A simple simulation model of tuberculosis epidemiology for use without large-scale computers

TUBERCULOSIS A simple simulation model of tuberculosis epidemiology for use without large-scale computers Y. AZUMA' A large-scale computer service is not always available in many countries witk tuberculosis problems needing epidemiological analysis. To facilitate work in such countries, a simple epidemiological model was made to calculate annual trends in the prevalence and incidence of tuberculosis and its infection, in tuberculosis mortality, and in BCG coverage, using average parameter values not specific for age groups or birth year cohorts. To test its- approximation capabilities and limits, the model was applied to epidemiological data from Japan, where sufficient information was available from repeated nation-wide sample surveys- and national statistics. The approximation wasfound to be satisfactory within certain limits. The model is best used with a desk-top computer, but the calculations can be performed with a small calculator or even by hand. Up-to-date epidemiological information is very desirable when planning a.tuberculosis control pro- gramme to meet existing demands. However, plan- ning often has to be based on old survey data, since it is not easy to repeat surveys even in countries that can afford it technically and financially. If the epi- demiological time trend can be estimated using old data, the present status can also be estimated within certain limits. There must be cases when it is worth using such estimates and risking some errors rather than relying or' "id data, since the epidemiological situation i'.nlikely tu be static for a long period in many crAtries. There are various models for simulating the time trend of tuberculosis epidemiology, such as that of Waaler (1-4). For applying these models a large-scale computer service is needed, and such a service is often inaccessible in countries with a tuberculosis problem. A simple model was made to enable calculations to be made without a large-scale computer. It was tried out on survey data from Japan, where informa- tion was sufficient both for the estimation of neces- sary parameter values and for a comparison of simulated with observed trends. The results were satisfactory within certain limits. 1 Chief, Department of Education, Research Institute of Tuberculosis, Japan Anti-Tuberculosis Association, Kiyose, Tokyo, Japan. For the sake of simplicity, the model was made toG apply average parameter values of the population instead of being age, sex, and cohort specific. Also, the waning effect after infection and vaccination was ignored and no importance was attached to exogen- ous reinfection in the development of the disease. DESCRIPTION OF THE MODEL The present model is intended for a closed com- munity, the population changing through births and deaths alone. The total population is divided into three groups, namely non-infected, BCG-vaccinated and TB-infect- ed. BCG-vaccinated includes a sub-group, BCG- protected. TB-infected includes a sub-group, TB, with a sub-sub-group, TB-treated. The population is continuously supplied with inflow from birth while losing a portion by outflow from its various com- ponents into death. The flow of the population between these categories is shown in Fig. 1. The assumptions were made, for the sake of simplification, that protection remains constant after effective BCG vaccination given before tuberculosis infection, that healing occurs only in the TB-treated group (this assumes that the cure rate is roughly proportional to the size of the TB-treated group), and that the general death rate in each group is equal to that in the total population. 