic methods for estimation. The average number of susceptible persons infected by each infectious (i.e., sputum-positive) case in a geographic location in 1 year was termed the "contagious parameter" by Styblo, who noted the correspondence between the prevalence rate of the disease and the annual risk for infection (25,26). Since we assume that new active cases are treated during their year of occurrence, our use of the contagious parameter is consistent with that of Styblo.

An important assumption made in the model about the spread of infection is that each ill person infects susceptible persons entirely within race ethnicity location-specific subgroups, with the number of infections determined by the contagious parameter. The age-sex distribution of newly infected persons in the model was based on national TB contact investigation data (27). Time trends for contagious parameters for the various race ethnicity location-specific subgroups were based on quadratic regression fits of the total cases of TB for those subgroup,; over time from 1980 through 1993. For illustration, we present in Table 1 baseline values for the contagious parameter for 17- to 25-year-old white men, by state, derived from the U.S. naval recruits data (28-35).

Table 1. Baseline values of contagious parameters, for 17- to 25-year-old white men, by state

                        New ACR(a)    ARI(b)    ARI/TB(c)

Alabama                    0.13        38.2       293.6
Arizona                    0.30        52.7       178.3
Arkansas                   0.15        39.7       261.6
California                 0.16        36.4       228.1
Colorado                   0.19        18.9        97.4
Connecticut                0.12        15.7       134.2
Delaware                   0.12        30.1       243.2
District of Columbia       0.15        55.6       376.1
Florida                    0.07        27.4       410.5
Georgia                    0.08        31.8       395.7
Idaho                      0.10        13.2       135.3
Illinois                   0.15        39.6       269.3
Indiana                    0.16        30.7       198.0
Iowa                       0.07         9.2       130.4
Kansas                     0.07        12.6       176.9
Kentucky                   0.28        50.6       177.9
Louisiana                  0.10        35.4       339.6
Maine                      0.14        25.2       179.9
Maryland                   0.22        36.4       164.4
Massachusetts              0.14        28.9       203.9
Michigan                   0.14        29.7       205.3
Minnesota                  0.10        17.3       181.8
Mississippi                0.09        30.3       334.3
Missouri                   0.14        32.9       231.8
Montana                    0.11        13.6       123.2
Nebraska                   0.09        16.4       174.5
Nevada                     0.07        53.8       747.9
New Hampshire              0.11        16.8       153.6
New Jersey                 0.15        23.7       161.6
New Mexico                 0.22        29.1       130.9
New York                   0.17        36.6       218.1
North Carolina             0.08        26.1       342.8
North Dakota               0.07         8.6       122.5
Ohio                       0.14        27.1       198.2
Oklahoma                   0.13        24.0       185.8
Oregon                     0.12        32.0       257.0
Pennsylvania               0.17        32.1       183.8
Rhode Island               0.12        27.9       235.7
South Carolina             0.04        26.7       615.2
South Dakota               0.06         9.2       163.8
Tennessee                  0.19        45.8       244.7
Texas                      0.21        29.2       138.7
Utah                       0.04         7.6       197.1
Vermont                    0.18        18.3        99.3
Virginia                   0.13        32.1       237.9
Washington                 0.11        25.7       231.9
West Virginia              0.21        39.1       189.4
Wisconsin                  0.08        22.1       260.1
Wyoming                    0.15        10.3        66.7
Total                      0.14        30.1       210.3

(a) ACR, Active case rate per 100,000 per year tuberculosis

(b) ARI, annual rate of infection per 100,000 per year

(c) Contagious parameter

The probability of transition from infected to ill is a product of host-parasite inter biologic features of these interactions. A person with a new Mycobacterium tube of [is less than] 1 year) is at much higher risk for clinical disease than someone with a [ILLEGIBLE TEXT] ongoing risk in later years (36). These risks vary with age. Although not [ILLEGIBLE TEXT] persons is allowed at the same rate as infections with the same characteristics in [ILLEGIBLE TEXT] persons with new and mature infections in the next year are based on the upper a earlier studies, and are adjusted to calibrate the model to the historical time series [ILLEGIBLE TEXT]

