EUROPEAN
Eur. J. Epidemiol. 0392-2990 July 1991, p. 349-357
Vol. 7, No. 4
JOURNAL
OF EPIDEMIOLOGY
THE USE OF AIDS SURVEILLANCE DATA FOR SHORT-TERM PREDICTION OF AIDS CASES IN MADRID, SPAIN G.F. MEDLEY*l, V. ZUNZUNEGUI**, R. BUENO*** and D. LOPEZ GAI*** * Department of Biology Imperial College of Science, Technology and Medicine Prince Consort Road - London SW7 2BB UK. ** Centro Universitario de Salud Publica General Oraa 39 Madrid 28006 Spain. *** Servicio de Epidemiologia Consejeria de Salud Comunidad de Madrid Spain.
Key words: AIDS - Epidemiology - Surveillance - Prediction - Forecasting The paper presents a preliminary attempt to predict the numbers of MDS cases in the Community of Madrid (CAM) up to 1992. Using AIDS case surveillance data gathered by CAM, and a statistical procedure that includes a distribution for reporting delays, the numbers of new diagnoses, reports, AIDS deaths and numbers of patients alive is predicted. Approximate confidence limits for the numbers of new diagnoses are given. We emphasise that these predictions are tentative given the nature of the reporting delays, and discuss the use of such predictions and the requirements for their improvement.
INTRODUCTION
This paper considers the incidence of AIDS cases in Madrid, and the use of surveillance data to determine trends and predict the likely future patterns of diagnoses. Of obvious interest is the natural history of the disease from infection to AIDS and death. This has been considered elsewhere in detail (12, 13) and we shall not consider it here, other than to point out that it is out that it is long and variable. The average incubation period (the time from infection to AIDS) is about 10 years. Consequently, the AIDS cases that are observed now are only a proportion of the eventual AIDS cases just from those already infected. The modes of transmission by which people in the Community of Madrid (CAM) have been infected are, in order of decreasing frequency: a) by injecting drug use involving the sharing of syringes and syringes contaminated with blood from an infected person; 1 Corresponding author.
b) by sexual contact between male homosexuals and bisexuals; c) by receiving blood or blood products contaminated with HIV; d) by sexual contact between women and men (heterosexual); e) by vertical transmission from seropositive pregnant women to their babies. Most women transmitting the infection in this way have been IDUs. The proportion of AIDS cases related to IDU is approximately 700/0 as of December 1988 (10). Data on seropositivity is collected from the reference laboratories in the hospitals of Madrid. There are about 472 new cases of seropositivity per month and this number remained fairly stable during 1988. Most of these newly infected people will go on to develop AIDS. According to the estimates of progression based on cohorts of homosexuals and transfusion associated cases, roughly 250/0 of the 20117 HIV positive people tested in the laboratories of Madrid, i.e. 5000 will develop AIDS during the next five years. This number does not include all those who 349
Medley G.F. et al.
are presently infected but who have not requested to be tested. At present, we do not have any good estimate for the number of HIV seropositives in the population of CAM. Our objective is to predict how many cases of AIDS will develop in the Community of Madrid over a 2-3 year period. The epidemic of AIDS will have a great impact on health services' delivery in the coming years and there is a need for financial planning and resource allocation. An estimate of how many cases of AIDS will develop every year during the next years is urgently needed, even when such an estimate is based on the scarce information on incidence of AIDS. Some attempts have been made on the prediction of AIDS cases in Spain. However, the models used have severe limitations. They do not take into account the influence of lag times between diagnosis and reports to the National Commission on AIDS (1) and/or they depend heavily on the distribution of the incubation period estimated from the USA transfusion associated cases (14). Several m e t h o d s have been developed for the short term predictions of AIDS cases. These methods fall broadly into two categories: those t h a t use estimates, of the incubation period (from infection to .disease)and those that extrapolate the observed trends. ~ The former method is known as "the back calculation" method (3, 5). The latter method is adopted here and involves the simple forecasting of observed trends (4, 6, 7, 8, 10, 15). The third method of prediction in the longer term (more than 5 years) requires that more detailed undei'st~tnding of the biology and epidemiology be inCtii'pdrated into a mathematical model of transmission dynamics
Eur. J. Epidemiol.
MATERIALS AND METHODS
Description of Data The register of AIDS cases at CAM contains information on all cases reported to and confirmed by the National Commission of AIDS on cases residing in CAM. Of the 819 cases reported up to September 1989, 16 had no date of diagnosis, and have been removed from the data, making a total of 803 cases. The data by year of diagnosis and year of report are shown in Table 1. This shows the increase in the rate of reporting and diagnosis over time. The comparison between the numbers diagnosed and reported in each year demonstrates the important effect that the lag between diagnosis and reporting has on the data (e.g.: up to 1987 389 diagnoses had been made, but only 264 notifications had been received). There is no effect of
TABLE 1. - AIDS Cases Notified to CAM by Year of Diagnosis and Report. Diagnoses Year
Number of Cases
82
1
83
7
84
15
85
44
86
115
87
207
(2).
88
301
The instruments to improve our knowledge of the scope of the epidemic in Spain are at hand. Advanced methods have been developed in some European and American countries and the results are applied in planning for services. Predictions based on this mathematical models are subject to some limitations. The number of predicted diagnosed cases is an understimate of the actual number of cases because it is computed under current conditions of reporting. Underreporting comes mainly from two sources: a) lack of precise diagnosis: many diseases associated with infection are not diagnosed as AIDS, even when they are serious infections which demand resources, such as pulmonary tuberculosis; b) lack of reporting: physicians may not be willing to report every diagnosed case to the National Commission of AIDS. An updated analysis in the UK estimates that between 60-800/0 of deaths related to HIV infection were reported to the Communicable Disease Surveillance Centre in 1987 (6). We have reasons to believe that these proportions will not be larger in Spain.
89
113
Total
803
Reports Year
Number of Cases
83
6
84
6
85
23
86
59
87
170
88
262
89
277
Total
803
Note that there is great variability between quarters.
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Prediction of AIDS in Madrid, Spain
calendar month on either reporting or diagnosis. Figure 1 shows the monthly incidence of diagnosis. Again, the effect of the reporting lag is demonstrated causing the downturn of the data towards the end of the observation period. Figure 2 shows the cumulative probability of reporting occurring for intervals after diagnosis. Note that over 80% of all the cases reported within one year of diagnosis, but some cases took up to five years to be reported.
