Model fitting and projection of the AIDS epidemic.

Previously it was possible to fit detailed models to incidence data (for example, of AIDS) only by trial and error and good judgment; the large number of parameters obstructed optimization of, for example, the (approximate) likelihood. Here, we analyze a model for the spread of AIDS in a homosexual population and identify a minimal set of primary components that dictate the dynamics of the Model: the initial growth rate theta, the basic reproductive ratio R0, and the heterogeneity coefficient S. It is then shown that it is sufficient to maximize the likelihood over these three primary components; further maximization over the remaining secondary parameters does not produce a significant improvement in the fit or affect the projection of the epidemic. This method also allows construction of confidence limits for the projected incidence curve, allowing us to quantify the uncertainties associated with such model fitting procedures. The method is tested on simulation data to analyze how the accuracy of estimates and projections changes as we gain more data.     

The overall distribution of survival times for U.K. AIDS patients

Data on the survival times of 997 U.K. AIDS patients are analysed with the aim of deriving a simple form for the overall survival distribution. The exponential and Weibull distributions are modified to accommodate specific features of the data, in particular, the recording of survival times to the nearest month and the occurrence of a significant proportion of cases recorded as having zero time on study. The final model has a probability 0.08 of underlying survival time being zero and, given non-zero survival time, takes the form of an exponential distribution with mean of 14.95 months. The results are in close agreement with those of a study of New York City patients as well as the empirical data.     

The AIDS epidemic: profiles of development

As all HIV-infected subjects become virus carriers, the epidemic will not attain a "steady state" until the number of deletions (from death and other factors) equals or outnumbers that of new cases, i.e. each HIV-infected subject transmits the infection to only one subject in the course of his lifespan. A full stop of all spreading of HIV will most likely require worldwide vaccination. By simple mathematical models it is shown that calculation of the number of HIV infected individuals based on the number of AIDS cases is very uncertain. The ratio of HIV infected subjects to AIDS cases is greatly influenced by the length of the incubation period and the case doubling time. Since the growth of the epidemic is exponential, all efforts to control the epidemic should be continuously intensified as single measures will only retard the rate of spread. The effect of saturation/deletion on the number of susceptible individuals is insignificant in this phase of the epidemic, except in small groups at special risk.     

Three-stage stochastic epidemic model: an application to AIDS.

A three-stage stochastic epidemic model extending the so-called classical epidemic process to one that includes time-dependent transition probabilities is described, and a solution to the appropriate set of forward differential-difference equations is given. When an individual can move from being a susceptible to one infected with the HIV virus to one diagnosed as having AIDS, we can use this general model to describe an AIDS epidemic process. We obtain expressions for the mean and variance of the number of AIDS cases for some special cases. By comparing these with actual data, it is suggested that, for some categories of cases (in particular, children), this model might be a plausible model to describe the underlying mechanism of the AIDS epidemic.     

Accommodating change due to treatment in the method of back projection for estimating HIV infection incidence

This note shows how the method of back projection, which is being widely applied to predict the incidence of HIV infection, can be extended to incorporate distributional changes due to a treatment effect, such as zidovudine (commonly known as AZT). By way of example we consider one of the approaches to back projection and apply the method to some Australian data.     

Modeling the AIDS epidemic in Mexico City

To study the future course of the AIDS epidemic in Mexico City, we use an open compartmental model to forecast new AIDS cases among homosexual and bisexual males and among heterosexual males and females. For each group three compartments are defined: uninfected persons, infected but asymptomatic persons, and persons diagnosed with AIDS. It is assumed that the AIDS epidemic will follow the propagation of infectious disease model, where spread of infection is proportional to the product of the number of healthy persons and the number of infected ones. The compartmental model is represented by a system of nonlinear differential equations describing the rate of change in the number of persons in each compartment. The impact of preventive measures is explored by decreasing the probability of HIV transmission, which is one of the model parameters representing behavioral patterns. By April 1989, 491 AIDS cases had been reported in Mexico City and classified as sexually related. Our model predicts that the AIDS incidence will continue to rise in Mexico City for the foreseeable future and will spread among the heterosexual population. Decreasing the transmission probability by 10% in all groups (through education programs) will result in a decrease of 18.1% in the number of accumulated cases over a 5-year period. A 20% decrease would prevent more than 31% of the cases. We conclude that mathematical models can be valuable in predicting the spread of the AIDS epidemic and the impact of behavioral change on its spread.     