The effects of the HIV epidemic and the emergence of multidrug-resistant TB [ILLEGIBLE TEXT] exogenous inputs for appropriate geographic locations and, within these location subgroups, as dictated by trends in prevalence and dual-infection rates (43-45). [ILLEGIBLE TEXT] dependent changes to the probabilities of transition from new or mature infection [ILLEGIBLE TEXT]

We established 1980 as the baseline year for all variables used in the model. The based on the Reports of Verified Cases of Tuberculosis (RVCT) files and [ILLEGIBLE TEXT] Centers for Disease Control and Prevention. Because the RVCT program was no [ILLEGIBLE TEXT] 1985, reporting for the years 1980 to 1984 was incomplete. To estimate the 1980 of missing data according to the multivariate structure observed in 1985. Baselin of mature infections prevalent in 1980 and counts of new infections in 1980. The applying difference equations relating the numbers of cases, new infections, and the numbers of cases, new infections, and mature infections in the previous year, figures for the 2 years were dependent on model parameters such as survival rate infection to disease, rates of transition from new infection to disease, and [ILLEGIBLE TEXT] difference equation pertained to a specific multivariately defined subpopulation, difference equations, it was possible to express the number of mature infections [ILLEGIBLE TEXT] for an arbitrary number of subsequent years for which actual data were available, examined for consistency and the most stable was selected. Because of the lack [ILLEGIBLE TEXT] attempt to address the underreporting of TB cases or underestimates of some [ILLEGIBLE TEXT]

Computer implementation of the model was carded out on a Sun SPARCstation memory and network access to a SPARCserver 1000E with approximately 100 [ILLEGIBLE TEXT] software was written in C++, an object-oriented programming language. A [ILLEGIBLE TEXT] (46,47) was used as the basis for the representation of fuzzy numbers in the soft [ILLEGIBLE TEXT]

Because of the extremely large sizes of the multidimensional arrays containing [ILLEGIBLE TEXT] year-to-year transition data, specialized data structures and algorithms were [ILLEGIBLE TEXT] characteristics of these data that allowed them to be stored in a more compact for population data, a height-balanced binary search tree structure (49,50) was used. Information and array subscript information were stored. Because the subscripts each node of the tree if stored conventionally, a 4-byte compressed [ILLEGIBLE TEXT] each node. For transition data, a simple linked-list structure was used in which [ILLEGIBLE TEXT] information and compressed subscript information. This approach resulted in a 9 required for population data and a 99% reduction of storage space required for [ILLEGIBLE TEXT]

Results

Figure 2 presents model projections of new cases of TB for the United States for through 2010 as well as actual data for white non-Hispanics, blacks, and [ILLEGIBLE TEXT] groupings used in TB case reporting in the United States. For these three groups, groupings used in TB case reporting in the United States. For these three groups, differences between actual counts and model predictions were generally [is less than] 1,000 [ILLEGIBLE TEXT] blacks and Hispanics and [is less than] 1,500 for white non-Hispanics.

[Figure 2 ILLUSTRATION OMITTED]

Table 3 presents actual and projected TB case rates per 100,000 for each 5 years 1980 to 2010 with results presented for racial, ethnic, and geographic groups. [ILLEGIBLE TEXT] the model performs best, within the 1980 to 1995 validation period, for large [ILLEGIBLE TEXT] with relatively high case rates. For example, it projects TB case rates to within 1 actual rates for the U.S. population at large. Its performance is not as good in [ILLEGIBLE TEXT] having lower case rates (e.g., there is a 23% projection error for non-Hispanic [ILLEGIBLE TEXT] 1995). In addition, projection errors are relatively high for smaller subgroups, [ILLEGIBLE TEXT] those in which case reporting and population reporting are subject to inaccuracy projection errors are consistently above 20% for Hispanics and nearly as high for American Indians and Asian/Pacific Islanders). The model's best performance [ILLEGIBLE TEXT] among the older population: the model predicts case rates to within 1% accuracy U.S. population aged 65 and over for 1990 and 1995. This is of significance in [ILLEGIBLE TEXT] highest TB case rates are among the elderly.

[TABULAR DATA 3 NOT REPRODUCIBLE IN ASCII]

Figure 3 presents model projections of the TB case rate quartiles, by state, for the 2010. Note that the projected case rate for Wyoming spuriously falls into the [ILLEGIBLE TEXT] quartile, as a result of its small population size. TB is projected to decrease in all the United States and remain largely a problem of states receiving large numbers immigrants.

