Charles J. Mode

Stochastic processes in epidemiology : HIV/AIDS, other infectious diseases and computers

This text deals with the mathematical and statistical techniques underlying the models used to understand the population dynamics of not only HIV/AIDS, but also of other infectious diseases. Attention is given to the development of strategies for the prevention and control of the international epidemic within the frameworks of the models. The text incorporates stochastic and deterministic formulations within a unifying conceptual framework.

A methodological study of a stochastic model of an AIDS epidemic

Abstract A model of an AIDS epidemic in a population of male homosexuals was formulated as a stochastic population process. The paper is a methodological study in the sense that computer-intensive methods were used to investigate some properties of the model statistically rather than relying solely on classical methods of deductive mathematics. Three factors of importance in the evolution of an AIDS epidemic were studied in a numerical factorial experiment. These factors were the distribution of the latent period of HIV, the probability of infection with HIV per sexual contact with an infected individual, and the distribution of the number of contacts per sexual partner per month. The numerical experiment suggested that the distribution of the latent period of HIV will have a decisive impact on the evolution of an AIDS epidemic but this impact will depend crucially on the levels of the other two factors. A Monte Carlo experiment suggested that if forecasts of an epidemic were made solely on the basis of deterministic nonlinear difference equations embedded in the stochastic population process, then predictions of the number of individuals infected with HIV and AIDS cases may be overly pessimistic.

An eight-parameter model of human mortality—The single decrement case

Developed in this paper is an eight-parameter model of human mortality. A step-wise nonlinear least-squares procedure for estimating the parameters from abridged life tables is also described and implemented. Used for purposes of illustration were nine period life tables, ranging from 1900 to 1977, for the United States white male population. The agreement between the observed and calculated survival functions in the nine life tables was very good. Apart from its phenomenological interest, the model provides an effective means for calculating interpolations and extrapolations of abridged life tables, which are useful making population projections and in computer graphics.

Estimating the transmission potential of supercritical processes based on the final size distribution of minor outbreaks.

Abstract Use of the final size distribution of minor outbreaks for the estimation of the reproduction numbers of supercritical epidemic processes has yet to be considered. We used a branching process model to derive the final size distribution of minor outbreaks, assuming a reproduction number above unity, and applying the method to final size data for pneumonic plague. Pneumonic plague is a rare disease with only one documented major epidemic in a spatially limited setting. Because the final size distribution of a minor outbreak needs to be normalized by the probability of extinction, we assume that the dispersion parameter (k) of the negative-binomial offspring distribution is known, and examine the sensitivity of the reproduction number to variation in dispersion. Assuming a geometric offspring distribution with k=1, the reproduction number was estimated at 1.16 (95% confidence interval: 0.97–1.38). When less dispersed with k=2, the maximum likelihood estimate of the reproduction number was 1.14. These estimates agreed with those published from transmission network analysis, indicating that the human-to-human transmission potential of the pneumonic plague is not very high. Given only minor outbreaks, transmission potential is not sufficiently assessed by directly counting the number of offspring. Since the absence of a major epidemic does not guarantee a subcritical process, the proposed method allows us to conservatively regard epidemic data from minor outbreaks as supercritical, and yield estimates of threshold values above unity.

On estimating critical population size for an endangered species in the presence of environmental stochasticity

Abstract A stationary second order autoregressive process with Gaussian noise, which was linked to survivorship and reproductive success by logistic transformations, was used as a model for an environmental process. Computer experiments in Monte Carlo integration, with the objective of exploring the sensitivity of estimates of mean critical population size to variations in the parameters of the environmental process, were then conducted. These experiments suggest that estimates of mean critical population size are very sensitive to the form of the autocorrelation function of the stationary environmental process. For the most part, those experiments in which the autocorrelation function was strictly positive not only resulted in the largest estimates of mean critical population size but also led to the highest levels of environmental stochasticity as measured by its coefficient of variation. As in previous work, these experiments suggest that concerted efforts should be made to model those environmental factors that are critical to the survivability of an endangered species in assessing its chances for continued existence.

A study of the impact of environmental stochasticity on extinction probabilities by Monte Carlo integration

Abstract An environmental process was characterized by a stationary second order autogressive process with Gaussian noise. This process was then linked to survivorship and reproductive success by logistic transformations. The sensitivity of extinction probabilities to variations in the parameters of the environmental process was studied by computer experiments in Monte Carlo integration. Against the background of the rather limited number of fertility and mortality levels studied in these experiments, the extinction probabilities were demonstrated to be quite sensitive to variations in the parameters of the environmental process. Although more extensive experiments will need to be carried out, those conducted so far suggest that concerted efforts should be made to model those environmental factors that are critical to the survivability of an endangered species in assessing its chances for continued existence.

Computational Methods for Renewal Theory and Semi-Markov Processes with Illustrative Examples

Abstract Some illustrative applications of semi-Markov processes in biostatistics, demography, and queuing theory are discussed. Algorithms for implementing such processes on a computer are also described, with suggestions for statistical inference based on models constructed from renewal theory and semi-Markov processes. An illustrative numerical example, based on a simple illness-death process of Fix and Neyman, is also provided.

A non-Markovian stochastic model for the Taichung medical IUD experiment

Abstract A non-Markovian stochastic model is developed and validated within the context of data from the Taichung Medical IUD Study. The principles of the more traditionally used semi-Markov processes are first set down, along with the presentation of a computing algorithm to handle a key calculation arising in connection with these processes. Then, the set of data from the Taichung study is described, and an indication is given as to why a Markovian process is inappropriate for modeling the behavior reported in these data. Using semi-Markov processes to model various portions of the reproductive process, an overall non-Markovian model is developed with a view towards the contraceptive-pregnancy history of an individual as a sequential decision process. Some calculations based on this model are presented along with empirical determinations from the data reflecting what actually occurred. A comparison of these calculations and determinations shows that the model seems to describe the behavior reported quite well.

Perspectives in stochastic models of human reproduction: a review and analysis.

Abstract The purpose of this paper is to review, to analyze, and to take steps toward synthesizing, two research areas in stochastic models of population dynamics. The first of these areas consists of stochastic models of human reproduction associated with the late Mindel C. Sheps and the second area is a class of stochastic models of population growth called generalized age-dependent branching processes, a class of models that shows promise of throwing more light on the classical mathematical demography of Lotka. The substantive material of the paper is arranged in eight sections ranging in content from a comparison of classical mathematical demography with generalized age-dependent branching processes, to suggestions for restructuring models of the Sheps school in quest of greater realism. The paper ends with a section on numerical examples illustrating applications in family planning evaluation and an appendix suggesting ways in which algebraic concepts are useful in short cutting computations.

An age-parity dependent stochastic model of human reproduction

Abstract A terminating age-parity dependent model of human reproduction is formulated and studied with a view toward the development of more realistic quantitative methods in family planning evaluation. The results reported in this paper, which extend those reported previously by Mode (1972, 1974), are divided into seven sections detailing the construction of the components of the model and computer algorithms. Four novel features of the paper are (i) the construction of a terminating Markovian renewal process, (ii) the incorporation of turning points into the reproductive process, (iii) the incorporation of risks of becoming sterile into the distributions of the waiting times to conception, and (iv) the development of an algorithm for computing the sum of an infinite geometric series in Markovian convolutions exactly in finitely many steps.