Emerging Infectious Diseases
[Volume 6 No.2 / March - April 2000]

Research

Multivariate Markovian Modeling of Tuberculosis: Forecast for the
United States

Sara M. Debanne,* Roger A. Bielefeld,* George M. Cauthen,
Thomas M. Daniel,* and Douglas Y. Rowland
*Case Western Reserve University, Cleveland, Ohio, USA;
Centers for Disease Control and Prevention, Atlanta, Georgia,
USA; and D.Y. Rowland Associates, Cleveland, Ohio, USA

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-Hispanicsa
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.

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.

Figure 1. Tuberculosis model states and possible transitions.
Thicker lines depict transitions considered in sensitivity analysis.

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 popula tion 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 subgroups 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).  The probability
of transition from infected to ill is a product of host-parasite
interactions and is determined by the biologic features of these
interactions. A person with a new Mycobacterium tuberculosis
infection (i.e., an infection of <1 year) is at much higher risk for
clinical disease than someone with a mature infection, who has
been at an ongoing risk in later years (36). These risks vary with
age. Although not explicitly modeled, reinfection of cured persons
is allowed at the same rate as infections with the same
characteristics in the population. The case rates among persons
with new and mature infections in the next year are based on the
upper and lower bounds of rates reported in earlier studies, and are
adjusted to calibrate the model to the historical time series (27,37-
42) (Table 2). The effects of the HIV epidemic and the emergence
of multidrug-resistant TB are modeled through time-dependent
exogenous inputs for appropriate geographic locations and, within
these locations, for specific demographic subgroups, as dictated by
trends in prevalence and dual-infection rates (43-45). These
calculations result in time-dependent changes to the probabilities
of transition from new or mature infection to clinical TB.

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

We established 1980 as the baseline year for all variables used in
the model. The initial values of these variables were based on the
Reports of Verified Cases of Tuberculosis (RVCT) files and
published reports of TB incidence from Centers for Disease
Control and Prevention. Because the RVCT program was not
implemented nationwide until 1985, reporting for the years 1980 to
1984 was incomplete. To estimate the 1980 baseline values, we
inferred values of missing data according to the multivariate
structure observed in 1985. Baseline information also included
counts of mature infections prevalent in 1980 and counts of new
infections in 1980. These values were estimated by applying
difference equations relating the numbers of cases, new infections,
and mature infections in a given year to the numbers new
infections, and mature infections in the previous year, where the
relationships between the figures for the 2 years were dependent on
model parameters such as survival rates, rates of transition from
mature infection to disease, rates of transition from new infection
to disease, and contagious parameters, and where each difference
equation pertained to a specific multivariately defined
subpopulation. Through repeated application of the difference
equations, it was possible to express the number of mature
infections in 1980 in terms of TB case counts for an arbitrary
number of subsequent years for which actual data were available.
Estimates thus obtained were examined for consistency and the
most stable was selected. Because of the lack of reliable
information, we did not attempt to address the underreporting of
TB cases or underestimates of some populations.

Table 2. Literature-based lower, best, and upper estimates of the
risk of developing tuberculosis for infected persons, by age and sex

Figure 2. Total number of new cases of tuberculosis in white non-
Hispanics, blacks, and Hispanics in the United States, 19802010.

Computer implementation of the model was carried out on a Sun
SPARCstation 5 with 80 megabytes of main memory and network
access to a SPARCserver 1000E with approximately 100 gigabytes
of online disk storage. All software was written in C++, an object-
oriented programming language. A library of C++ objects and
functions (46,47) was used as the basis for the representation of
fuzzy numbers in the software. Because of the extremely large
sizes of the multidimensional arrays containing cross-sectional
population data and year-to-year transition data, specialized data
structures and algorithms were developed to take advantage of
characteristics of these data that allowed them to be stored in a
more compact form (48). For cross-sectional population data, a
height-balanced binary search tree structure (49,50) was used.
Within each tree node, both cell information and array subscript
information were stored. Because the subscripts would occupy at
least 9 bytes within each node of the tree if stored conventionally, a
4-byte compressed representation of the subscripts was stored
within each node. For transition data, a simple linked-list structure
was used in which each node contained both cell information and
compressed subscript information. This approach resulted in a 96%
reduction of storage space required for population data and a 99%
reduction of storage space required for transition data (48).