[Figure 3 ILLUSTRATION OMITTED]

In Figure 4, the model projection for the United States for 1980 through 2010 is presented together with actual numbers of newly reported TB cases for 1980 [ILLEGIBLE TEXT] 1997. Also depicted is the sensitivity of the model to changes in key parameters, thicker lines in Figure 1. Specifically, changes were allowed for the contagious parameters and for rates of transition from healthy to infected and infected to ill.

[Figure 4 ILLUSTRATION OMITTED]

Specifically, above and below the central projection are lines representing [ILLEGIBLE TEXT] from simultaneous 10% increases and decreases in these key parameters for each the projection period. A simultaneous decrease of 10% in the parameters for each the projection period results in a decrease of 27% (from 13,458 to 9,861) in [ILLEGIBLE TEXT] cases in 2010. A simultaneous increase of 10% yields an increase of 44% (from to 19,381) in projected cases in 2010. Not shown in the figure, a simultaneous [ILLEGIBLE TEXT] of 5% in the same parameters yields a decrease of 15% (from 13,458 to 11,459) [ILLEGIBLE TEXT] projected cases in 2010, and a simultaneous increase of 5% yields an increase of (from 13,458 to 16,063) in 2010.

Conclusions

The studies of Waaler and colleagues, who carefully noted and credited other [ILLEGIBLE TEXT] workers in this field, mark the beginning of a modern approach to modeling and thoughtful consideration of transition probabilities in modeling (51,52). ReVelle, and Feldman, using many of the assumptions of Waaler and his colleagues, [ILLEGIBLE TEXT] developing countries as a result of BCG vaccination (53). Azuma modeled TB in AIDS prevalence were used by Styblo (55), Schulzer and colleagues (56), and [ILLEGIBLE TEXT] in sub-Saharan Africa. Modeling was used by Murray, Styblo, and Rouillon to [ILLEGIBLE TEXT] in developing countries (58). Blower and colleagues used a differential [ILLEGIBLE TEXT] of TB epidemics and emphasize the importance of effective case treatment in TB colleagues used differential equation-based models to compare the global impact (61,62). Brewer and colleagues used simulation techniques to compare TB [ILLEGIBLE TEXT]

Mathematical modeling ,of TB in the United States probably began in 1939 with age cohort analyses of secular trends to predict the future decline of TB (64). [ILLEGIBLE TEXT] the United States based on data from 1963 and used it to emphasize the potential chemoprophylaxis to TB control (65). The model we present is substantially [ILLEGIBLE TEXT] from her early work by disaggregating the American population. Given the mark the United States and the corresponding diversity of TB case rates in various [ILLEGIBLE TEXT] approach offers major advantages over aggregate modeling both in improved [ILLEGIBLE TEXT] in output that can be disaggregated to relate to segments of the population with [ILLEGIBLE TEXT]

A highly disaggregated model incorporating population dynamics is appropriate. population dynamics are aptly described by a linear model, particularly when [ILLEGIBLE TEXT] The model we constructed is Markovian in that it is independent of its history [ILLEGIBLE TEXT] computer implementation of the theoretical model is clearly Markovian because [ILLEGIBLE TEXT] Markov chain, in the sense that the initial state was the state of our system in [ILLEGIBLE TEXT] year was generated from the previous year's state and 1-year transition rates. Sin previous year's state and 1-year transition rates are available to the computer soft the Markovian nature is established.

Our model for the intermediate years predicted (on the basis of 1980 baseline [ILLEGIBLE TEXT] ensuing decade and decline again. This prediction reasonably approximated the [ILLEGIBLE TEXT] does include changes in TB incidence related to the HIV epidemic and to [ILLEGIBLE TEXT] incorporate changes in TB control measures, whose impacts are only addressed [ILLEGIBLE TEXT] transition probabilities in the model. The latter could be addressed by a submode subgroups, and could affect projections.

The projections presented here suggest that the number of new cases and [ILLEGIBLE TEXT] decrease to a level approximately half the present level nationally. Measuring the efforts contributed to the return of the TB epidemic to its formerly declining [ILLEGIBLE TEXT] analysis.