Epidemiological and statistical aspects of the AIDS epidemic: introduction

In 1988, a government working party studied estimates of incidence and prevalence of numbers of acquired immunodeficiency syndrome (AIDS) cases. They investigated a series of epidemiological, statistical and mathematical problems associated with predicting trends in incidences of AIDS. This paper introduces a series of papers that give a fuller and more technical exposition of the appendixes of that working party report. The papers provide a brief background to the current state of knowledge on the epidemiology of the infection and the disease; a deterministic model for human immunodeficiency virus (HIV) transmission in the male homosexual community in England and Wales is introduced. Back-projection methods are studied in two papers, following the distribution of the incubation period of the disease. The concept of minimum size of the epidemic is introduced. Mathematical functions to describe the spread of HIV infection are refined by using past trends in the incidence of AIDS to estimate values for some parameters. Survival times for AIDS patients from the point of diagnosis are considered and evidence for changes in male homosexual sexual behaviour is studied; lag-time from the point of diagnosis to the report of the case is also examined. There is a comparative analysis of the AIDS epidemic in various European countries. The incubation period of HIV in patients with haemophilia A and B infections and the problems associated with making predictions for different at-risk groups or small subgroups based on geographical area are discussed. Reasons for fluctuation between the number of reported cases from month to month are provided.     

Monitoring acid waters in the U.K.: an overview of the U.K. Acid Waters Monitoring Network and summary of the first interpretative exercise

1. The U.K. Acid Waters Monitoring Network (AWMN) was established in 1988. It comprises eleven lake and eleven stream sites located throughout the U.K. in areas sensitive to acidification. The principal objective of the AWMN is to provide a long-term, high-quality chemical, biological and palaeolimnological record which, in conjunction with the U.K. precipitation monitoring networks, will facilitate the assessment of trends within U.K. surface waters.
2. The first interpretation of results generated by the AWMN (April 1988–March 1993) has recently been completed and is summarized. During this period there have been no sustained changes in atmospheric deposition in the U.K. Trends are recognized in certain chemical and/or biological parameters at individual AWMN sites, but despite some (regional) patterns no regional trends toward increasing or decreasing acidification are apparent. The data comprise an excellent baseline with which future changes may be compared.
3. The methodology of the AWMN is reviewed and some amendments are suggested in the light of the first 5 years experience. These include: additional determinations for nitrogen species and total phosphorus; changes to the frequency of macrophyte and palaeolimnological sampling; new emphasis on data generated by sediment traps at lake sites; and the addition of extra sites to address better the full extent of acid-sensitive surface waters revealed by the national critical loads mapping programme, and to pay particular attention to the acidifying role of nitrogen.
4. The contribution of AWMN data to other national and international environmental monitoring programmes is highlighted. In addition, AWMN sites provide data to develop and validate critical loads models and dynamic models of acidification, and could also be used to monitor the impact of other air pollutants such as trace metals and persistent organic compounds on the freshwater environment.     

Politics and pellagra: the epidemic of pellagra in the U.S. in the early twentieth century.

The epidemic of pellagra in the first half of this century at its peak produced at least 250,000 cases and caused 7,000 deaths a year for several decades in 15 southern states. It also filled hospital wards in other states, which had a similar incidence but refused to report their cases. Political influences interfered, not only with surveillance of the disease, but also in its study, recognition of its cause, and the institution of preventive measures when they became known. Politicians and the general public felt that it was more acceptable for pellagra to be infectious than for it to be a form of malnutrition, a result of poverty and thus an embarrassing social problem. Retrospectively, a change in the method of milling cornmeal, degermination, which began shortly after 1900, probably accounted for the appearance of the epidemic; such a process was suggested at the time, but the suggestion was ignored.