Results

Figure 2 presents model projections of new cases of TB for the
United States for 1980 through 2010 as well as actual data for
white non-Hispanics, blacks, and Hispanics, the groupings used in
TB case reporting in the United States. For these three groups,
differences between actual counts and model predictions were
generally <1,000 for blacks and Hispanics and <1,500 for white
non-Hispanics. Table 3 presents actual and projected TB case rates
per 100,000 for each 5 years from 1980 to 2010 with results
presented for racial, ethnic, and geographic groups. Generally, the
model performs best, within the 1980 to 1995 validation period, for
large subgroups with relatively high case rates. For example, it
projects TB case rates to within 12% of actual rates for the U.S.
population at large. Its performance is not as good in subgroups
having lower case rates (e.g., there is a 23% projection error for
non-Hispanic whites in 1995). In addition, projection errors are
relatively high for smaller subgroups, especially those in which
case reporting and population reporting are subject to inaccuracy
(e.g., projection errors are consistently above 20% for Hispanics
and nearly as high for American Indians and Asian/Pacific
Islanders). The model's best performance tends to be among the
older population: the model predicts case rates to within 1%
accuracy for the U.S. population aged 65 and over for 1990 and
1995. This is of significance in that the highest TB case rates are
among the elderly.

Table 3. Actual and projected tuberculosis rates per 100,000 for
selected population groups, 1980 to 2010

Figure 4. Total number of new cases of tuberculosis in the United
States, 19802010 and sensitivity of model projections.

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

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 through 1997. Also depicted is the
sensitivity of the model to changes in key parameters, seen in
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. Specifically, above and below the
central projection are lines representing projections from
simultaneous 10% increases and decreases in these key parameters
for each year in the projection period. A simultaneous decrease of
10% in the parameters for each year of the projection period results
in a decrease of 27% (from 13,458 to 9,861) in projected TB cases
in 2010. A simultaneous increase of 10% yields an increase of 44%
(from 13,458 to 19,381) in projected cases in 2010. Not shown in
the figure, a simultaneous decrease of 5% in the same parameters
yields a decrease of 15% (from 13,458 to 11,459) for projected
cases in 2010, and a simultaneous increase of 5% yields an
increase of 19% (from 13,458 to 16,063) in 2010.

Conclusions

The studies of Waaler and colleagues, who carefully noted and
credited other early workers in this field, mark the beginning of a
modern approach to modeling and the thoughtful consideration of
transition probabilities in modeling (51,52). ReVelle, Lynn, and
Feldman, using many of the assumptions of Waaler and his
colleagues, projected declining TB case rates in developing
countries as a result of BCG vaccination (53). Azuma modeled TB
incidence in Japan (54). Estimates of AIDS prevalence were used
by Styblo (55), Schulzer and colleagues (56), and Heymann (57) to
project TB incidence in sub-Saharan Africa. Modeling was used by
Murray, Styblo, and Rouillon to examine the economic impact of
TB in developing countries (58). Blower and colleagues used a
differential equation-based model to examine the decline of TB
epidemics and emphasize the importance of effective case
treatment in TB control (59,60). Murray and colleagues used
differential equation-based models to compare the global impact of
different control strategies (61,62). Brewer and colleagues used
simulation techniques to compare TB control policies (63).