The handling of geographic location necessarily relies on politically defined [ILLEGIBLE TEXT] collected. Since disease patterns do not necessarily follow these boundaries, the geographic location. This may be seen in Table 3 where the TB case rate in 1995 and underestimated for Miami. However, the very fine age structure imposed has within 20% accuracy for age 25 and older (Table 3). Refinement of the [ILLEGIBLE TEXT] corresponding increases in accuracy.

The model projections are smoother over time than the actual counts, as should [ILLEGIBLE TEXT] fluctuations are not built into the model (Figures 2,3). In Figure 4 the time series lower 10% bound. This may be a temporary fluctuation (as in the mid-1980s) but transitions in the real-world disease process.

[Figure 4 ILLUSTRATION OMITTED]

Errors of a projection model may arise from shortcomings of model structure, [ILLEGIBLE TEXT] or from errors in the values of key model parameters. The structure of the Marko by highly detailed population dynamics. Each part of the mechanistic process [ILLEGIBLE TEXT] refinements and discussions of variability; notably, occurrence of infection [ILLEGIBLE TEXT] (68,69), and involvement of HIV (70) and immigrant status (71). The greatest [ILLEGIBLE TEXT] initial conditions arises from estimates of the 1980 pool of prevalent infections, consequential with passing time, as persons infected before 1980 die. So, model projection period are least affected by errors in the numbers of those infected [ILLEGIBLE TEXT] have the obvious role of governing transitions for various population subgroups, dependency, are used to depict exogenous phenomena such as the HIV epidemic, efforts, and changes in immigration patterns. Geographically focused changes of after 1993 related to directly observed therapy could result in overestimation in [ILLEGIBLE TEXT]

Although the model does not allow mixing of groups from one location to [ILLEGIBLE TEXT] adjusted in the calculations of transition probabilities since these are based on act movements of healthy, infected, and ill groups. The other type of mixing not [ILLEGIBLE TEXT]location among different race-ethnicity subgroups. The most serious projection e subgroup of uninfected persons lives in the same location as a subgroup of [ILLEGIBLE TEXT] arising in the former subgroup would not be predicted by our model. This type of locations with very few active TB cases. Otherwise, the projection errors arise on cross-spread of infection fails to cancel out (i.e., the net extent to which a [ILLEGIBLE TEXT] number of the latter being infected by the former).

Although HIV status is addressed in this model, adding structure that more [ILLEGIBLE TEXT] rates of HIV-infected persons might also improve model projections. The model cases will occur in Hispanics, and this prediction is supported by the most recent decline in the number of cases in Hispanics is slower than for white [ILLEGIBLE TEXT]

The foreign born account :for a large fraction of the new cases seen in the United regarding the future contribution of the foreign born to the U.S. population are [ILLEGIBLE TEXT] projections; therefore, this component of the population may be thought of as for which the model projections are made. Departure from the patterns of [ILLEGIBLE TEXT] could cause two kinds of errors in model projections. A surge of immigration [ILLEGIBLE TEXT] would soon be followed by increases in TB cases and case rates in the United Stated the model. A different effect would be seen as a consequence of unexpected [ILLEGIBLE TEXT] Regardless of the TB status of the countries of origin, an increase in the number first in an artificial lowering of case rates because of the increase in denominator increase in numbers of cases. The model could not recognize these changes within potential errors may be avoided as additional structure better [ILLEGIBLE TEXT] the model so that unexpected influxes may be modeled as exogenous inputs.

[TABULAR DATA 2 NOT REPRODUCIBLE IN ASCII]

Acknowledgment

The authors thank Philip J. Smith for helpful comments.

This work was supported in part by a contract with the Centers for Disease [ILLEGIBLE TEXT] cooperative agreement with the Association of Teachers of Preventive Medicine.

References

(1.) Bloch AB, Rieder HL, Kelly GD, Cauthen GM, Hayden CH, Snider DE Jr in the United States. Clin Chest Med 1989;10:297-313.

(2.) Ellner JJ, Hinman AR, Dooley SW, Fischl MA, Sepkowitz KA, Goldberge symposium: emerging problems and promise. J Infect Dis 1993;168:537-[ILLEGIBLE TEXT]

(3.) Cantwell MF, Snider DE Jr, Cauthen GM, Onorato IM. Epidemiology of [ILLEGIBLE TEXT] 1985 through 1992. JAMA 1994;272:535-9.