3355 313 - BULL. WORLD HEALTH ORGAN., Vol. 52,1975 Y. AZUMA POPULATION NON INFECTED BCG-VACCINATED q -CG-PROTECTED - TB-INFECTED TB-TX1 T-TREATED Table 1. Equations for the simulation model of tuber- culosis epidemiology DET ,---+___ TB_EAT Fig. 1. Population flow between different categories of the epidemiology model. The model consists of a set of 15 equations based on the flow chart of Fig. 1, as shown in Table 1. Equations 1-4 give, respectively, the one-year trend from year 0 to year 1 of the population size (POP), the number of TB-infected (INF), the number of BCG-vaccinated (BCG), and the number of tuber- culosis cases (TB) by the balance between the annual number of births (B), annual number of deaths (D), primary infections (inf), primary vaccinations given in the non-infected group (bcg), tuberculosis deaths (TBD), tuberculosis cures (HEAL), and annual inci- dence (inc). Equations 5-11 determine the value of each vari- able causing the above-mentioned trends. Equations 12-15 give, respectively, the annual trends of the regularity of tuberculosis treatment (REG), the treatment coverage in existing tuberculosis cases (COV), the birth rate, and the death rate. It is assumed that the number of primary infec- tions is proportional to that of non-treated tuber- culosis cases and to the relative frequency of non- infected, non-BCG-protected persons in the popula- tion as expressed by equation 5; that the tuberculosis death rate is constant (T) in non-treated cases and also constant (T') in the treated (equation 7); and that the tuberculosis incidence rate in the tuberculo- sis infected but not diseased population is con- stant (i) (equation 9). The total number of deaths in tuberculosis cases is approximated by the sum of tuberculosis deaths and the general death rate in equation 4. This is based on a study by Kihara (5) who revealed, by analysing data from nation-wide follow-up sample surveys in Japan, that the increase in the death rate among patients with tuberculosis compared with that of the general population was about equal to deaths from tuberculosis. In Table 1, subscripts 0 and 1 indicate the values of year 0 and year 1, respectively. When the values of the variables at year 0 and the value of each constant POP1 = POPO-(1-Do+Bo) INF1 = INFO.(1-Do)+infO BCG1 = BCG0(1 -Do) +bcgo TB, = TB0(1-D0)-TBDo-HEALO+inco info = k.(TB0-TRO)-(POPO-INFo- BCG PO)/POPO BCGPO = BCGo-p TBDo = (TBO-TRO).T+TRO^T' HEALO = TROREGO^C/2 inco = i-(INFO-TBO) TRo = TBOCOVO bcgo = (POPO-BCGO).(1 -DO+Bo).(1 -A)+BCGOBo REG1 = 1 -(1 -REGo).f COV, COVO+(1-COVO).g B, = Bo^b D1 = Do^d POP: population size INF: number of TB-infected BCG: number of BCG- vaccinated TB: number of TB cases BCGP: number of BCG-protected inf: number of primary infec- tions bcg: number of primary vac- cinations TBD: number of TB deaths HEAL: number of cures inc: number of new TB cases B: annual birth rate D: annual death rate k: ratio of risk of infection to prevalence of non-treated TB cases T: TB death rate in non-treated cases T': TB death rate in treated cases p: protection effect of BCG vaccination C: cure rate in regularly treated cases i:TB incidence in infected non-TB population A: annual decrease in pre- valence of non-vaccinated g: annual decrease in pre- valence of non-treated in TB cases TR: number of treated TB cases f: annual reduction coefficient of treatment irregularity REG: relative frequency of re- b: annual rate of decrease in gular cases in the treated birth rate population COV: treatment coverage d: annual rate of decrease in death rate are given, those at year 1 can be obtained by simple arithmetic. Thus the trend with those variables can be estimated for any given year, starting with a set of initial conditions, under an assumption that the conditions determining the value of those constants remain unchanged. E7j3 (1) (2) (3) (4) (5) (6) (7) (8) (13) (14) 314 SIMULATION MODEL OF TUBERCULOSIS EPIDEMIOLOGY ESTIMATION OF INITL CONDITIONS AND PARAMETER VALUES "Tuberculosis " was defined as bacteriologically confirmable pulmonary tuberculosis in the study, since tuberculosis prevalence as determined by X-ray is limited in its use as an epidemiological index; this is due to the inconsistency of the criterion for " active tuberculosis " and to the variability of prog- nosis and infectivity of " active tuberculosis ", caused by the changing distribution pattern of patients by type and extent of X-ray