Mathematical modeling of TB in the United States probably began
in 1939 with the early work of Frost, who used age cohort analyses
of secular trends to predict the future decline of TB (64). Ferebee
developed a model for TB in the United States based on data from
1963 and used it to emphasize the potentially important
contribution of chemoprophylaxis to TB control (65). The model
we present is substantially more complex than Ferebee's and
differs from her early work by disaggregating the American
population. Given the marked heterogeneity of the population of
the United States and the corresponding diversity of TB case rates
in various American population groups, our approach offers major
advantages over aggregate modeling both in improved accuracy of
estimation of input data and in output that can be disaggregated to
relate to segments of the population with very different TB control
problems.

A highly disaggregated model incorporating population dynamics
is appropriate. When applied to small groups, population dynamics
are aptly described by a linear model, particularly when natural
constraints are stated as totals. The model we constructed is
Markovian in that it is independent of its history prior to its last
known state. Our computer implementation of the theoretical
model is clearly Markovian because its operation parallels the
nature of a Markov chain, in the sense that the initial state was the
state of our system in 1980, and the state in each subsequent year
was generated from the previous year's state and 1-year transition
rates. Since no information other than the previous year's state and
1-year transition rates are available to the computer software to
generate the state for a year, the Markovian nature is established.
Our model for the intermediate years predicted (on the basis of
1980 baseline data) that TB would increase late in the ensuing
decade and decline again. This prediction reasonably approximated
the subsequent events. While the model does include changes in
TB incidence related to the HIV epidemic and to immigration, it
does not specifically incorporate changes in TB control measures,
whose impacts are only addressed indirectly through net changes of
transition probabilities in the model. The latter could be addressed
by a submodel for appropriate population subgroups, and could
affect projections. The projections presented here suggest that the
number of new cases and incidence rates of TB will now continue
to decrease to a level approximately half the present level
nationally. Measuring the extent to which increased control efforts
contributed to the return of the TB epidemic to its formerly
declining phase will require more detailed analysis.

The handling of geographic location necessarily relies on
politically defined entities which determine how data are collected.
Since disease patterns do not necessarily follow these boundaries,
the model is not uniformly accurate by geographic location. This
may be seen in Table 3 where the TB case rate in 1995 is
overestimated for Los Angeles and underestimated for Miami.
However, the very fine age structure imposed has resulted in
projections by age group within 20% accuracy for age 25 and older
(Table 3). Refinement of the geographic characterization could
result in corresponding increases in accuracy. The model
projections are smoother over time than the actual counts, as
should be expected because random fluctuations are not built into
the model (Figures 2,3). In Figure 4 the time series of actual cases
is moving toward the lower 10% bound. This may be a temporary
fluctuation (as in the mid-1980s) but could also indicate changes in
transitions in the real-world disease process. Errors of a projection
model may arise from shortcomings of model structure, from
inaccuracies in initial conditions, or from errors in the values of
key model parameters. The structure of the Markov chain model is
mechanistic, driven by highly detailed population dynamics. Each
part of the mechanistic process may prompt suggestions for model
refinements and discussions of variability; notably, occurrence of
infection (66,67), time from infection to disease (68,69), and
involvement of HIV (70) and immigrant status (71). The greatest
potential for model inaccuracy due to initial conditions arises from
estimates of the 1980 pool of prevalent infections. Errors thus
introduced become less consequential with passing time, as
persons infected before 1980 die. So, model projections in the
latter part of the projection period are least affected by errors in the
numbers of those infected before 1980. Model parameters not only
have the obvious role of governing transitions for various
population subgroups, but also, as reflected by their time
dependency, are used to depict exogenous phenomena such as the
HIV epidemic, implementation of TB control efforts, and changes
in immigration patterns. Geographically focused changes of TB
control efforts implemented after 1993 related to directly observed
therapy could result in overestimation in projections in those areas.