(4.) Comstock GW. Variability of tuberculosis trends in a time of resurgence.

(5.) Centers for Disease Control and Prevention. Tuberculosis in the United States

(6.) Census of Population and Housing, 1990: Public use microdata sample U. machine-readable data file/prepared by the Bureau of the Census. [ILLEGIBLE TEXT]

(7.) 1990 Census of Population and Housing, 5% and 1% public use microdata equivalency files machine-readable data files / prepared by the Bureau of [ILLEGIBLE TEXT] 1994.

(8.) Census of Population and Housing, 1980; summary tape file 3A machine-[ILLEGIBLE TEXT] Bureau of the Census. Washington: The Bureau; 1982.

(9.) Census of Population and Housing, 1980; summary tape file 3B machine-[ILLEGIBLE TEXT] Bureau of the Census. Washington: The Bureau; 1982.

(10.) Census of Population and Housing, 1980; summary tape file 3C machine-[ILLEGIBLE TEXT] Bureau of the Census. Washington: The Bureau; 1982.

(11.) Census of Population and Housing, 1980; summary tape file 3 technical do User Services Division, Bureau of the Census. Washington: The Bureau; 1

(12.) Census of Population and Housing, 1990: summary tape file 3 on CD-ROM prepared by the Bureau of the Census. Washington: The Bureau; 1992.

(13.) Intercensal estimates of the population of counties by age, sex, and race: 1 files / prepared by the Bureau of the Census. Washington: The Bureau; [ILLEGIBLE TEXT]

(14.) Population projections for states, by age, sex, race, and hispanic origin: 19 file / prepared by the Bureau of the Census Population Division. [ILLEGIBLE TEXT]

(15.) U.S. Bureau of the Census. Current population reports, P25-1111, population race, and hispanic origin: 1993 to 2020, by Paul R. Campbell. Washington [ILLEGIBLE TEXT]

(16.) Tuberculosis in the United States, for years 1980 through 1997, Atlanta [ILLEGIBLE TEXT] Prevention, Division of Tuberculosis Control, 1982-1997.

(17.) AIDS Public Information Data Set, Statistics and Data Management [ILLEGIBLE TEXT] 1993, Atlanta, GA.

(18.) Reports of Verified Cases of Tuberculosis (RVCT), CDC, Surveillance [ILLEGIBLE TEXT] of Tuberculosis Control, 1980-1993, Atlanta, GA.

(19.) Daniel TM, Debanne SM. Estimation of the annual risk of tuberculous [ILLEGIBLE TEXT] States. J Infect Dis 1997;175:1535-7.

(20.) Markov AA. Investigation of a noteworthy case of dependent trials. Izv [ILLEGIBLE TEXT]

(21.) Markov AA. The calculus of probabilities. 4th ed. Moscow: GIZ; 1924.

(22.) Takacs LF. Stochastic processes. London: Methuen, 1960.

(23.) Liaw K-L, Frey WH. Destination choices of the 1985-90 young adult [ILLEGIBLE TEXT] importance of race, educational attainment and labor market forces. Intern Geography 1998;4:49-61.

(24.) Muench H. Derivation of rates from summation data by the catalytique [ILLEGIBLE TEXT] Statistical Association 1934;29:25-38.

(25.) Styblo K. The relationship between the risk of tuberculous infection and [ILLEGIBLE TEXT] tuberculosis. Bulletin of the International Union against Tuberculosis 1985 [ILLEGIBLE TEXT]

(26.) Styblo K. Epidemiology of tuberculosis. Selected papers. Vol 24. The [ILLEGIBLE TEXT] Netherlands Tuberculosis Association (KNCV); 1991. p. 49-50.

(27.) Tuberculosis Program Management in the United States, 1984. Atlanta [ILLEGIBLE TEXT] Prevention, Center for Prevention Services, Division of Tuberculosis [ILLEGIBLE TEXT]

(28.) Edwards LB, Acquaviva FA, Livesay VT, Cross FW, Palmer CE. An atlas and histoplasmin in the United States. Am Rev Respir Dis 1969;99 Suppl: [ILLEGIBLE TEXT]

29. Edwards LB, Smith DT. Community-wide tuberculin testing in Pamlico [ILLEGIBLE TEXT] Respir Dis 1965;92:43-54.

(30.) Rosenthal SR, Afremow ML, Nikurs L, Loewinsohn E, Leppmann M, [ILLEGIBLE TEXT] tuberculosis in students of nursing. Am J Nurs 1963;63:88-93.

