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A problem of importance in modelling epidemics of sexually transmitted diseases is the development of mathematical structures accommodating sexual and other contacts among members of a population. Because these models may be complex, it is often necessary to use computer intensive methods in their analysis, which raises questions on the design of computer models. In this paper a new approach to designing models sexual contacts is presented within the context of a stochastic model accommodating the formation and dissolution of partnerships in heterosexual populations. Emphasis will be placed on the development of algorithms with a view towards developing software to implement computer intensive methods. Unlike previous formulations, rather than using rejection methods in Monte Carlo simulations to impose necessary constraints on random functions describing partnership formation, in the new formulation all constraints are satisfied with probability one.

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In this paper, branching process approximations to non-linear stochastic partnership models for sexually transmitted diseases in heterosexual populations were used to find points in the parameter space such that an epidemic would occur. At selected points in the parameter space, samples of Monte Carlo realizations of the process were computed and analyzed statistically to gain insights into the stochastic evolution of epidemics seeded by one infective single female and male. Non-linear difference equations were embedded in the stochastic processes, making it possible to compare trajectories computed according to the deterministic model with those computed from samples of Monte Carlo realizations. From these trajectories it was shown that stochastic fluctuations may have a profound effect on the long-term evolution of an epidemic, and examples demonstrate that an investigator may be misled if a deterministic model alone were used to project an epidemic, particularly when there is a significant probability of extinction.

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A simple stochastic model describing an epidemic of a sexually transmitted disease, accommodating the formation and dissolution of partnerships, was formulated within a two-type Crump-Mode-Jagers (CMJ) process in continuous time. A submodel, describing the formation and dissolution of partnerships, was formulated in terms of a semi-Markov process with a finite state space and death as an absorbing state. This model was then linked to a two-type CMJ process through offspring distributions. Numerical values of threshold parameters greater than one suggests that either the epidemic dies out with a probability significantly different form zero or it spreads explosively in the population.

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Many deterministic models of sexually transmitted diseases, as well as population models in general, contain elements of stochastic or statistical reasoning. An example of such a model is that of Dietz and Hadeler (1988) concerning sexually transmitted diseases in which there is partnership formation and dissolution. Among the interesting formulas in this paper, which enter into the analysis of the model, are those for the expected number of partners a male or female has during a lifetime. To a probabilist such formulas suggest the possibility that some stochastic process may be constructed so as to yield these formulas as well as others that may be of interest. The principal purpose of this paper is to demonstrate that such a stochastic process does indeed exist in the form of a three state semi-Markov process in continuous time with stationary laws of evolution and with a one-step density matrix determined by four parameters which were interpreted as constant latent risk functions in the classical theory of competing risks. This construction of a semi-Markov process not only provides a framework for the systematic derivation of the formulas of Dietz and Hadeler but also suggests pathways for extensions to the age-dependent case.

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A two-sex age-dependent demographic model is formulated within the framework of a stochastic population process, including both time-homogeneous and time-inhomogeneous laws of evolution. An outline of the parametric components of the system, which expedite computer implementation and experimentation, is also given. New features of the model include a component for couple formation, using the class of Farlie-Morgenstern bivariate distributions to accommodate age preferences in selecting marriage partners, a component for couple dissolution due to separation or divorce, and an outline of techniques for initializing a two-sex projection given scanty information. For the case of time-homogeneous laws of evolution, stability properties of two-sex models that are analogs of those for one-sex models are difficult to prove mathematically due to nonlinearities. But computer experiments in this case suggest that these properties continue to hold for two-sex models for such widely used demographic indicators as period crude birth rates, period rates of natural increase, and period age distributions, which converge to constant forms in long-term projections. The values of the stable crude birth rate, rate of natural increase, and quantiles of the stable age distribution differ markedly among projections that differ only in selected values of parameters governing couple formation and dissolution. Such experimental results demonstrate that two-sex models are not merely intellectual curiosities but exist in their own right and lead to insights not attainable in simpler one-sex formulations.

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The goodnesses of fit over the entire life span of four models of mortality are compared using life tables from Australia and the United States. The results indicate that the five-parameter Siler model fits considerably better than the more complex eight-parameter Heligman-Pollard and Mode-Busby models. On the other hand, the ten-parameter model proposed by Mode and Jacobson fits human mortality patterns better than the Siler model. We conclude that the Heligman-Pollard and Mode-Busby models are probably misspecified. Additional research is necessary to determine (1) whether the Heligman-Pollard model can be improved by specifying it as a true hazard model and (2) whether the respecified Heligman-Pollard and Mode-Jacobson models are statistically robust, particularly with abridged life tables.