Although the model does not allow mixing of groups from one
location to another, this limitation is in large measure adjusted in
the calculations of transition probabilities since these are based on
actual data that implicitly reflect the movements of healthy,
infected, and ill groups. The other type of mixing not accounted for
in the model occurs in a location among different race-ethnicity
subgroups. The most serious projection errors would then occur
when a subgroup of uninfected persons lives in the same location
as a subgroup of persons with active TB, for then any cases arising
in the former subgroup would not be predicted by our model. This
type of situation is likely to arise only in locations with very few
active TB cases. Otherwise, the projection errors arise only from
the extent to which the cross-spread of infection fails to cancel out
(i.e., the net extent to which a subgroup infects others, over and
above the number of the latter being infected by the former).

Although HIV status is addressed in this model, adding structure
that more precisely reflects changes in transition rates of HIV-
infected persons might also improve model projections. The model
predicts that an increasing portion of cases will occur in Hispanics,
and this prediction is supported by the most recent data, which
show that the rate of decline in the number of cases in Hispanics is
slower than for white non-Hispanics and for black non-Hispanics.

The foreign born account for a large fraction of the new cases seen
in the United States (71). The model assumptions regarding the
future contribution of the foreign born to the U.S. population are
from the U.S. Bureau of the Census projections; therefore, this
component of the population may be thought of as forming part of
the backdrop against which the model projections are made.
Departure from the patterns of immigration projected by the
Census Bureau could cause two kinds of errors in model
projections. A surge of immigration from countries with high rates
of TB would soon be followed by increases in TB cases and case
rates in the United States that would not be projected by the model.
A different effect would be seen as a consequence of unexpected
large increases in immigrants. Regardless of the TB status of the
countries of origin, an increase in the number of immigrants would
be reflected first in an artificial lowering of case rates because of
the increase in denominators, followed by a longer-term slow
increase in numbers of cases. The model could not recognize these
changes within its current structure. These potential errors may be
avoided as additional structure better reflecting immigrant status
becomes incorporated into the model so that unexpected influxes
may be modeled as exogenous inputs.

Acknowledgment

The authors thank Philip J. Smith for helpful comments.

This work was supported in part by a contract with the Centers for
Disease Control and Prevention under its cooperative agreement
with the Association of Teachers of Preventive Medicine.

Dr. Debanne is an associate professor of epidemiology and
biostatistics at the School of Medicine, Case Western Reserve
University. She specializes in the construction of mathematical
models of infectious diseases and has worked extensively in the
area of tuberculosis.

Address for correspondence: Sara M. Debanne, Department of
Epidemiology and Biostatistics, School of Medicine, Case Western
Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106-
4945, USA; fax: 216-368-3970; e-mail: smd@hal.cwru.edu.