shadows. The initial values in 1953 were estimated for the variables by the results of the national tuberculosis prevalence surveys and follow-up surveys made dur- ing 1953-1973 (6), since some of the values had to be estimated by applying the time trends revealed by the surveys. The estimates are given in Table 2. The trend in the number of annual prmiary vac- cinations is expressed by equation 11 so as to estimate the trend in the prevalence of BCG-vacci- nated observed by the surveys under the influence of annual birth and death rates. The protection effect of BCG vaccination was set at 80% following the results of the BMRC trial (7). Table 2. Initial conditions in 1953 Population Annual birth rate Annual death rate Prevalence of non-infected Prevalence of TB-infected Prevalencp of BCG-vaccinated Prevalence of tuberculosis Treatment coverage in TB cases Risk of infection Incidence of tuberculosis TB mortality in non-treated cases TB mortality in treated cases BCG primary vaccinations 87.033 million 1.8370 % 0.8273 % 21.1 % 44.8% 31.4% 0.7447 % 20.76 % 2.9% 0.05 % 10.15 % 6.30% 1.9175 million Annual reduction rates: treatment irregularity 11 % non-treated/TB cases 2.8 % birth rate 0.0930 % death rate 1.2787 % The trend in the population of the country was estimated by applying the birth and death rates for 1953 (1.837% and 0.827 %, respectively) to the initial population of 87.033 million given by the 1953 national statistics, and decreasing by annual rates of 0.093% and 1.2787 %, respectively. Further, the initial value of treatment regularity was set at 60%, with irregularity decreasing annually by a rate of 11%, with 99% of regularly-treated cases being cured every year. The number of cases cured annually is calculated with equation 8, assum- ing that about half of the cases under treatment begins treatment each year, that one year of treat- ment is required to cure the majority of cases, and that in the majority of cases the effect of more than one year of treatment is negligible. In the calculation, the overall effect of the treatment was simulated so that the annual cure rate in treated cases increased from 29.7% in 1953 to 47.57% in 1973 with a gradu- ally falling rate of increase. COMPARISON OF THE SIMULATED AND OBSERVED TRENDS Table 3 gives the simulated 20-year trends for the period 1953-1973 for population size, tuberculosis prevalence, tuberculosis mortality rate, prevalence of tuberculosis infection, BCG vaccination coverage in the population, and treatment coverage in tuberculo- sis cases, and compares them with data obtained from the national statistics and the national preva- lence surveys. The simulated population size tends to be a little larger in later years, but the approximation is considered good enough for calculations with this simple model. The simulated tuberculosis mortality rate is com- pared with that for all types of tuberculosis given by the national statistics, ignoring a slight difference caused by extrapulmonary tuberculosis deaths in- cluded in the latter. As is also shown in Fig. 2, the yearly trend is estimated quite well by the simula- tion, the difference remaining between 1.1 and 6.1 per 100 000 since 1957. The simulated prevalence of tuberculosis infection is compared with the prevalence of non-vaccinated tuberculin-positive reactors estimated by the nat- ional surveys. The simulation values are slightly higher than the survey data but the difference is below 2.5%. This is due to 20% of the vaccinated population being treated as non-protected in the simulation. The simulation of BCG vaccination coverage is slightly higher than the survey data, the 315 Y. AZUMA Table 3. Comparison of the simulation results with observed estimates Population TB prevalence TB mortality Prevalence of BCG coverage (%) Treatment coverage Year (millions) (per 1 000) (per 100000) TB infection (%) (%) observed simulated observed simulated observed simulated observed simulated observed simulated observed simulated 1953 87.033 87.033 7.447 7.447 66.5 69.6 44.8 44.80 34.1 34.20 20.76 1954 88.293 87.911 