Description of Method Unfortunately, the estimation of the lag distribution and the actual numbers of diagnoses made
is not simple. This is because of the bias caused by the increasing epidemic. For example, presumably most AIDS cases in the data set have been diagnosed in 1988, but the longest lag we can observe with these data is only one year. The data with reporting lags longer than 4 years must have been diagnosed prior to 1984, when few cases were found. We use a method that allows simultaneous estimation of the incidence function of new diagnoses and the distribution of reporting delays between a case being diagnosed and the report arriving at CAM (4). These t w o functions are fitted to the data using the date of diagnosis, and the length of time elapsed between diagnosis and report. A technical description is given in Cox & Medley (4). It involves maximising the following log-likelihood with respect to the
50
45
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40
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30 +
425
l:l M
+ 7
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+ +
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0 82
+
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ii
~ . . A - - ~ + ++ ~ i I I
84,
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I
86
88
90
YNAR
Figure 1. - This figure shows the monthly total of diagnoses reported to the end of September 1989. The upper line is the estimated incidence function, I(t), with parameter estimates: c%= 481.71, cq = 29.673, c~2= 0.2779 and u3 = 0.9239. The lower line is the number of diagnoses expected to have been reported, calculated using the incidence function I(t), and the lag distribution, f(x), with parameter estimates: O0 = 0.4406, ®1 = 7.9863 and O2 = 2.0597. The distance between the two lines represents the estimate of those diagnoses that have been made, but not reported.
351
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Eur. J. Epidemiol.
value o f the log-likelihood, although specific statistics are not available unless the models are nested. The incidence function chosen for prediction was the "linear-logistic" form used elsewhere (4, 6, 15):
parameters o f the incidence function o f new diagnoses, I(t), and the probability distribution function of reporting lags, f(x): .
a
l= 1
%
i= 1
I(0=
0
1 + ao~gze -a3t
where to is the date o f the last observation, ti and x~ (i = 1..n) are the observed times of diagnosis and corresponding reporting delay for the data, and F(x) is the cumulative distribution of the reporting delay distribution. The incidence of diagnoses was assumed to start when the first diagnosis was made. Different choices for I(t) and f(x) can be compared with the
0.9
gO+li1 t
where I(t) is the instantaneous incidence of new diagnoses per year, t is time since the beginning o f the epidemic (October 1982) and the (x are parameters to be estimated. This function is chosen because it mirrors the epidemiological understanding o f how the epidemic will develop (see Figure 3).
+ +~,/~.. X X X J":.. X x~ /~X xx'"
++ 0.8
+
l
m 0
0.7
-
H
0.6
-
I~
0.5
-
H ~
0.4
--
0.3
-
0.2
-
0 . i
-
H
m M
i
I
/ +
I
0
I
I
i0
20 REGISTRATION
LAG
I
I
30
I
40
(MONTH)
Figure 2. - The figure compares the cumulative lag distribution calculated directly from the data (points) with that obtained from the model (line) using the parameter estimates listed in Figure 1. The upper points (+) are from the total data set (803 observations) and the lower set of points from those cases diagnosed prior to 1988 (381 points). The distribution from the total data set should produce a shorter lag distribution than that actually operating because of the bias which prevents the most recent diagnoses (the majority of all diagnoses) from exhibiting long reporting lags. Those cases diagnosed prior to 1988 should be less biased in this respect, but should show a slower reporting lag than that operating now (see discussion). The estimated line falls between these two. For the estimated distribution the median time between diagnosis and report is 5.128 mths, and the mean is 7.839 mths. 352
Prediction of AIDS in Madrid, Spain
Vol. 7, 1991
A variety of distributions for the reporting lag were tried. The distribution used was a mixture of two gamma distributions:
,/(x)=0o01Xe-elx+(1-0o)0zxe-°z~
were f(x) is the probability density function of lags of length x and the O are parameters to be estimated. We attach no significance to the form of this distribution it is simply a good empirical description. Following Cox & Medley (4), the delays of Omth, 1 ruth and 2 ruth were estimated as the cumulative distribution associated with this distribution. Those lags greater
than 2yrs (32 cases) where omitted from the fitting procedure. The distribution and longest tag included where chosen by comparison of the log-likelihoods and by direct inspection of the fitted distribution and observed data (see Figure 2). Approximate confidence intervals for the predictions were obtained by evaluating the loglikelihood for systematic combinations of parameter values and plotting them against the predictions. The limits are given by the minimum and maximum predicted values that fall on the log-likelihood less the appropriate value from the distribution with one degree of freedom. This method is computationaUy expensive (over 1000 evaluations are required for each interval calculated), and can only be approximate (4, 6).
70
60
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50
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-
~o
-
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u H U
+
+ N
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+
~" +++4+ ++ + ~44-
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+
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"~I ~' I I 8'11,
i
I
86
i
88
''
i
i
90
i
i
92
xrE/k.R
Figure 3. - This figure shows the monthly total of diagnoses (as Figure 1), and the estimated incidence function extrapolated beyond the data. Note that the incidence function is exponential at the beginning of the epidemic, and then becomes linear beyond the end of the data. Parameter estimates as figure 1.
353
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Eur. J. Epidemiol.
RESULTS
DISCUSSION
The results of the estimation are given in Figures 1 and 2. In Figure 1, the lower line (going through the points) is the expected pattern of notifications. The upper line is the fitted incidence function. The gap between the two lines represents the reporting delay, i.e. the difference between the actual numbers of cases diagnosed and those reported. The fit appears to give an adequate description of the current data. However, this is not necessarily a guarantee that predictions will be accurate. Tables 2 to 4 give the predictions associated with the parameters estimated from the data for 1990 and 1991. We have not provided predictions beyond this point, because they become increasingly unreliable. Table 2 shows the predicted monthly, quarterly and annual numbers of diagnoses (regardless of time of report) and numbers of reports (regardless of time of diagnosis). Up to 31st December 1988, 697 cases of AIDS had been reported to the Register at CAM. The current prediction suggests that 536 will be diagnosed throughout 1989, 646 throughout 1990 and 719 throughout 1991. The number of new notifications predicted for the same years are 440, 572 and 667 respectively. Thus, we expect the incidence of new notifications always to be below the incidence of new diagnoses. Table 2 also shows the actual number of new notifications received in 1989 was 446 (compared with the prediction of 440), and to July 1990 317 have been received (282 predicted). Table 3 shows the expected numbers of AIDS patients alive at the end of each quarter. Two estimates are given under different assumptions about the distribution of survival times of patients from diagnosis of AIDS to death. The predicted numbers of cases alive as of March 1990 falls between 566 and 895, in agreement with the observed value of 641 (9). By the end of December 1991, the number of living AIDS patients is predicted to have risen to between 797 and 1402. This implies that by the end of 1991 the resources allocated to care of AIDS patients should be approximately doubled in order to maintain the present level of care. The 90% confidence limits for the numbers of new diagnoses per year are shown in Table 4. The predicted number of diagnoses during the period 19891991 is 1901 with 90% confidence interval 1395 to 2800. Note that the limits are skewed such that they extend further above the estimate than they do below. These limits are calculated from variability in the data, and not from assumptions about the incidence function (which may not be a good description of the future epidemic). The lower estimate suggests a virtually constant number of cases per year. In spite of the wide confidence interval, it seems useful to know that during this three year period the number of diagnoses is likely to be equal to the total number diagnosed to the end of 1988.
