a Department of Applied Mathematics, National Chung-Hsing University, Taichung, Taiwan.
b Departamento Ecuaciones Diferenciales, Facultad Mathematica-Computacion, Universidad de la Habana, San Lazaro y L Habana 4, Cuba.
c Department of Statistics, Feng-Chia University, Taichung, Taiwan.
Ying-Hen Hsieh, Department of Applied Mathematics, National Chung-Hsing University, Taichung, Taiwan 402. E-mail: yhhsieh{at}dragon.nchu.edu.tw
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Abstract |
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Methods The generalized removal model for open populations is utilized for the estimation. The total number of known HIV-infected Cubans at each sampling time is used in the prior to provide more reasonable approximations.
Results We estimated a yearly survival rate of 93%. The median estimates for the number of all living asymptomatic HIV-positive Cubans, infected by sexual contact, tripled from 714 in 1991 to 2170 in 2000. The number of unknown HIV-positive Cubans infected sexually is estimated to range from 174 in 1991 to 401 in 2000.
Conclusions A consistent increase in the number of sexually infected HIV-positive individuals in Cuba from 1991 to 2000 is evident from the estimates. From 1996 onwards more sexually active homosexual/bisexual contacts were traced and consequently more sexually-infected HIV-positives were detected. A consequence of increased detection is the levelling off and subsequent decrease in the number of unknown HIV-positives during this time period. The estimation procedure is useful in estimating prevalent population sizes of epidemiological and public health interest.
Keywords Cuba, epidemiology, HIV/AIDS, Latin America, sexual contact, Bayes statistics, contact tracing
Accepted 11 January 2002
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Introduction |
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Another important aspect of the Cuban national AIDS programme is the Partner Notification Programme (PNP) based on contact tracing and screening of the sexual partners of known HIV-positives. This also began in 1986 and the idea is to search out asymptomatic carriers before they develop AIDS. Indeed, the result is impressive given that 55% of those detected with HIV in Cuba have not developed AIDS. General screening and subsequent admission to a sanatorium have been reduced in recent years due to economic constraints and an evolving sanatorium policy and so the PNP has gradually assumed added importance. Since approximately 90% of the reported AIDS cases in Cuba by the end of 1997 were acquired by sexual (hetero-, homo-, or bisexual) contact,5 the number of HIV-positives detected via contact tracing should give a good indicator of the size of the HIV-infected population in general. Furthermore, recent growth in tourism has led to a re-emergence of prostitutes in the last few years.6 Perhaps understandably, recent data have shown an increase in the number of HIV-positives, starting in 1996.7,8 Hence it is especially worthwhile from the public health point of view to focus our attention on estimating the population size of HIV-positives infected through sexual activity.
Recent AIDS data have shown that approximately 14% of AIDS cases were unknown to the Health Authority before developing AIDS symptoms. To obtain an estimate of the size of unknown HIV-positive population, Arazoza7 and Lounes8 applied a mathematical model to compute the theoretical numbers of the known and unknown HIV-positives infected by sexual contact in Cuba. Their results indicate that roughly 20% to 30% of the HIV asymptomatic carriers have not been detected. Recently a generalized removal model for open populations was proposed by Hsieh et al.9 which uses an empirical Bayes approach to estimate the number HIV-infected people in a hidden, hard-to-count population without any knowledge of the population size. The method was employed in a preliminary study of recent trends in HIV infections in Cuba.10 Here we use this method to estimate the known and unknown numbers of HIV-positives in Cuba infected via sexual contact. We will make use of HIV seroprevalence data from the PNP from 1991 to 2000. In contrast to Hsieh et al.,9 we have additional information in the Cuban data set, i.e. the total number of known individuals in the HIV-infected population at each sampling time. We will employ this additional knowledge in our priors to improve our estimates.
The paper is organized as follows. We briefly describe the data and the statistical method used. We then give the results of our estimates, followed by concluding remarks and comments. Statistical details are given in the Appendix.
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Data and Methods |
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One of the main focuses of this study is to obtain an estimate for the number of unknown HIV-positives in the sexually active population in recent years who have not developed AIDS. Table 1 gives the accumulated yearly number of known HIV-positives living in Cuba at the end of each year from 1991 to 2000. Table 2
is the seroprevalence data from 1991 to 2000 with number of contacts tested, number of HIV-positives detected, and the percentage of positive tests for each year listed.