6.662 62.4 61.7 44.78 35.75 22.98 1955 a 89.275 88.807 5.942 52.3 54.5 44.63 37.28 25.14 1956 90.259 89.719 5.289 48.6 48.1 44.37 38.76 27.25 1957 91.088 90.648 4.702 46.9 42.4 44.03 40.21 29.29 1958 92.010 91.595 5.616 4.181 39.4 37.4 42.0 43.62 38.2 41.63 31.3 31.27 1959 92.971 92.559 3.723 35.5 33.0 43.15 43.01 33.20 1960 a 93.418 93.541 3.324 34.2 29.2 42.64 44.36 35.08 1961 94.285 94.541 2.979 29.6 26.0 42.10 45.68 36.90 1962 95.178 95.560 2.684 29.3 23.2 41.54 46.96 38.67 1963 96.156 96.596 1.914 2.432 24.2 20.9 38.5 40.95 47.1 48.22 40.4 40.39 1964 97.186 97.652 2.217 23.6 18.9 40.36 -49.45 42.07 1965 a 98.274 98.726 2.035 22.8 17.2 39.76 50.64 43.69 1966 99.056 99.820 1.879 20.3 15.8 39.16 51.81 45.27 1967 99.637 100.933 1.747 17.8 14.5 38.55 52.96 46.81 1968 100.794 102.066 0.924 1.633 16.8 13.5 35.6 37.94 52.9 54.07 48.3 48.30 1969 102.022 103.218 1.534 16.1 12.6 37.34 55.16 49.75 1970 a 102.736 104.392 1.449 15.5 11.8 36.74 56.22 51.16 1971 104.345 105.585 1.373 13.0 11.1 36.15 57.26 52.54 1972 105.742 106.800 1.307 11.9 10.5 35.56 58.27 53.87 1973 108.079 108.035 1.203 1.247 11.1 10.0 34.98 58.5 59.26 44.4 55.16 a Census year. difference being about 1% except in 1958 when it was 3.4%. Treatment coverage is very well approximated by the simulation up to 1968, but it is higher than the survey value by almost 11 % in 1973, when coverage by the survey drops suddenly. This may be due to random fluctuation owing to too small a sample size: coverage was estimated in the survey in only 59 patients with infectious tuberculosis. Simulated tuberculosis prevalence shows some difference from the survey data as shown in Fig. 3, the difference being between 0.004% and 0.144%. The "observed values" of tuberculosis prevalence are actually estimates calculated from the weighted aver- age of the bacteriology-positive rate in various cate- gories of X-ray finding. Owing to the low prevalence, the standard error had to be relatively large, ranging between 6% and 20% of the estimate in the surveys. Furthermore, the estimate of tuberculosis prevalence in 1958 was based on 334 positive cases detected, among which 22 out of 308 examined strains were non-tuberculosis acid-fast bacilli. Concerning the 1963 and 1968 prevalence rates, laryngeal swabs were mainly used instead of sputum specimens for bacteriological examinations in these two survey years; this is suspected to have resulted in a lower positive rate. Taking into consideration the above- mentioned discrepancies the simulation may be con- sidered satisfactory. Tuberculosis incidence was estimated at 0.05%, 316 80 70 c 8 _e- : 5. 60 50 40 30 20 10 Fig. 2. Trend of tuberculosis mortality by simulation with various schemes. 8 ,-I r 6 5 4 3 1 -- 1953 1958 1963 1968 1973 1978 1983 Fig. 3. Trend of tuberculosis prevalence by simulation with various schemes. 8~ c -n t, A. 318 60 50 000 8 cu Cz H 40 30 20 10 0 1953 1958 1963 1968 Y. AZUMA 1973 1978 1983 Fig. 4. Trend of tuberculosis incidence by simulation with various schemes. 0.03 %, and 0.02% per year, respectively, for 1953/54, 1958/59 and 1963/64 by the one-year fol- low-up surveys. As these were estimated from a few cases occurring in a sample population of about 20 000, it may be concluded only that there was a declining trend in tuberculosis incidence during the period from an initial value of about 0.05% in a range of the same order. In the simulation, incidence declines gradually from 0.05% in 1953 to 0.04% in 1973, as shown by the lowest curve in Fig. 4. Thus, the model was found suitable for the simu- lation of epidemiological trends in tuberculosis, at least in respect of data from Japan. After the model was found usable, it was further evaluated on the effect of the national tuberculosis control programme by inserting a few more items into the calculation programme to reveal the cumu- lative sum of annual workloads of BCG vaccination and treatment. ESTIMATION OF EPIDEMIOLOGICAL EFFECT OF TUBERCULOSIS PROGRAMME The epidemiological trend resulting from no tuberculosis programme was calculated by entering zero values for BCG vaccination and treatment in 100. 