The prediction of AIDS cases is a difficult problem, and as with all attempts at extrapolating into the future, is subject to many possible flaws. However, the provision of health care for people with AIDS and HIV infection is an important issue which is already demanding a great deal of health services, and is likely to demand more. Consequently, some attempts at prediction, with all the appropriate caveats, are required. It is only when such predictions are attempted, and then compared with observations, that some estimate of the reliability of predictions will be obtained. All forecasting methods produce wide confidence intervals for predictions, but future experience may suggest that these are unduly pessimistic. The current results in Tables 2-4 are a useful starting point for planning, where we do not need exact figures but approximated estimates. These are, as far as we are aware, the best attempts at predicting cases in Madrid and Spain. As under-reporting is not included, and the confidence limits of such predictions are skewed, for planning purposes, it may be better to multiply the estimates by some factor (say, 120%). Also, over-allocation of resources to a problem, although not good planning, is significantly better than under allocation which may leave patients without adequate health care. There are a number of concerns that we have regarding the validity of the predictions presented here, and we discuss each in turn. The main limitation of the register of AIDS cases at CAM is under-reporting. Some hospitals have stopped reporting for a period of time or are reporting less cases than those diagnosed. Some causes for under-reporting are: a) excessive demand for care on the health personnel, understaffed units of care; b) organizational problems of work; c) lack of awareness of the use of surveillance data in health planning. Reporting has not been uniform across hospitals and along time. Often there is a period of under-reporting followed by a period during which old cases are reported following medical history reviewing. Factors such as changes in personnel and organizational structure have originated a skewed distribution of lag times (times between diagnosis and reports). While most of the cases are reported within six months after diagnosis, a considerable proportion of cases are reported more than two years after diagnosis. Although the fit of the estimated distribution to the observations (Fig. 2) seems reasonable, we are not convinced that it is an accurate description. It is likely that there was an increase in reporting efficiency during 1987, which coincided with the formation of the CAM register of AIDS cases and an increased awareness of the importance of prompt reporting of cases. Also, the fitting procedure assumes that the diagnoses are reported independently and continuously, but anecdotal evidence suggests that the reports from the eight hospitals do not follow these assumptions. The collection of reports into "batches" at the hospitals is one problem, as is the fact that some hospitals do not report continuously. Table 5 shows the numbers of
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Vol. 7, 1991
Prediction of AIDS in Madrid, Spain
TABLE 2. - Monthly Predictions of AIDS cases by Diagnosis and Repoa for CAM. Monthly Diags. Reps.
Year
Month
1989
1 2 3 4 5 6 7 8 9 10 11 12
39 40 42 43 43 44 45 46 47 48 49 50
31 32 33 34 35 36 37 38 39 40 41 42
1 2 3 4 5 6 7 8 9 10 11 12
50 51 52 52 53 54 54 55 55 56 57 57
43 44 45 46 47 47 48 49 50 50 51 52
1 2 3 4 5 6 7 8 9 10 11 12
58 58 58 59 59 60 60 61 61 61 62 62
52 53 54 54 55 55 56 56 57 57 58 58
1990
1991
Qua~efly Diags. Reps.
121
96 ( 83)
130
105 (188)
138
115 (73)
147
124 (102)
153
132 (150)
159
140 (167)
164
147 (183)
170
153 (165)
174
159
178
165
182
169
185
174'
Annual Diags.
Reps.
536
440 (446)
646
572 (665)
719
667
The above table shows the predictions for the expected number of AIDS cases diagnosed (Diags.) and reports (Reps.) received for each month, quarter and year from 1989 to 1991using the data gathered to September 1989 for the calculations. Not~that the number of diagnoses is irrespective of time report, and is the estimated number of new cases requiring medical attention discovered each month. The quarterly numbers or reports actually received are given in parentheses following the estimate.
355
Medley G.F. et aL
Eur. J. Epidemiol.
T A B L E 3, - Quarterly Predictions of the N u m b e r o f A I D S Patients Alive in CAM.
T A B L E 5. - A I D S Cases Notified to C A M by Hospital and Year.
Year Quarter Pessimistic Number Optimistic Number Deaths Living Deaths Living 89
1 2 3 4
78 87 96 104
397 440 483 525
51 57 63 70
592 665 741 817
90
1 2 3
112 120 127
566 605 643
76 82 89
895 971 1047
4
134
678
95
1122
1 2
141 147
711 741
101 107
1195 1266
3 4
153 158
770 797
112 118
1335 1402
91
Hospital
T A B L E 4. - Approximate 90% Confidence Limits for A I D S Diagnoses in CAM. Lower
Estimate
Upper
1989 1990 1991
425 470 500
536 646 719
700 950 1150
1988
1989
1990
Gregorio Maranon Clinico La Principessa Ramon y Cajal La Paz
41 16 11 22 21
3 1 36 46 43
104 69 20 10 37
64 9 5 1 18
Del Ray
38
97
91
20
Data to the end of March 1990. This is a partial list of the data available (9). It shows the variability in the reporting efficiency both between hospitals and between years within the same hospital. This is due not only to variability in the number of AIDS diagnoses made, but also to disruption of the reporting system. See discussion.
This table shows the estimated number of AIDS patients alive, and therefore demanding health-care resources, at the end of each quarter in 1989 to 1991. Two estimates are given: pessimistic assuming that the median survival of patients is about 9mths from diagnosis and optimistic assuming that the median is about 18mths. The survival distributions used are taken from a recent study in the UK, and reflect the better medical care available to AIDS patients (15), and are as follows: p(t) = 0.92 exp (-0.7320 (pessimistic) P(0 = 0.94 exp (-0.312t) (optimistic), where t is the time in year and p(t) is the proportion surviving at time t.