(31.) Hanzel GD, Rogers KD. Multiple-puncture and Mantoux tuberculin tests [ILLEGIBLE TEXT] comparative study. JAMA 1964;190:1038-42.

(32.) Rhoades ER, Alexander CP. Reactions to the tuberculin time test in air for tests. Am Rev Respir Dis 1968;98:837-41.

(33.) Comstock GW. Frost revisited: the modern epidemiology of tuberculosis.

(34.) Reichman LB, O'Day R. Tuberculous infection in a large urban population 12.

(35.) Engel A, Roberts J. Tuberculin skin test reaction among adults 25-74 years Publication No. (HRA) 77-1649.

(36.) Comstock GW, Cauthen GM. In: Reichman LB, Hershfield ES, editors. [ILLEGIBLE TEXT] international approach. New York: Marcel Dekker, Inc.; 1993. p. 23-48.

(37.) Zeidberg LD, Gass RS, Dillon A, Hutcheson RH. The Williamson County year epidemiologic :study. Am Rev Respir Dis 1963;87 Suppl:1-88.

(38.) Ferebee SH. Controlled chemoprophylaxis trials in tuberculosis. A general Research 1970;17:28-106.

(39.) Devadatta S, Dawson JJY, Fox W, Janardhanam B, Radhakrishna S, Rama tuberculosis in a 5-year period among close family contacts of tuberculous with isoniazid plus PAS or isoniazid alone. Bull World Health Organ 1970 [ILLEGIBLE TEXT]

(40.) Morris SI. Tuberculosis as an occupational hazard during medical training: up study, with effective control program for tuberculosis in women [ILLEGIBLE TEXT] Tuberculosis 1946;54:140-58.

(41.) Myers JA. Tuberculosis. A half century of study and conquest. St. Louis [ILLEGIBLE TEXT] p. 166-99.

(42.) Ferebee SH, Mount FW. Tuberculosis morbidity in a controlled trial of the household contacts. Am Rev Respir Dis 1962;85:490-521.

(43.) Daley CL, Small PM, Schecter GF, Schoolnik GK, McAdam RA, Jacobs tuberculosis with accelerated progression among persons infected with the Engl J Med 1992;326:231-5.

(44.) Di Perri G, Cruciani M, Danzi MD, Luzatti R, De Checchi G, Malena M, [ILLEGIBLE TEXT] tuberculosis among HIV-infected patients. Lancet 1989;2:1502-4.

(45.) Pitchenik AE, Burr J, Laufer M, Miller G, Cacciatore R, Bigler WJ, et al. tuberculosis an AIDS center. Lancet 1990;336:440-1.

(46.) Bielefeld RA. Fuzzy representation of uncertainty in disease progression [ILLEGIBLE TEXT] Epidemiology and Biostatistics. Cleveland (OH): Case Western Reserve [ILLEGIBLE TEXT]

(47.) Bielefeld RA. Estimation of HIV infection and AIDS onset times using [ILLEGIBLE TEXT] Proceedings of the Seventh World Congress on Medical Informatics. [ILLEGIBLE TEXT] 921-7.

(48.) Bielefeld RA, Debanne SM, Rowland DY. Storing sparse and repeated [ILLEGIBLE TEXT] of tuberculosis spread. Computers and Biomedical Research 1996;29:85-[ILLEGIBLE TEXT]

(49.) Adel'son-Vel'skii GM, Landis EM. An algorithm for the organization of in Math 1962;146:263-6.

(50.) Standish TA. Data structure techniques. Reading (MA): Addison-Wesley;

(51.) Waaler H, Geser A., Anderson S. The use of mathematical models in the [ILLEGIBLE TEXT] tuberculosis. Am J Public Health 1962;52:1002-13.

(52.) Waaler HT, Piot MA. The use of an epidemiologic model for estimating [ILLEGIBLE TEXT] control measures: sensitivity of the effectiveness of tuberculosis control [ILLEGIBLE TEXT] population. Bull World Health Organ 1969;41:75-93.