8.0 6.0 4.0 2.0 1.0 0.8 0.6 '_ 0.40 CL 0.2 0.1 0.06 II i U.Uw 0.01 L 195 -__ NO TB PROGRAMME BCG PROGRAMMIE ....... ........................ NO TB PROGRAMME TREATMENT PROGRAMME TREATMENT+BCG PROGR. 1993 2003 ffi I[[ = II T 1963 1973 1983 Fig. 5. Trend of annual risk of tuberculosis infection by simulation with various schemes. A ~~~~I II ' __o TB PROGRA NoD TB PR0 +--ZBP_4 qn - I 4 1 It I I I4- I I4. I- 4. l+ F 4 I l- I- 1988 1993 1998 2003 U.wU _ Uux L .'\ 1-.. i3 SIMULATION MODEL OF TUBERCULOSIS EPIDEMIOLOGY the simulation model, with all other initial condi- tions and parameter values unchanged, as shown by the curve No TBprogramme in Fig. 2-5; the trend of the national tuberculosis programme is given by the curve Treatment + BCG programme. Reduction in the tuberculosis problem is indicated by the area between those two curves in the figures. In this case the " time-perspective" as discussed by Waaler & Piot (4) was not taken into account for the sake of simplicity. The trends resulting from no tuberculosis programme include the remaining effects of the tuberculosis programme carried out before 1953. The cumulative sums of tuberculosis case-years, tuberculosis deaths, and new tuberculosis cases are given in Table 4 for periods of 25 and 50 years from 1953. The problem reduction rate is expressed as the percentage decrease compared with no tuberculosis programme. The epidemiological effect of the na- tional tuberculosis programme in Japan was thus estimated with problem reduction rates of 46.5%, 13.0%, and 55.6%, respectively, for case-years, new cases, and tuberculosis deaths for the 25-year period from 1953. Also given in Fig. 2-5 are the trends for treatment alone from 1953, for BCG alone from 1953, and for stopping the national programme ofBCG and treat- ment in 1978. As shown by these figures, the treat- ment programme has a remarkable effect on tuber- culosis mortality and prevalence, and also on the risk of infection, owing to its rapid effect. As regards incidence, the effect of BCG vaccination is largely due to the long-term replacement of the infected population with the vaccinated population. The effect of the combined programme is smaller than the sum of the two owing to some overlapping of effects. An estimated 2.397 million treatment-years and 52.114 million primary BCG vaccinations are needed Table 4. Tuberculosis problem reduction rate by various schemes 1953-1978 1953-2003 cumulative problem cumulative problem sum reduction sum reduction (million) (%) (million) (%) Tuberculosis cases no TB programme 12.858 - 27.290 - national TB programme 6.882 46.47 9.665 64.58 (treatment + BCG) treatment alone 6.959 45.87 9.941 63.57 BCG alone 12.499 2.79 23.761 12.93 no programme after 1978 6.882 46.47 14.493 46.89 Tuberculosis deaths no TB programme 1.312 - 2.768 - national TB programme 0.582 55.59 0.759 72.56 (treatment + BCG) treatment alone 0.588 55.18 0.777 71.91 BCG alone 1.278 2.58 2.421 12.52 no programme after 1978 0.582 55.59 1.340 51.57 Tuberculosis incidence no TB programme 1.259 - 2.962 - national TB programme 1.095 12.96 2.081 29.74 (treatment + BCG) treatment alone 1.122 10.86 2.183 26.29 BCG alone 1.172 6.84 2.373 19.89 no programme after 1978 1.095 12.96 2.178 26.45 319 Y. AZUMA Table 5. The efficiency of treatment and BCG vaccination programmes in tuberculosis problem reduction Reductions: No. of vaccinations per 100 per 100 equivalent to one treatment-years vaccinations treatment-year 1953-1978 1953-2003 1953-1978 1953-2003 1953-1978 1953-2003 TB-years 242.75 408.30 0.68 3.02 352 135 TB deaths 29.79 46.85 0.06 0.29 457 157 TB incidence 5.63 18.33 0.16 0.50 34 36 to achieve the above-mentioned effects during the 25 years from 1953, excluding revaccination and treat- ment of non-bacillary patients. The efficiency of the BCG programme and that of the treatment pro- gramme in the country may be estimated by dividing the problem reduction by the total workload re- quired for each programme. As shown in Table 5, the efficiency of a single programme