Year
1987
The approximate 90% confidence limits for the predicted annual numbers of AIDS cases diagnosed in Madrid for 1989, 1990 and 1991.
reports at C A M from the different hospitals, which varies over time. For example, the reduction in reports received f r o m Gregorio Maranon in 1988 was due to a change in staff rather in the n u m b e r of A I D S diagnoses m a d e at the hospital. There are a variety o f graphical and statistical methods for estimating the lag distributions that m a y be m o r e reliable (e.g.: 11, 16). However, for the present purpose, the parametric f o r m chosen here should capture the broad shape, and be acceptable to produce predictions. It is also worth noting that this m e t h o d cannot make any estimate
about the extent or effect o f under-reporting either due to A I D S cases not diagnosed or those diagnosed but not reported. T h e predictions presented here depend completely on the choice o f incidence function, I(t). W e have no way o f knowing the true function, if, indeed, the incidence o f diagnoses can be described by any simple parametric function. The f o r m used is a result of considering the dynamics of the A I D S epidemic, which shows an exponential increase, followed by a more linear growth. Only comparison between prediction and observations will show whether a particular choice is good or not. Future consideration of short-term predictions should separate the different risk-groups within the epidemic (IDU, homosexual men, heterosexuals etc), and should also make use o f the m a n y methods available for prediction. A combination o f methods produces a better consensus of the future for planning purposes (3, 6, 15). This preliminary report suggests that several steps need to be taken to improve the surveillance data for A I D S in Madrid (& Spain), if the data are to provide worthwhile in future attempts at prediction. 1) The reporting system should be quickened. 2) T h e level o f under reporting should be decreased. W e hope that this work will encourage physicians to report cases, as they can see the benefit o f their efforts in terms of adequate allocation of resources. 3) There should be adequate statistical expertise available to analyze surveillance data. With the available data, it should be possible to use other methods of prediction and to estimate the prevalence o f H I V infection in the main high risk groups, prevalence o f A I D S at any given calendar year, n u m b e r of deaths associated to H I V infection, and distributions o f survival times. 4) A study on transmission dynamics o f H I V infection a m o n g I D U could be undertaken taking into account drug using behaviour.
356
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Prediction of AIDS in Madrid, Spain
5) Some studies should be undertaken to try to estimate the incidence of diagnoses which does not depend on the notification lag. Such a direct m e t h o d would give some indication whether the predictions o f diagnoses in Table 2 were reasonable, as they are unobservable with the present data. Even using data and methodology with all the limitations discussed above, we believe that we have enough evidence to suggest that resource allocation to care for AIDS cases should be at least doubled up to the beginning o f 1992 in order to provide the current level o f care. W e hope that this suggestion will be of guiding value to health authorities when taking decisions on funding requests from clinicians and other health professionals, who are currently under pressure from the increasing burder of A I D S and H I V related disease.
6. Department of Health and Social Services (1988): Short-term Prediction of HIV Infection and AIDS in England and Wales, Report of a Committee. London, HMSO. 7. Downs A.M., Ancelle R.M., Jager J.C. and Brunet J.B. (1987): AIDS in Europe; current trends and shortterm predictions estimated from surveillance data, January 1981-June 1986. - AIDS 1: 53-57. 8. Epidemiological Surveillance of AIDS/HIV in Community of Madrid (Vigilancia Epidemiologica del SIDA/VIH en la CAM), No. 4, Octubre 1989. Communidad de Madrid, Consejeria de Salud, Servico Regional de Salud.
Acknowledgements
9. Epidemiological Surveillance of AIDS/HIV in Community of Madrid (Vigilancia Epidemiologica del SIDA/VIH en la CAM), No. 5, Abril 1990. Communidad de Madrid, Consejeria de Salud, Servico Regional de Salud.
GFM is a University Research Fellow of the Royal Society of London. We thank the health care personnel in CAM for their efforts in collecting these data. We also thank an anonymous referee for helpful comments.
10. Healy M.J.R. and Tillett H.E. (1988): Short-term extrapolation of the AIDS epidemic (with discussion). - Journal of the Royal Statistical Society, Series A 151: 50-61.
REFERENCES
11. Heisterkamp S.H., Jager J.C., Ruitenberg E.J., van Druten J.A.M. and Downs A.M. (1989): Correcting reported AIDS incidence: a statistical approach. Stat. Med. 8: 963-976.
1. Alvarez V.A., Laguarta B.A., Artalejo F.R. et aL (1988): E1 SIDA en Espana: prediccion de nuevos casos mediante el uso de modelos matematicos. - Rev. Clin. Esp. 183: 60-63. 2. Anderson R.M. (1989): Mathematical and statistical studies on the epidemiology of HIV. - AIDS 3: 333346. 3. Brookmeyer R. and Damiano A. (1989): Statistical methods for short-term projections of AIDS incidence. - Statistics in Medicine 8: 23-34.
12. Lifson A.R., Rutherford G. W. and Jaffe H. (1988): The natural history of Human Immune Deficiency Virus infection. - J. Infect. Dis. 158: 1360-1367. 13. Moss A.R. and Bachetti P. (1989): Natural history of HIV infection. - AIDS 3: 55-61. 14. Peruga A. (1988): Provision de casos nuevos de SIDA en la Comunidad de Madrid 1982-1995. - Working paper. Consejeria de Salud. Comunidad de Madrid.
Cox D.R. and Medley G.F. (1989): A process of events with notification delay and the forecasting of AIDS. Phiilosophical Transactions of the Royal Society of London, Series B 325: 135-145.
15. Public Health Laboratory Services (1990): Acquired Immune Deficiency Syndrome in England and Wales to End 1993; Projections Using Data to End September 1989. - PHLS Communicable Disease Surveillance Centre, January.
5. De Gruttola V. and Lagakos S. IV. (1989): The value of AIDS incidence data in assessing the spread of HIV infection. - Statistics in Medicine 8: 53-65.
16. RosenbergP. (1990): A simple correction procedure of AIDS surveillance data for reporting delays. - J. AIDS 3: 49-54.
4.