References

1. Bloch AB, Rieder HL, Kelly GD, Cauthen GM, Hayden CH,
Snider DE Jr. The epidemiology of tuberculosis in the United
States. Clin Chest Med 1989;10:297-313.
2. Ellner JJ, Hinman AR, Dooley SW, Fischl MA, Sepkowitz KA,
Goldberger MJ, et al. Tuberculosis symposium: emerging
problems and promise. J Infect Dis 1993;168:537-51.
3. Cantwell MF, Snider DE Jr, Cauthen GM, Onorato IM.
Epidemiology of tuberculosis in the United States, 1985 through
1992. JAMA 1994;272:535-9.
4. Comstock GW. Variability of tuberculosis trends in a time of
resurgence. Clin Infect Dis 1994;19:1015-22.
5. Centers for Disease Control and Prevention. Tuberculosis in the
United States, 1997. Atlanta: July 1998:10-11.
6. Census of Population and Housing, 1990: Public use microdata
sample U.S. technical documentation and machine-readable data
file/prepared by the Bureau of the Census. Washington: The
Bureau; 1992.
7. 1990 Census of Population and Housing, 5% and 1% public use
microdata samples-U.S. geographic equivalency files machine-
readable data files / prepared by the Bureau of the Census.
Washington: The Bureau; 1994.
8. Census of Population and Housing, 1980; summary tape file 3A
machine-readable data file / prepared by the Bureau of the Census.
Washington: The Bureau; 1982.
9. Census of Population and Housing, 1980; summary tape file 3B
machine-readable data file / prepared by the Bureau of the Census.
Washington: The Bureau; 1982.
10. Census of Population and Housing, 1980; summary tape file
3C machine-readable data file / prepared by the Bureau of the
Census. Washington: The Bureau; 1982.
11. Census of Population and Housing, 1980; summary tape file 3
technical documentation / prepared by the Data User Services
Division, Bureau of the Census. Washington: The Bureau; 1982.
12. Census of Population and Housing, 1990: summary tape file 3
on CD-ROM machine-readable data files / prepared by the Bureau
of the Census. Washington: The Bureau; 1992.
13. Intercensal estimates of the population of counties by age, sex,
and race: 1980-1990 machine-readable data files / prepared by the
Bureau of the Census. Washington: The Bureau; 1994.
14. Population projections for states, by age, sex, race, and
hispanic origin: 1993 to 2020 machine-readable data file / prepared
by the Bureau of the Census Population Division. Washington: The
Bureau; 1994.
15. U.S. Bureau of the Census. Current population reports, P25-
1111, population projections for states by age, sex, race, and
hispanic origin: 1993 to 2020, by Paul R. Campbell. Washington:
Government Printing Office; 1994.
16. Tuberculosis in the United States, for years 1980 through 1997,
Atlanta (GA): Centers for Disease Control and Prevention,
Division of Tuberculosis Control, 1982-1997.
17. AIDS Public Information Data Set, Statistics and Data
Management Branch, Division of HIV/AIDS, CDC, 1993, Atlanta,
GA.
18. Reports of Verified Cases of Tuberculosis (RVCT), CDC,
Surveillance System for TB (SURVS-TB), Division of
Tuberculosis Control, 1980-1993, Atlanta, GA.
19. Daniel TM, Debanne SM. Estimation of the annual risk of
tuberculous infection for white men in the United States. J Infect
Dis 1997;175:1535-7.
20. Markov AA. Investigation of a noteworthy case of dependent
trials. Izv Akad Nauk Ser Biol 1907;1.
21. Markov AA. The calculus of probabilities. 4th ed. Moscow:
GIZ; 1924.
22. Takcs LF. Stochastic processes. London: Methuen, 1960.
23. Liaw K-L, Frey WH. Destination choices of the 1985- 90
young adult immigrants to the United States: importance of race,
educational attainment and labor market forces. International
Journal of Population Geography 1998;4:49-61.
24. Muench H. Derivation of rates from summation data by the
catalytique curve. Journal of the American Statistical Association
1934;29:25-38.
25. Styblo K. The relationship between the risk of tuberculous
infection and the risk of developing infectious tuberculosis.
Bulletin of the International Union against Tuberculosis