is calculated for 1953-1978 as 242.8 case-years, 29.8 tuberculosis deaths, and 5.6 new tuberculosis cases prevented per 100 treatment-years, and 0.7 case-years, 0.1 tubercu- losis deaths, and 0.2 new cases prevented per 100 primary BCG vaccinations. This means that, in the prevention of case-years, tuberculosis deaths, and new cases 352, 457, and 34 primary vaccinations, respectively, were equivalent to one treatment-year. It is evident that the efficiency of BCG vaccination in public health is, despite its slow effect, very remark- able, particularly when the cost per vaccination is considered, which in Japan is about 0.05%Y of the cost of one year of treatment. PROJECTION OF FUTURE TRENDS IN TUBERCULOSIS EPIDEMIOLOGY Simulation was made for a further 25-year period from 1978 as already mentioned, assuming un- changed conditions, to determine parameter values and trends of treatment coverage and BCG vaccina- tions (Fig. 2-5). In the simulation tuberculosis prevalence and mortality continue to decline but more slowly, de- creasing to 1/7 and 1/10, respectively, in the first 25 years from 1953, whereas both fell to only 1/2 in the following 25 years. Tuberculosis incidence, however, does not show such a marked slowing down, de- creasing to about 7/10 of its initial value throughout the 50 years. Accordingly, the problem reduction rate over the full 50 years is 2.3 times that in the first 25 years in the case of tuberculosis incidence, 1.4 times as regards case-years, and 1.3 times as regards tuberculosis deaths. As shown in Fig. 5, the risk of infection declines at a decreasing rate and is thus not linear on the semilogarithmic scale. The risk is reduced to 1/15 of its 1953 value in the first 25 years, but only to 1/4 of its 1978 value in the second 25 years. A further simulation was tried to see what would happen if all tuberculosis programme activities cease in 1978. Prevalence, mortality, incidence, and risk of infection would increase rapidly, leaving only the after-effect of the BCG programme. The situation would regress 10 years in 5 years and 15 years in 10 years, approaching the trend of the BCG programme alone in 25 years. The tuberculosis prevalence and mortality rates have been declining rapidly since 1953 in Japan, and have begun to show a deceleration in recent years. This might give the impression that the reduction in the problem brought about by the national tubercu- losis programme is approaching its limit and that the declining trend would continue even were the tuber- culosis programme to be ended. However, the above-mentioned simulation results suggest that the tuberculosis problem would deteriorate should the programme be abandoned. A spontaneous reduction in the tuberculosis prob- lem can occur without tuberculosis programmes, since the trend is determined by the balance between incidence, death, and healing which can be affected by various factors other than tuberculosis control measures, even by a sudden change in the trend of population growth. The results of this simulation indicate that conditions are not yet favourable enough to allow the trend to continue without programmes and that the present trend in Japan is 320 SIMULATION MODEL OF TUBERCULOSIS EPIDEMIOLOGY being maintained by the pressure of the national tuberculosis programme. DISCUSSION The model described was found suitable, insofar as it has been tested on the data from the nation- wide sample surveys of tuberculosis prevalence in Japan, for estimating the time trend of tuberculosis epidemiology and also future trends based on a number of assumptions. Each step of the calculation is simple. The epi- demiological indexes can be calculated for successive points in time with a given time interval, depending on the unit of time used. The stepwise calculation can be made with a simple calculator or even by hand. A calculation table can easily be made with several lines for calculations and a column for each year or unit of time. If a desk-top computer is available with a minimum capacity of about 50 words of memory and about 