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Methods
The generalized removal model for open populations9 proposed recently allows only recruitment (of new HIV-infected individuals) and deaths (removal of HIV-infected individuals due to development of AIDS) to occur during the sampling. In this paper, where the number to be estimated is the yearly number of HIV-infected individuals within the sexually active population in Cuba, there is no recapture of those HIV-positive individuals detected in previous samplings since it is reasonable to assume that those tested positive will not be tested again. Hence the removal model is the appropriate choice of model to work with. In each sample, a number of subjects (in our case, those with recent contact with known HIV-positives) are selected for testing. Moreover, the sample-taking would exclude anyone who has already developed AIDS symptoms, hence the estimate we obtain is the number of HIV-infected individuals who have not developed AIDS. It does not hinder public health assessment of the AIDS scenario because the size of population with AIDS symptoms can be easily counted from clinical records.
Since it is not possible to obtain a valid estimate of the HIV-infected population using maximum likelihood estimation, we propose a Bayesian estimation procedure. Bayesian inference of a population size for various models has been proposed in the literature (e.g. refs 1315). The detailed derivation of the model is given in Hseih et al.9 Differing from Hseih et al., there is extra information in this data set for Cuba. Namely, we know the total number, Rj of known subjects in the HIV population infected by sexual contact (i.e. 90% of the known number of HIV-positives) just before time tj. Intuitively, we must have Nj Rj; i.e. the number of HIV-infected subjects known to the health authority cannot exceed the number of all HIV-infected subjects. We will make use of this additional knowledge in our priors. Using this extra information our posterior estimates could provide more reasonable approximations.
Statistical details of the method are given in the Appendix. An empirical Bayes analysis of the model is implemented using the Gibbs sampler, a Markov chain Monte Carlo (MCMC) method. Detailed discussions can be found in Casella and George.16 The Bayes estimates are based on Monte Carlo samples from the Gibbs sampler run of 6000 iterations after 2000 burn-in, and selecting every 5th sampled value.
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Results |
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Discussion and Concluding Remarks |
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We wanted to estimate the number of HIV-positives in the sexually (both heterosexual and homo-bisexual) active population. Hence we consider the sexual contacts of known HIV-positives who have been traced as a random sample of the sexually active population since they evidently are sexually active. However, as those traced and tested are known to have had contact with at least one HIV-positive in the past, the capture probability might be higher than it would be in a truly random sample. Consequently, this may result in some overestimation of the true numbers. On the other hand, it is generally unknown whether the contacts are made when the HIV-positive person is already infective, due to variance in infection time and progression of disease. Therefore the actual effect of this uncertainty on the estimate is not clear. In this respect, a possible future research direction is to improve the method by considering the detailed individual contact tracing data of the HIV-positives in Cuba. That would require a much more complicated and difficult model which is beyond our scope.
A full discussion of the advantages as well as the drawbacks and limitations of the generalized removal model is given in Hsieh et al.9 It suffices to point out the difficulty in obtaining information regarding hidden and elusive populations such as the sexually active population in a society. In practice, the dilemma has proved to be even more challenging in the context of HIV epidemic. This work and Hsieh et al.'s previous paper,9 in which we estimated the number of HIV-infected people in elusive, hard-to-count population groups, demonstrate the usefulness of the generalized removal model, not just in estimating the HIV-infected population sizes, but any prevalent population size of epidemiological and public health interest. As long as two or more (non-overlapping) random samplings of the prevalence data are obtained, one can use it to make inference of the prevalent population size.
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Appendix |
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The likelihood function can be obtained as follows:
| ((1)) |
Suppose that the prior distribution of (N,P) where N = (N1,...Ns) is given by (N,P) =
(N1,...Ns)
(P). This asserts that N and P are a priori independent. We assume that the priors of Pj's are a priori independent and follow a Beta distribution Be(
1,
2). In addition, let
![]() | ((2)) |
The assumption on the prior of N in (2) is appropriate since intuitively Nj must be larger than Rj.
Such priors lead to conditional posteriors of the forms:
| ((3)) |
| ((4)) |
![]() | ((5)) |
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Acknowledgments |
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References |
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