1985;60:117-9.
26. Styblo K. Epidemiology of tuberculosis. Selected papers. Vol
24. The Hague (The Netherlands): Royal Netherlands Tuberculosis
Association (KNCV); 1991. p. 49-50.
27. Tuberculosis Program Management in the United States, 1984.
Atlanta (GA): Centers for Disease Control and Prevention, Center
for Prevention Services, Division of Tuberculosis Control; 1986.
28. Edwards LB, Acquaviva FA, Livesay VT, Cross FW, Palmer
CE. An atlas of sensitivity to tuberculin, PPD-B, and histoplasmin
in the United States. Am Rev Respir Dis 1969;99 Suppl:1-132.
29. Edwards LB, Smith DT. Community-wide tuberculin testing in
Pamlico County, North Carolina. Am Rev Respir Dis 1965;92:43-
54.
30. Rosenthal SR, Afremow ML, Nikurs L, Loewinsohn E,
Leppmann M, Katele E, et al. BCG vaccination and tuberculosis in
students of nursing. Am J Nurs 1963;63:88-93.
31. Hanzel GD, Rogers KD. Multiple-puncture and Mantoux
tuberculin tests in high school students. A comparative study.
JAMA 1964;190:1038-42.
32. Rhoades ER, Alexander CP. Reactions to the tuberculin time
test in air force recruits. An analysis of 193,856 tests. Am Rev
Respir Dis 1968;98:837-41.
33. Comstock GW. Frost revisited: the modern epidemiology of
tuberculosis. Am J Epidemiol 1975;101:363-82.
34. Reichman LB, O'Day R. Tuberculous infection in a large urban
population. Am Rev Respir Dis 1978;117:705-12.
35. Engel A, Roberts J. Tuberculin skin test reaction among adults
25-74 years, United States, 1971-1972. DHEW Publication No.
(HRA) 77-1649.
36. Comstock GW, Cauthen GM. In: Reichman LB, Hershfield ES,
editors. Tuberculosis: a comprehensive international approach.
New York: Marcel Dekker, Inc.; 1993. p. 23-48.
37. Zeidberg LD, Gass RS, Dillon A, Hutcheson RH. The
Williamson County tuberculosis study. A twenty-four-year
epidemiologic study. Am Rev Respir Dis 1963;87 Suppl:1-88.
38. Ferebee SH. Controlled chemoprophylaxis trials in
tuberculosis. A general review. Advances in Tuberculosis Research
1970;17:28-106.
39. Devadatta S, Dawson JJY, Fox W, Janardhanam B,
Radhakrishna S, Ramakrishna CVL, et al. Attack rate of
tuberculosis in a 5-year period among close family contacts of
tuberculous patients under domiciliary treatment with isoniazid
plus PAS or isoniazid alone. Bull World Health Organ
1970;42:337-51.
40. Morris SI. Tuberculosis as an occupational hazard during
medical training: report of case-finding and follow-up study, with
effective control program for tuberculosis in women medical
students. American Review of Tuberculosis 1946;54:140-58.
41. Myers JA. Tuberculosis. A half century of study and conquest.
St. Louis (MO): Warren H. Green, Inc.; 1970. p. 166-99.
42. Ferebee SH, Mount FW. Tuberculosis morbidity in a controlled
trial of the prophylactic use of isoniazid among household contacts.
Am Rev Respir Dis 1962;85:490-521.
43. Daley CL, Small PM, Schecter GF, Schoolnik GK, McAdam
RA, Jacobs WR Jr, et al. An outbreak of tuberculosis with
accelerated progression among persons infected with the human
immunodeficiency virus. N Engl J Med 1992;326:231-5.
44. Di Perri G, Cruciani M, Danzi MD, Luzatti R, De Checchi G,
Malena M, et al. Nosocomial epidemic of active 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. Outbreaks of drug-resistant tuberculosis in an AIDS
center. Lancet 1990;336:440-1.
46. Bielefeld RA. Fuzzy representation of uncertainty in disease
progression [Ph.D. dissertation]. Department of Epidemiology and
Biostatistics. Cleveland (OH): Case Western Reserve University;
1992.
47. Bielefeld RA. Estimation of HIV infection and AIDS onset
times using fuzzy arithmetic techniques. In: Proceedings of the
Seventh World Congress on Medical Informatics. Amsterdam
(North-Holland); 1992. p. 921-7.
48. Bielefeld RA, Debanne SM, Rowland DY. Storing sparse and
repeated data in multivariate Markovian models of tuberculosis
spread. Computers and Biomedical Research 1996;29:85-92.
49. Adel'son-Vel'skii GM, Landis EM. An algorithm for the