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A non-age-dependent model, describing the evolution of a bisexual population, is developed in this paper and applied to projecting an AIDS epidemic in a heterosexual population. Included in the formulation are frequency- and non-frequency-dependent rules of partnership formation as well as five states of HIV disease, affecting the probability of infection per sexual contact. Results from computer experiments, designed to study the development of an AIDS epidemic in a heterosexual population fed by single males with a 50% prevalence of HIV infection prior to becoming active in heterosexual partnerships, are reported. In these experiments, the only source of HIV infection for females was sexual contacts with infected males within partnerships. Data on the probability of infection per sexual contact with an infected partner and the number of sexual contacts per month were incorporated into the model. However, the numbers used for the initial population of singles, couples, and those becoming sexually active per month were hypothetical. Even though the prevalence of HIV infection among males entering heterosexual partnerships was high, after 30 years the projected prevalence of HIV infection among females ranged from about 10 to 15% depending in part on the expected duration of partnerships and on whether the frequency- or non-frequency-dependent model was used. In these experiments, solutions of the embedded, nonlinear, deterministic equations for the incidence of HIV infection and the cumulative number of deaths due to AIDS proved to be good measures of central tendency for the sample functions of the stochastic population process.

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We designed and implemented stochastic methods for short term projections of HIV disease at the local level, that accommodate various states or stages of the disease. We gave particular attention to projection of the number of patients with HIV disease who need care, when durations of stay in these various states depend on current methods for treating opportunistic infections. We consider two types of data as input to these projections. One concerns seroprevalence surveys conducted over time and from which we can obtain time series estimates of the numbers of HIV-infected individuals. The other is a reported time series of AIDS cases adjusted for delays in reporting. Several projections, with data from the City of Philadelphia, illustrate this method. In addition, we consider a Monte Carlo method for computing confidence bounds on a projection.

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A nonlinear stochastic model of an AIDS epidemic with recruitment of infectives, susceptibles, and AIDS cases into a randomly mixing population of male homosexuals was formulated and studied from a methodological point of view through intensive computer experimentation. Probability generating functions were used to formulate a model for the monthly probability that a susceptible individual becomes infected with HIV, under the assumption that the probability of infection per sexual contact varies as a function of the duration of infection. A method for taking into account the use of condoms to prevent infection with HIV was also introduced. Nonlinear difference equations, resembling deterministic epidemic models, were embedded in the stochastic population process by iterating an initial conditional expectation. Examples of Monte Carlo experiments are presented, illustrating that solutions of these nonlinear difference equations are not always good measures of central tendency for variations in the sample functions of the process. Two important substantive conclusions drawn from the Monte Carlo experiments were that efforts should be made to collect quantitative information on the probability of infection per sexual contact as a function of duration of infection and the frequency of condom use within and among risk categories in a population.

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The continuing revolution in personal computers has placed computing power rivaling that of some computational laboratories on the desktops of many investigators. Contained in this article are some suggestions for taking advantage of this enhanced computing power in analyzing data in which an investigator does not have control over his sample. Sample reuse methods, called the jackknife and the bootstrap, are discussed, along with possible applications of time-series methods. Some suggestions for getting help from mathematical scientists are also included.

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"Algorithms for a stochastic population process, based on assumptions underlying general age-dependent branching processes in discrete time with time inhomogeneous laws of evolution, are developed through the use of a new representation of basic random functions involving birth cohorts and random sums of random variables. New algorithms provide a capability for computing the mean age structure of the process as well as variances and covariances, measuring variation about means. Four exploratory population projections, testing the implications of the algorithms for the case of time-homogeneous laws of evolution, are presented. Formulas extending mean and variance functions for unit population projections...are also presented. These formulas show that, in population processes with non-random laws of evolution, stochastic fluctuations about the mean function are negligible when initial population size is large. Further extensions of these formulas to the case of randomized laws of evolution suggest that stochastic fluctuations about the mean function can be significant even for large initial populations."

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Recent developments in a class of stochastic population processes were used to study the impact of demographic heterogeneity on the uncertainty of population projections. Selected for study by computer simulation were population projections for East Africa designed to quantify opinions regarding expected fertility and mortality declines. Since both fertility and mortality declined in these projections, their laws of evolution may be described as time inhomogeneous. The computer simulation studies reported in this paper strongly suggest that randomized laws of evolution should be taken into account in further developments of population projection methodologies designed formally and computationally to accommodate uncertainty. Variability in fecundability, the kind of demographic heterogeneity studied in this paper, is only one aspect of these randomized laws.