357
Eur. J. Epidemiol. 0392-2990 July 1991, p. 349-357
Vol. 7, No. 4
JOURNAL
OF EPIDEMIOLOGY
THE USE OF AIDS SURVEILLANCE DATA FOR SHORT-TERM PREDICTION OF AIDS CASES IN MADRID, SPAIN G.F. MEDLEY*l, V. ZUNZUNEGUI**, R. BUENO*** and D. LOPEZ GAI*** * Department of Biology Imperial College of Science, Technology and Medicine Prince Consort Road - London SW7 2BB UK. ** Centro Universitario de Salud Publica General Oraa 39 Madrid 28006 Spain. *** Servicio de Epidemiologia Consejeria de Salud Comunidad de Madrid Spain.
Key words: AIDS - Epidemiology - Surveillance - Prediction - Forecasting The paper presents a preliminary attempt to predict the numbers of MDS cases in the Community of Madrid (CAM) up to 1992. Using AIDS case surveillance data gathered by CAM, and a statistical procedure that includes a distribution for reporting delays, the numbers of new diagnoses, reports, AIDS deaths and numbers of patients alive is predicted. Approximate confidence limits for the numbers of new diagnoses are given. We emphasise that these predictions are tentative given the nature of the reporting delays, and discuss the use of such predictions and the requirements for their improvement.
INTRODUCTION
This paper considers the incidence of AIDS cases in Madrid, and the use of surveillance data to determine trends and predict the likely future patterns of diagnoses. Of obvious interest is the natural history of the disease from infection to AIDS and death. This has been considered elsewhere in detail (12, 13) and we shall not consider it here, other than to point out that it is out that it is long and variable. The average incubation period (the time from infection to AIDS) is about 10 years. Consequently, the AIDS cases that are observed now are only a proportion of the eventual AIDS cases just from those already infected. The modes of transmission by which people in the Community of Madrid (CAM) have been infected are, in order of decreasing frequency: a) by injecting drug use involving the sharing of syringes and syringes contaminated with blood from an infected person; 1 Corresponding author.
b) by sexual contact between male homosexuals and bisexuals; c) by receiving blood or blood products contaminated with HIV; d) by sexual contact between women and men (heterosexual); e) by vertical transmission from seropositive pregnant women to their babies. Most women transmitting the infection in this way have been IDUs. The proportion of AIDS cases related to IDU is approximately 700/0 as of December 1988 (10). Data on seropositivity is collected from the reference laboratories in the hospitals of Madrid. There are about 472 new cases of seropositivity per month and this number remained fairly stable during 1988. Most of these newly infected people will go on to develop AIDS. According to the estimates of progression based on cohorts of homosexuals and transfusion associated cases, roughly 250/0 of the 20117 HIV positive people tested in the laboratories of Madrid, i.e. 5000 will develop AIDS during the next five years. This number does not include all those who 349
Medley G.F. et al.
are presently infected but who have not requested to be tested. At present, we do not have any good estimate for the number of HIV seropositives in the population of CAM. Our objective is to predict how many cases of AIDS will develop in the Community of Madrid over a 2-3 year period. The epidemic of AIDS will have a great impact on health services' delivery in the coming years and there is a need for financial planning and resource allocation. An estimate of how many cases of AIDS will develop every year during the next years is urgently needed, even when such an estimate is based on the scarce information on incidence of AIDS. Some attempts have been made on the prediction of AIDS cases in Spain. However, the models used have severe limitations. They do not take into account the influence of lag times between diagnosis and reports to the National Commission on AIDS (1) and/or they depend heavily on the distribution of the incubation period estimated from the USA transfusion associated cases (14). Several m e t h o d s have been developed for the short term predictions of AIDS cases. These methods fall broadly into two categories: those t h a t use estimates, of the incubation period (from infection to .disease)and those that extrapolate the observed trends. ~ The former method is known as "the back calculation" method (3, 5). The latter method is adopted here and involves the simple forecasting of observed trends (4, 6, 7, 8, 10, 15). The third method of prediction in the longer term (more than 5 years) requires that more detailed undei'st~tnding of the biology and epidemiology be inCtii'pdrated into a mathematical model of transmission dynamics
Eur. J. Epidemiol.
MATERIALS AND METHODS
Description of Data The register of AIDS cases at CAM contains information on all cases reported to and confirmed by the National Commission of AIDS on cases residing in CAM. Of the 819 cases reported up to September 1989, 16 had no date of diagnosis, and have been removed from the data, making a total of 803 cases. The data by year of diagnosis and year of report are shown in Table 1. This shows the increase in the rate of reporting and diagnosis over time. The comparison between the numbers diagnosed and reported in each year demonstrates the important effect that the lag between diagnosis and reporting has on the data (e.g.: up to 1987 389 diagnoses had been made, but only 264 notifications had been received). There is no effect of
TABLE 1. - AIDS Cases Notified to CAM by Year of Diagnosis and Report. Diagnoses Year
Number of Cases
82
1
83
7
84
15
85
44
86
115
87
207
(2).
88
301
The instruments to improve our knowledge of the scope of the epidemic in Spain are at hand. Advanced methods have been developed in some European and American countries and the results are applied in planning for services. Predictions based on this mathematical models are subject to some limitations. The number of predicted diagnosed cases is an understimate of the actual number of cases because it is computed under current conditions of reporting. Underreporting comes mainly from two sources: a) lack of precise diagnosis: many diseases associated with infection are not diagnosed as AIDS, even when they are serious infections which demand resources, such as pulmonary tuberculosis; b) lack of reporting: physicians may not be willing to report every diagnosed case to the National Commission of AIDS. An updated analysis in the UK estimates that between 60-800/0 of deaths related to HIV infection were reported to the Communicable Disease Surveillance Centre in 1987 (6). We have reasons to believe that these proportions will not be larger in Spain.
89
113
Total
803
Reports Year
Number of Cases
83
6
84
6
85
23
86
59
87
170
88
262
89
277
Total
803
Note that there is great variability between quarters.
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calendar month on either reporting or diagnosis. Figure 1 shows the monthly incidence of diagnosis. Again, the effect of the reporting lag is demonstrated causing the downturn of the data towards the end of the observation period. Figure 2 shows the cumulative probability of reporting occurring for intervals after diagnosis. Note that over 80% of all the cases reported within one year of diagnosis, but some cases took up to five years to be reported.
