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.     

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.     

Forecasting the obesity epidemic in the aging U.S. population

Objective: The objective was to forecast BMI distribution in the U.S. population along with demographic changes based on past race‐, sex‐, and birth cohort‐specific secular trends. Research Methods and Procedures: We compiled data from 44,184 subjects from 4 National Health and Nutrition Examination Surveys (NHANES; 1971 to 2004). By race and sex, we fit regression models to create smoothed mean BMI curves by age for 1970 to 2010. Linking corresponding birth cohorts across age‐ and year‐specific mean BMI projections, we estimated the trajectory of relative BMI throughout each cohort's lifetime. These projections were validated using actual cohorts in the Nurses’ Health Study and Health Professionals Follow‐up Study. Combined with U.S. census, we predicted BMI distributions in 2010 and examined the joint impact of the obesity epidemic and population aging. Results: BMI secular trends in the past 3 decades differ significantly by birth cohort, sex, and race. If these trends continue, the prevalence of obesity is expected to reach 35%, 36%, 33%, and 55% in 2010 among white men, white women, black men, and black women, respectively, far from the Healthy People 2010 goal of 15%. Such forecasts translate into 9.3 million more obese adults 20 to 74 years of age than in 2000, 8.3 million of whom would be 50 years of age or older, and 8.5 million of whom would be white. The mean age among obese men and women is also expected to rise from 47 to 49 years among whites and from 43 to 44 years among blacks. Discussion: As the baby boom generation approaches retirement age, the continuing obesity epidemic signals a likely expansion in the population with obesity‐related comorbidities. A framework to combine BMI and demographic trends is essential in evaluating the burden and disparity associated with the epidemic in the aging U.S. population.     

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.     

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.