### Abstract

A methodology is proposed for obtaining short-term projections of the acquired immunodeficiency syndrome (AIDS) epidemic by projecting the number of cases from those already infected with the AIDS virus. This is a lower bound on the size of the AIDS epidemic, because even if future infections could be prevented, one could still anticipate this number of cases. The methodology is novel in that no assumptions are required about either the number of infected individuals in the population or the probability of an infected individual eventually developing AIDS. The methodology presupposes knowledge of the incubation distribution, however, among those destined to develop AIDS. Although the method does not account for new infections, it may produce accurate short-term projections because of the relatively long incubation period from infection to clinically diagnosed AIDS. The estimation procedure “back-calculates” from AIDS incidence data to numbers previously infected. The number of cases diagnosed in each calendar period has a multinomial distribution with cell probabilities that can be expressed as a convolution of the density of infection times and the incubation distribution. The problem is shown to reduce to estimating the size of a multinomial population. A simple EM algorithm is developed for obtaining maximum likelihood estimates when the density of infection times is parameterized as a step function. The methodology is applied to AIDS incidence data in the United States, to obtain short-term projections and an estimate of the minimum size of the epidemic by assuming no new infections in 1987 and after. The sensitivity of the projections to the assumed incubation distribution is investigated. It was found that short-term projections are not nearly as sensitive to the assumed incubation distribution as long-term projections. Although some information on the incubation distribution has been reported from an analysis of transfusion-associated AIDS cases, we discuss several issues and caveats associated with estimates of the incubation distribution from such data. An important point is that short-term projections are considerably less sensitive to the assumed incubation-period distribution and the parametric model for the density of infection times than long-term projections. There is a temptation to use back-calculation to estimate current human immunodeficiency virus seroprevalence, and to make longer-term projections; however, such estimates are highly uncertain both because they are sensitive to model assumptions and because the AIDS incidence data provide more information about infections in the distant past than the near past.

Original language | English (US) |
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Pages (from-to) | 301-308 |

Number of pages | 8 |

Journal | Journal of the American Statistical Association |

Volume | 83 |

Issue number | 402 |

DOIs | |

State | Published - Jun 1988 |

### Keywords

- Epidemiology
- Infectious diseases
- Multinomial distribution

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

*Journal of the American Statistical Association*,

*83*(402), 301-308. https://doi.org/10.1080/01621459.1988.10478599