Description of Method Unfortunately, the estimation of the lag distribution and the actual numbers of diagnoses made
is not simple. This is because of the bias caused by the increasing epidemic. For example, presumably most AIDS cases in the data set have been diagnosed in 1988, but the longest lag we can observe with these data is only one year. The data with reporting lags longer than 4 years must have been diagnosed prior to 1984, when few cases were found. We use a method that allows simultaneous estimation of the incidence function of new diagnoses and the distribution of reporting delays between a case being diagnosed and the report arriving at CAM (4). These t w o functions are fitted to the data using the date of diagnosis, and the length of time elapsed between diagnosis and report. A technical description is given in Cox & Medley (4). It involves maximising the following log-likelihood with respect to the
50
45
-
40
-
+ -
35
N
0 N N
30 +
425
l:l M
+ 7
g: N
+
+
+ +
+ -H-
++ i5
-
i0
-
+-H-
5
+
-
+ +
-i ,~--rl
0 82
+
II
ii
~ . . A - - ~ + ++ ~ i I I
84,
,
I
86
88
90
YNAR
Figure 1. - This figure shows the monthly total of diagnoses reported to the end of September 1989. The upper line is the estimated incidence function, I(t), with parameter estimates: c%= 481.71, cq = 29.673, c~2= 0.2779 and u3 = 0.9239. The lower line is the number of diagnoses expected to have been reported, calculated using the incidence function I(t), and the lag distribution, f(x), with parameter estimates: O0 = 0.4406, ®1 = 7.9863 and O2 = 2.0597. The distance between the two lines represents the estimate of those diagnoses that have been made, but not reported.
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Eur. J. Epidemiol.
value o f the log-likelihood, although specific statistics are not available unless the models are nested. The incidence function chosen for prediction was the "linear-logistic" form used elsewhere (4, 6, 15):
parameters o f the incidence function o f new diagnoses, I(t), and the probability distribution function of reporting lags, f(x): .
a
l= 1
%
i= 1
I(0=
0
1 + ao~gze -a3t
where to is the date o f the last observation, ti and x~ (i = 1..n) are the observed times of diagnosis and corresponding reporting delay for the data, and F(x) is the cumulative distribution of the reporting delay distribution. The incidence of diagnoses was assumed to start when the first diagnosis was made. Different choices for I(t) and f(x) can be compared with the
0.9
gO+li1 t
where I(t) is the instantaneous incidence of new diagnoses per year, t is time since the beginning o f the epidemic (October 1982) and the (x are parameters to be estimated. This function is chosen because it mirrors the epidemiological understanding o f how the epidemic will develop (see Figure 3).
+ +~,/~.. X X X J":.. X x~ /~X xx'"
++ 0.8
+
l
m 0
0.7
-
H
0.6
-
I~
0.5
-
H ~
0.4
--
0.3
-
0.2
-
0 . i
-
H
m M
i
I
/ +
I
0
I
I
i0
20 REGISTRATION
LAG
I
I
30
I
40
(MONTH)
Figure 2. - The figure compares the cumulative lag distribution calculated directly from the data (points) with that obtained from the model (line) using the parameter estimates listed in Figure 1. The upper points (+) are from the total data set (803 observations) and the lower set of points from those cases diagnosed prior to 1988 (381 points). The distribution from the total data set should produce a shorter lag distribution than that actually operating because of the bias which prevents the most recent diagnoses (the majority of all diagnoses) from exhibiting long reporting lags. Those cases diagnosed prior to 1988 should be less biased in this respect, but should show a slower reporting lag than that operating now (see discussion). The estimated line falls between these two. For the estimated distribution the median time between diagnosis and report is 5.128 mths, and the mean is 7.839 mths. 352
Prediction of AIDS in Madrid, Spain
Vol. 7, 1991
A variety of distributions for the reporting lag were tried. The distribution used was a mixture of two gamma distributions:
,/(x)=0o01Xe-elx+(1-0o)0zxe-°z~
were f(x) is the probability density function of lags of length x and the O are parameters to be estimated. We attach no significance to the form of this distribution it is simply a good empirical description. Following Cox & Medley (4), the delays of Omth, 1 ruth and 2 ruth were estimated as the cumulative distribution associated with this distribution. Those lags greater
than 2yrs (32 cases) where omitted from the fitting procedure. The distribution and longest tag included where chosen by comparison of the log-likelihoods and by direct inspection of the fitted distribution and observed data (see Figure 2). Approximate confidence intervals for the predictions were obtained by evaluating the loglikelihood for systematic combinations of parameter values and plotting them against the predictions. The limits are given by the minimum and maximum predicted values that fall on the log-likelihood less the appropriate value from the distribution with one degree of freedom. This method is computationaUy expensive (over 1000 evaluations are required for each interval calculated), and can only be approximate (4, 6).
70
60
--
50
-
4o
-
~o
-
.•+
u H U
+
+ N
2O
+
~" +++4+ ++ + ~44-
lo
t
_
82
+ ~ + + / ~ + I-+
.+ a-a~ i
+
÷
+
÷
+
+
+1-
"~I ~' I I 8'11,
i
I
86
i
88
''
i
i
90
i
i
92
xrE/k.R
Figure 3. - This figure shows the monthly total of diagnoses (as Figure 1), and the estimated incidence function extrapolated beyond the data. Note that the incidence function is exponential at the beginning of the epidemic, and then becomes linear beyond the end of the data. Parameter estimates as figure 1.
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Eur. J. Epidemiol.