organization of information. Dokl Acad Nauk SSSR Math
1962;146:263-6.
50. Standish TA. Data structure techniques. Reading (MA):
Addison-Wesley; 1980. p. 108-18.
51. Waaler H, Geser A, Anderson S. The use of mathematical
models in the study of the epidemiology of tuberculosis. Am J
Public Health 1962;52:1002-13.
52. Waaler HT, Piot MA. The use of an epidemiologic model for
estimating the effectiveness of tuberculosis control measures:
sensitivity of the effectiveness of tuberculosis control measures to
the coverage of the population. Bull World Health Organ
1969;41:75-93.
53. ReVelle CS, Lynn WR, Feldman F. Mathematical models for
the economic allocation of tuberculosis control activities in
developing countries. Am Rev Respir Dis 1967;96:893-909.
54. Azuma Y. A simple simulation model of tuberculosis
epidemiology for use without large-scale computers. Bull World
Health Organ 1975;52:313-22.
55. Styblo K. The global aspects of tuberculosis and HIV infection.
Bulletin of the International Union against Tuberculosis and Lung
Disease 1990;65:28-32.
56. Schulzer M, Fitzgerald JM, Enarson DA, Grzybowski S. An
estimate of the future size of the tuberculosis problem in sub-
Saharan Africa resulting from HIV infection. Tuber Lung Dis
1992;73:52-8.
57. Heymann SJ. Modeling the efficacy of prophylactic and
curative therapies for preventing the spread of tuberculosis in
Africa. Trans R Soc Trop Med Hyg 1993;87:406-11.
58. Murray CJL, Styblo K, Rouillon A. Tuberculosis in developing
countries: burden, intervention and cost. Bull Internat Union
against Tuberc 1990;65:2-20.
59. Blower SM, McLean AR, Porco TC, Small PM, Hopewell PC,
Sanchez MA, Moss AR. The intrinsic transmission dynamics of
tuberculosis epidemics. Nature Medicine 1995;1:815-21.
60. Blower SM, Small PM, Hopewell PC. Control strategies for
tuberculosis epidemics: New models for old problems. Science
1996;273:497-500.
61. Murray CJL, Salomon JA. Expanding the WHO tuberculosis
control strategy: rethinking the role of active case-finding.
International Journal of Tuberculosis and Lung Disease 1998;2:S9-
15.
62. Murray CJL, Salomon JA. Modeling the impact of global
tuberculosis control strategies. Proc Natl Acad Sci U S A
1998;95:13881-6.
63. Brewer TF, Heymann SJ, Colditz GA, Wilson ME, Auerbach
K, Kane D, Fineberg HV. Evaluation of tuberculosis control
policies using computer simulation.
JAMA 1996;276:1898-903. 64. Frost WH. The age selection of
mortality from tuberculosis in successive decades. American
Journal of Hygiene 1939;30:91-6.
65. Ferebee SH. An epidemiological model of tuberculosis in the
United States. Bulletin of the National Tuberculosis Association
1967 (January):4-7.
66. Rieder HL. Methodological issues in the estimation of the
tuberculosis problem from tuberculin surveys. Tuber Lung Dis
1995;76:114-21.
67. Gammaitoni L, Nucci MC. Using a mathematical model to
evaluate the efficacy of TB control measures. Emerg Infect Dis
1997;3:335-42.
68. MacIntyre CR, Plant AJ. Preventability of incident cases of
tuberculosis in recently exposed contacts. International Journal of
Tuberculosis and Lung Disease 1998;2:56-61.
69. Salpeter EE, Salpeter SR. Mathematical model for the
epidemiology of tuberculosis, with estimates of the reproductive
number and infection-delay function. Am J Epidemiol
1998;142:398-406.
70. West RW, Thompson JR. Modeling the impact of HIV on the
spread of tuberculosis in the United States. Math Biosci
1997;143:35-60.
71. Liu Z, Shilkret KL, Tranotti J, Freund CG, Finelli L. Distinct
trends in tuberculosis morbidity among foreign-born and U.S.-born
persons in New Jersey, 1986 through 1995. Am J Public Health
1998;88:1064-7.

Emerging Infectious Diseases
National Center for Infectious Diseases
Centers for Disease Control and Prevention
Atlanta, GA

URL: http://www.cdc.gov/ncidod/EID/vol6no2/ascii/debanne.txt

Please note that figures, equations, and tables are not available 
in ASCII format.  Many symbols do not convert to ASCII.