(53.) ReVelle CS, Lynn WR, Feldman F. Mathematical models for the [ILLEGIBLE TEXT] activities in developing countries. Am Rev Respir Dis 1967;96:893-909.

(54.) Azuma Y. A simple simulation model of tuberculosis epidemiology for us [ILLEGIBLE TEXT] World Health Organ 1975;52:313-22.

(55.) Styblo K. The global aspects of tuberculosis and HIV infection. Bulletin of Tuberculosis and Lung Disease 1990;65:28-32.

(56.) Schulzer M, Fitzgerald JM, Enarson DA, Grzybowski S. An estimate of [ILLEGIBLE TEXT] problem in sub-Saharan Africa resulting from HIV infection. Tuber Lung [ILLEGIBLE TEXT]

(57.) Heymann SJ. Modeling the efficacy of prophylactic and curative therapies tuberculosis in Africa. Trans R Soc Trop Med Hyg 1993;87:406-11.

(58.) Murray CJL, Styblo K, Rouillon A. Tuberculosis in developing countries: Internat Union against Tuberc 1990;65:2-20.

(59.) Blower SM, McLean AR, Porco TC, Small PM, Hopewell PC, Sanchez [ILLEGIBLE TEXT] transmission dynamics of tuberculosis epidemics. Nature Medicine 1995; [ILLEGIBLE TEXT]

(60.) Blower SM, Small PM, Hopewell PC. Control strategies for tuberculosis [ILLEGIBLE TEXT] problems. Science 1996;273:497-500.

(61.) Murray CJL, Salomon JA. Expanding the WHO tuberculosis control [ILLEGIBLE TEXT] case-finding. International Journal of Tuberculosis and Lung Disease 1998

(62.) Murray CJL, Salomon JA. Modeling the impact of global tuberculosis [ILLEGIBLE TEXT] S A 1998;95:13881-6.

(63.) Brewer TF, Heymann SJ, Colditz GA, Wilson ME, Auerbach K, Kane D, tuberculosis control policies using computer simulation. JAMA 1996;276:[ILLEGIBLE TEXT]

(64.) Frost WH. The age selection of mortality from tuberculosis in successive [ILLEGIBLE TEXT] Hygiene 1939;30:91-6.

(65.) Ferebee SH. An epidemiological model of tuberculosis in the United State Tuberculosis Association 1967 (January):4-7.

(66.) Rieder HL. Methodological issues in the estimation of the tuberculosis [ILLEGIBLE TEXT] Lung Dis 1995;76:114-21.

(67.) Gammaitoni L, Nucci MC. Using a mathematical model to evaluate the [ILLEGIBLE TEXT] Emerg Infect Dis 1997;3:335-42.

(68.) MacIntyre CR, Plant AJ. Preventability of incident cases of tuberculosis in International Journal of Tuberculosis and Lung Disease 1998;2:56-61.

(69.) Salpeter EE, Salpeter SR. Mathematical model for the epidemiology of [ILLEGIBLE TEXT] reproductive number and infection-delay function. Am J Epidemiol 1998; [ILLEGIBLE TEXT]

(70.) West RW, Thompson JR. Modeling the impact of HIV on the spread of [ILLEGIBLE TEXT] Biosci 1997;143:35-60.

(71.) Liu Z, Shilkret KL, Tranotti J, Freund CG, Finelli L. Distinct trends in [ILLEGIBLE TEXT] born and U.S.-born persons in New Jersey, 1986 through 1995. Am J [ILLEGIBLE TEXT]

Sara M. Debanne,(*) Roger A. Bielefeld,(*) George M. Cauthen,([dagger]) Thomas M. Daniel,(*) and Douglas Y. Rowland([double dagger])

(*) Case Western Reserve University, Cleveland, Ohio, USA; ([dagger]) Centers for Disease Control and Prevention, Atlanta, Georgia, USA; and ([double dagger]) D.Y. Rowland Associates, Cleveland, Ohio, USA

Dr. Debanne is an associate professor of epidemiology and biostatistics at the [ILLEGIBLE TEXT] Reserve University. She specializes in the construction of mathematical models extensively in the area of tuberculosis.

Address for correspondence: Sara M. Debanne, Department of Epidemiology an Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106-[ILLEGIBLE TEXT] mail: [e-mail]