RESULTS
DISCUSSION
The results of the estimation are given in Figures 1 and 2. In Figure 1, the lower line (going through the points) is the expected pattern of notifications. The upper line is the fitted incidence function. The gap between the two lines represents the reporting delay, i.e. the difference between the actual numbers of cases diagnosed and those reported. The fit appears to give an adequate description of the current data. However, this is not necessarily a guarantee that predictions will be accurate. Tables 2 to 4 give the predictions associated with the parameters estimated from the data for 1990 and 1991. We have not provided predictions beyond this point, because they become increasingly unreliable. Table 2 shows the predicted monthly, quarterly and annual numbers of diagnoses (regardless of time of report) and numbers of reports (regardless of time of diagnosis). Up to 31st December 1988, 697 cases of AIDS had been reported to the Register at CAM. The current prediction suggests that 536 will be diagnosed throughout 1989, 646 throughout 1990 and 719 throughout 1991. The number of new notifications predicted for the same years are 440, 572 and 667 respectively. Thus, we expect the incidence of new notifications always to be below the incidence of new diagnoses. Table 2 also shows the actual number of new notifications received in 1989 was 446 (compared with the prediction of 440), and to July 1990 317 have been received (282 predicted). Table 3 shows the expected numbers of AIDS patients alive at the end of each quarter. Two estimates are given under different assumptions about the distribution of survival times of patients from diagnosis of AIDS to death. The predicted numbers of cases alive as of March 1990 falls between 566 and 895, in agreement with the observed value of 641 (9). By the end of December 1991, the number of living AIDS patients is predicted to have risen to between 797 and 1402. This implies that by the end of 1991 the resources allocated to care of AIDS patients should be approximately doubled in order to maintain the present level of care. The 90% confidence limits for the numbers of new diagnoses per year are shown in Table 4. The predicted number of diagnoses during the period 19891991 is 1901 with 90% confidence interval 1395 to 2800. Note that the limits are skewed such that they extend further above the estimate than they do below. These limits are calculated from variability in the data, and not from assumptions about the incidence function (which may not be a good description of the future epidemic). The lower estimate suggests a virtually constant number of cases per year. In spite of the wide confidence interval, it seems useful to know that during this three year period the number of diagnoses is likely to be equal to the total number diagnosed to the end of 1988.
The prediction of AIDS cases is a difficult problem, and as with all attempts at extrapolating into the future, is subject to many possible flaws. However, the provision of health care for people with AIDS and HIV infection is an important issue which is already demanding a great deal of health services, and is likely to demand more. Consequently, some attempts at prediction, with all the appropriate caveats, are required. It is only when such predictions are attempted, and then compared with observations, that some estimate of the reliability of predictions will be obtained. All forecasting methods produce wide confidence intervals for predictions, but future experience may suggest that these are unduly pessimistic. The current results in Tables 2-4 are a useful starting point for planning, where we do not need exact figures but approximated estimates. These are, as far as we are aware, the best attempts at predicting cases in Madrid and Spain. As under-reporting is not included, and the confidence limits of such predictions are skewed, for planning purposes, it may be better to multiply the estimates by some factor (say, 120%). Also, over-allocation of resources to a problem, although not good planning, is significantly better than under allocation which may leave patients without adequate health care. There are a number of concerns that we have regarding the validity of the predictions presented here, and we discuss each in turn. The main limitation of the register of AIDS cases at CAM is under-reporting. Some hospitals have stopped reporting for a period of time or are reporting less cases than those diagnosed. Some causes for under-reporting are: a) excessive demand for care on the health personnel, understaffed units of care; b) organizational problems of work; c) lack of awareness of the use of surveillance data in health planning. Reporting has not been uniform across hospitals and along time. Often there is a period of under-reporting followed by a period during which old cases are reported following medical history reviewing. Factors such as changes in personnel and organizational structure have originated a skewed distribution of lag times (times between diagnosis and reports). While most of the cases are reported within six months after diagnosis, a considerable proportion of cases are reported more than two years after diagnosis. Although the fit of the estimated distribution to the observations (Fig. 2) seems reasonable, we are not convinced that it is an accurate description. It is likely that there was an increase in reporting efficiency during 1987, which coincided with the formation of the CAM register of AIDS cases and an increased awareness of the importance of prompt reporting of cases. Also, the fitting procedure assumes that the diagnoses are reported independently and continuously, but anecdotal evidence suggests that the reports from the eight hospitals do not follow these assumptions. The collection of reports into "batches" at the hospitals is one problem, as is the fact that some hospitals do not report continuously. Table 5 shows the numbers of
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Vol. 7, 1991
Prediction of AIDS in Madrid, Spain
TABLE 2. - Monthly Predictions of AIDS cases by Diagnosis and Repoa for CAM. Monthly Diags. Reps.
Year
Month
1989
1 2 3 4 5 6 7 8 9 10 11 12
39 40 42 43 43 44 45 46 47 48 49 50
31 32 33 34 35 36 37 38 39 40 41 42
1 2 3 4 5 6 7 8 9 10 11 12
50 51 52 52 53 54 54 55 55 56 57 57
43 44 45 46 47 47 48 49 50 50 51 52
1 2 3 4 5 6 7 8 9 10 11 12
58 58 58 59 59 60 60 61 61 61 62 62
52 53 54 54 55 55 56 56 57 57 58 58
1990
1991
Qua~efly Diags. Reps.
121
96 ( 83)
130
105 (188)
138
115 (73)
147
124 (102)
153
132 (150)
159
140 (167)
164
147 (183)
170
153 (165)
174
159
178
165
182
169
185
174'
Annual Diags.
Reps.
536
440 (446)
646
572 (665)
719
667
The above table shows the predictions for the expected number of AIDS cases diagnosed (Diags.) and reports (Reps.) received for each month, quarter and year from 1989 to 1991using the data gathered to September 1989 for the calculations. Not~that the number of diagnoses is irrespective of time report, and is the estimated number of new cases requiring medical attention discovered each month. The quarterly numbers or reports actually received are given in parentheses following the estimate.
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Eur. J. Epidemiol.
T A B L E 3, - Quarterly Predictions of the N u m b e r o f A I D S Patients Alive in CAM.
T A B L E 5. - A I D S Cases Notified to C A M by Hospital and Year.
Year Quarter Pessimistic Number Optimistic Number Deaths Living Deaths Living 89
1 2 3 4
78 87 96 104
397 440 483 525
51 57 63 70
592 665 741 817
90
1 2 3
112 120 127
566 605 643
76 82 89
895 971 1047
4
134
678
95
1122
1 2
141 147
711 741
101 107
1195 1266
3 4
153 158
770 797
112 118
1335 1402
91
Hospital
T A B L E 4. - Approximate 90% Confidence Limits for A I D S Diagnoses in CAM. Lower
Estimate
Upper
1989 1990 1991
425 470 500
536 646 719
700 950 1150
1988
1989
1990
Gregorio Maranon Clinico La Principessa Ramon y Cajal La Paz
41 16 11 22 21
3 1 36 46 43
104 69 20 10 37
64 9 5 1 18
Del Ray
38
97
91
20
Data to the end of March 1990. This is a partial list of the data available (9). It shows the variability in the reporting efficiency both between hospitals and between years within the same hospital. This is due not only to variability in the number of AIDS diagnoses made, but also to disruption of the reporting system. See discussion.