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This paper provides algorithms for projection of mean and covariance functions for stochastic population processes governed by time-homogeneous laws of fertility and mortality. The theoretical foundation of the algorithms is general age-dependent branching processes in discrete time. The algorithms are employed in several illustrative projections, based on 1982 Chinese data, of a population experiencing an abrupt transition to below replacement fertility. Methods of constructing confidence limits for total population size are illustrated. Also developed are procedures for projecting mean and variance functions for populations which may be heterogeneous with respect to mortality or fertility. The projections performed yield two observations. First, the coefficient of variation in population size appears to be inversely related to the Malthusian parameter of population growth. Second, the coefficient of variation for population size is negligible for large homogeneous initial populations. But when the initial population is heterogeneous with respect to fertility or mortality, then substantial coefficients of variation, exceeding 0.4, are observed in some of the projections performed.

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A parametric algorithm was developed for computing model cohort and period survival functions used in making projections of human populations. Two levels of parameters were used in developing the algorithm. The algorithm provides a method for calculating model period survival functions as a function of an expectation of life at birth; model cohort survival functions are calculated as a function of a time series of period expectations of life at birth. Expectations of life at birth ranging from about 35 to 110 years in both sexes may be accommodated by the algorithm.

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In many studies of longevity in experimental animals, sample sizes are small enough to warrant the use of statistical tests of significance in making judgments of scientific interest. Illustrated in this paper is a nonparametric test procedure for deciding on the statistical significance of observed differences in two empirical survival functions. This procedure takes an experimenter one step beyond comparing means and medians. For experimenters who wish to pursue the subject further, a brief entry into the extensive literature on survival analysis is also provided.

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A mathematical overview of a stochastic computer simulation model of maternity histories is provided. Various components of human reproduction are accommodated in the model through distributions of waiting times among live births. Included in these components are distributions of age at first marriage in a cohort of women, waiting times to pregnancy for fecundable women, and the lengths of infecundable periods following live births. Probabilities that pregnancies end in either a live birth, induced abortion, or some other type of outcome are also included. Elements of renewal theory and semi-Markov processes in discrete time were the basic mathematical concepts used in the construction of the model. A brief description of an interactive software package called MATHIST, which may be used to implement the model on a computer, is also included. Four illustrative computer runs with MATHIST, pertinent to the operation of family planning programmes in Africa, are also described and discussed.

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PIP: This paper describes and tests a model for constructing data-based estimates of probabilities of pregnancy following initial acceptance of a contraceptive method. The report outlines the basic features of the model, describes the sampling framework of the 1973 U.S. National Survey of Family Growth (NSFG) against whose data the model is tested, and explains estimation procedures. The model summarizes the behavior of a cohort of continuously married couples from acceptance of a contraceptive method, including nonuse, pill, other methods, or IUD, until the 1st subsequent pregnancy, marital dissolution, or sterility. Estimated probabilities of these outcomes are presented and compared to 1-step transition probabilities and directly observed NSFG data on pregnancy, marital dissolution, or contraceptive sterilization in order to assess the validity of the model and the biases resulting from its use. The model was found to be more successful for transitions to marital dissolution or contraceptive sterilization than for transitions to pregnancy. Possible reasons for this include small sample size in some categories and inappropriateness of some assumptions underlying the model.^ieng

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PIP: The demographic effectiveness of a family planning program must measure the difference between actual fertility and the "gross" potential fertility, i.e., the natural fertility which would have been achieved without use of contraception. The various methods of measuring "gross" fertility which have been used are described. It is understood that users of contraception are never a random sampling of the general population; they tend to be higher in fecundability and lower in proportion of sterile. For this reason, the best strategm of measurement involves utilization of preacceptance fertility rates over some given period of time among those reporting no contraceptive usage during that period. A procedure of measurement is proposed whereby natural fecundability is estimated from null segments. Possible biases are taken into account. Application of the method is made to 2 sets of data collected in Taiwan. Results of the application of the method to the Taiwanese data are tabulated. Fecundability functions derived from unrestricted null segments are different from those constructed from restricted subsets of "closed" null segments. Choice of whether to use restricted or unrestricted null segments will depend on the nature of the data available.^ieng

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The robustness of the product life table estimator of the survival function was studied for large populations under perturbations in the age distribution, changing levels of mortality and changing patterns of fertility. A macrosimulation system, based on a class of stochastic population models called generalised age-dependent branching processes, was used to carry out the numerical investigations. Aside from drastic perturbations in the age distribution and changes in levels of mortality, the product life table estimator of the survival function was found to be robust in large populations, under a variety of conditions.

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