500 programme steps the simulation can be made very easily with a programme formulated by breaking down and rearranging the equations in Table I in accordance with the machine's character- istics. The only problem with the model is the collection of information for supplying the model with the initial conditions and parameter values listed in the lower part of Table 1. Of these the annual trends of BCG and treatment coverages, birth and death rates, and treatment regularity can be expressed differently according to the country's available information. Tuberculosis death rates in treated and non-treated cases can be estimated if the annual tuberculosis death rate, the tuberculosis prevalence, and the treatment coverage are available for a number of years. A follow-up study of treated cases would give useful information on this, and for the estimation of the cure rate as well. Some of the parameter values may need to be based on assumptions, but these could be adjusted by trial simulation. As already mentioned, age- or cohort-specific parameter values are not used in the present model. However, it has been observed in various countries that the overall pattern of tuberculosis is changing, together with its age- and cohort-specific patterns. This may result in changes in some of the parameter values used in the present model, such as incidence in the infected population, and particularly in the tuberculosis death rate. Therefore, when the model is applied to a fairly long-term trend it should be carefully checked by comparing, for instance, simu- lated tuberculosis mortality with the reported value, to see whether the parameter values are deviating too much and are resulting in unrealistic estimates. UME MODELE SIMPLE POUR LA SIMULATION DE L EPIDEMIOLOGIE DE LA TUBERCULOSE UTILISABLE SANS GROS ORDINATEUR Un modele simple pour la simulation de l'evolution epidemiologique de la tuberculose, utilisable sans gros ordinateur, a ete construit. Pour des raisons de simplicite, il est conqu pour s'appliquer a des valeurs parametriques moyennes de la population, et non a tel et tel groupe d'age, sexe ou cohorte. Il n'a d'autre part pas e tenu compte de I'attenuation possible avec le temps des effets de la vaccination par le BCG et de l'infection tubercu- leuse. Le modele a e teste sur des donnees demographiques et epidemiologiques provenant du Japon, ofi des enquetes nationales par sondage sur la prevalence de la tuberculose ont et faites tous les cinq ans de 1953 ia 1973, et ofu l'on dispose de renseignements suffisants sur la population et sur la mortalite tuberculeuse. Les tendances simulees ont ete comparees a celles qui avaient ete determinees d'apres les statistiques et les enquetes nationales en ce qui concerne: l'effectif de la population; la mortalite, la pr&e valence et l'incidence tuberculeuses; la prevalence de l'infection; le risque d'infection; et la couverture de la vaccination par le BCG et du traitement antituberculeux. La concordance s'est revelee satisfaisante dans certaines limites. Le modele a d'autre part ete utilise pour evaluer l'efficacite du programme national de lutte contre la tuber- culose et pour etablir une projection des tendances epid&e miologiques futures. Le modele est utilisable avec un ordinateur de bureau ayant une memoire minimale d'environ 50 mots et per- mettant quelque 500 phases de calcul. Les calculs peuvent egalement etre operes au moyen d'un petit calculateur, voire a la main. 321 322 Y. AZUMA REFERENCES 1. WAALER, H. T. ET AL. American journal of public health, 52: 1002-1013 (1962). 2. WAALR, H. T. American review ofrespiratory disease, 98: 591-600 (1968). 3. WAALER, H. T. & PIOT, M. A. Bulletin of the World Health Organization, 41: 75-93 (1969). 4. WAALER, H. T. & PIOT, M. A. Bulletin of the World Health Organization, 43: 1-16 (1970). 5. KmIARA, K. Kekkaku, 47: 3-7 (1972). 6. MINSTRY OF HEALTH AND WELFARE, JAPAN. Tuberculosis prevalence surveys I-VIII (1955-1975). 7. MRC TUBERCULOSiS VACCINES CLINICAL TRIALS ComrrEE. Bulletin of the World Health Organization, 46: 371-385 (1972).