This table shows the estimated number of AIDS patients alive, and therefore demanding health-care resources, at the end of each quarter in 1989 to 1991. Two estimates are given: pessimistic assuming that the median survival of patients is about 9mths from diagnosis and optimistic assuming that the median is about 18mths. The survival distributions used are taken from a recent study in the UK, and reflect the better medical care available to AIDS patients (15), and are as follows: p(t) = 0.92 exp (-0.7320 (pessimistic) P(0 = 0.94 exp (-0.312t) (optimistic), where t is the time in year and p(t) is the proportion surviving at time t.
Year
1987
The approximate 90% confidence limits for the predicted annual numbers of AIDS cases diagnosed in Madrid for 1989, 1990 and 1991.
reports at C A M from the different hospitals, which varies over time. For example, the reduction in reports received f r o m Gregorio Maranon in 1988 was due to a change in staff rather in the n u m b e r of A I D S diagnoses m a d e at the hospital. There are a variety o f graphical and statistical methods for estimating the lag distributions that m a y be m o r e reliable (e.g.: 11, 16). However, for the present purpose, the parametric f o r m chosen here should capture the broad shape, and be acceptable to produce predictions. It is also worth noting that this m e t h o d cannot make any estimate
about the extent or effect o f under-reporting either due to A I D S cases not diagnosed or those diagnosed but not reported. T h e predictions presented here depend completely on the choice o f incidence function, I(t). W e have no way o f knowing the true function, if, indeed, the incidence o f diagnoses can be described by any simple parametric function. The f o r m used is a result of considering the dynamics of the A I D S epidemic, which shows an exponential increase, followed by a more linear growth. Only comparison between prediction and observations will show whether a particular choice is good or not. Future consideration of short-term predictions should separate the different risk-groups within the epidemic (IDU, homosexual men, heterosexuals etc), and should also make use o f the m a n y methods available for prediction. A combination o f methods produces a better consensus of the future for planning purposes (3, 6, 15). This preliminary report suggests that several steps need to be taken to improve the surveillance data for A I D S in Madrid (& Spain), if the data are to provide worthwhile in future attempts at prediction. 1) The reporting system should be quickened. 2) T h e level o f under reporting should be decreased. W e hope that this work will encourage physicians to report cases, as they can see the benefit o f their efforts in terms of adequate allocation of resources. 3) There should be adequate statistical expertise available to analyze surveillance data. With the available data, it should be possible to use other methods of prediction and to estimate the prevalence o f H I V infection in the main high risk groups, prevalence o f A I D S at any given calendar year, n u m b e r of deaths associated to H I V infection, and distributions o f survival times. 4) A study on transmission dynamics o f H I V infection a m o n g I D U could be undertaken taking into account drug using behaviour.
356
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Prediction of AIDS in Madrid, Spain
5) Some studies should be undertaken to try to estimate the incidence of diagnoses which does not depend on the notification lag. Such a direct m e t h o d would give some indication whether the predictions o f diagnoses in Table 2 were reasonable, as they are unobservable with the present data. Even using data and methodology with all the limitations discussed above, we believe that we have enough evidence to suggest that resource allocation to care for AIDS cases should be at least doubled up to the beginning o f 1992 in order to provide the current level o f care. W e hope that this suggestion will be of guiding value to health authorities when taking decisions on funding requests from clinicians and other health professionals, who are currently under pressure from the increasing burder of A I D S and H I V related disease.
6. Department of Health and Social Services (1988): Short-term Prediction of HIV Infection and AIDS in England and Wales, Report of a Committee. London, HMSO. 7. Downs A.M., Ancelle R.M., Jager J.C. and Brunet J.B. (1987): AIDS in Europe; current trends and shortterm predictions estimated from surveillance data, January 1981-June 1986. - AIDS 1: 53-57. 8. Epidemiological Surveillance of AIDS/HIV in Community of Madrid (Vigilancia Epidemiologica del SIDA/VIH en la CAM), No. 4, Octubre 1989. Communidad de Madrid, Consejeria de Salud, Servico Regional de Salud.
Acknowledgements
9. Epidemiological Surveillance of AIDS/HIV in Community of Madrid (Vigilancia Epidemiologica del SIDA/VIH en la CAM), No. 5, Abril 1990. Communidad de Madrid, Consejeria de Salud, Servico Regional de Salud.
GFM is a University Research Fellow of the Royal Society of London. We thank the health care personnel in CAM for their efforts in collecting these data. We also thank an anonymous referee for helpful comments.
10. Healy M.J.R. and Tillett H.E. (1988): Short-term extrapolation of the AIDS epidemic (with discussion). - Journal of the Royal Statistical Society, Series A 151: 50-61.
REFERENCES
11. Heisterkamp S.H., Jager J.C., Ruitenberg E.J., van Druten J.A.M. and Downs A.M. (1989): Correcting reported AIDS incidence: a statistical approach. Stat. Med. 8: 963-976.
1. Alvarez V.A., Laguarta B.A., Artalejo F.R. et aL (1988): E1 SIDA en Espana: prediccion de nuevos casos mediante el uso de modelos matematicos. - Rev. Clin. Esp. 183: 60-63. 2. Anderson R.M. (1989): Mathematical and statistical studies on the epidemiology of HIV. - AIDS 3: 333346. 3. Brookmeyer R. and Damiano A. (1989): Statistical methods for short-term projections of AIDS incidence. - Statistics in Medicine 8: 23-34.
12. Lifson A.R., Rutherford G. W. and Jaffe H. (1988): The natural history of Human Immune Deficiency Virus infection. - J. Infect. Dis. 158: 1360-1367. 13. Moss A.R. and Bachetti P. (1989): Natural history of HIV infection. - AIDS 3: 55-61. 14. Peruga A. (1988): Provision de casos nuevos de SIDA en la Comunidad de Madrid 1982-1995. - Working paper. Consejeria de Salud. Comunidad de Madrid.
Cox D.R. and Medley G.F. (1989): A process of events with notification delay and the forecasting of AIDS. Phiilosophical Transactions of the Royal Society of London, Series B 325: 135-145.
15. Public Health Laboratory Services (1990): Acquired Immune Deficiency Syndrome in England and Wales to End 1993; Projections Using Data to End September 1989. - PHLS Communicable Disease Surveillance Centre, January.
5. De Gruttola V. and Lagakos S. IV. (1989): The value of AIDS incidence data in assessing the spread of HIV infection. - Statistics in Medicine 8: 53-65.
16. RosenbergP. (1990): A simple correction procedure of AIDS surveillance data for reporting delays. - J. AIDS 3: 49-54.
4.
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