Heterogeneous sexual mixing in populations with arbitrarily connected multiple groups.

"A simple procedure for constructing [social/sexual] mixing models for arbitrarily classified (e.g. by sex, age, geographical location, sexual preference) populations is outlined, including a scheme for finding the number of independent mixing parameters required, and a simple (linear algebra) means for finding the values of the dependent mixing parameters. Various worked examples are presented, including the two-sex problem and structured and selective mixing." The use of the models for analyzing mixing structures for AIDS transmission is assessed. (SUMMARY IN FRE)

Empirical methods for the estimation of the mixing probabilities for socially structured populations from a single survey sample.

"The role of variability of sexual behavior in the transmission dynamics of HIV and AIDS has been illustrated, through the use of mathematical models, by several investigators.... In this paper we describe some practical methods for estimating the deviations from random mixing from a single survey sample.... We include a description of the role of the estimated mixing probabilities in models for the spread of HIV, a discussion of alternatives and possible extensions of the methods described in this article, and an outline of future directions of research." (SUMMARY IN FRE)

Toward a unified theory of sexual mixing and pair formation.

Sexually transmitted diseases such as gonorrhea, syphilis, herpes, and AIDS are driven and maintained in populations by epidemiological and sociological factors that are not completely understood. One such factor is the way in which people mix sexually. In this paper, we outline a unified approach to modeling sexual mixing structures, where such structures are defined in terms of a set of axioms for a finite number of distinct groups of people. Theorems for homosexual, heterosexual, and arbitrary group mixing are presented, leading to a representation of all mixing structures defined by the axioms. The representation and its parameters are interpreted in terms of intergroup affinities for sexual mixing. The use of the approach in sexually transmitted disease modeling is discussed.

Like-with-like preference and sexual mixing models.

Two new general methods for incorporating like-with-like preference into one-sex mixing models in epidemiology are presented. The first is a generalization of the preferred mixing equation, while the second comprises a transformation of a general preference function for partners of similar sexual activity levels. Both methods satisfy the constraints implicit in a mixing model. The behavior of the transformation preference method is illustrated, and it is compared with the standard proportionate mixing model.

The transmission dynamics of the human immunodeficiency virus type 1 in the male homosexual community in the United Kingdom: the influence of changes in sexual behaviour.

This paper examines the transmission dynamics of human immune deficiency virus type 1 (HIV-1) in the male homosexual population in the U.K. via numerical studies employing a mathematical model representing the principal epidemiological process. The model is based on an assumption of proportionate mixing between different sexual-activity classes (defined by the rate of sexual partner change per unit of time) and incorporates heterogeneity in sexual activity, distributed infection and incubation periods and the recruitment of susceptibles to the sexually active population. The sensitivity of model predictions to various assumptions and parameter assignments is examined. Numerical studies of model behaviour focus on the influence of changes in the magnitudes of the transmission parameters, associated with three periods of infectiousness during the incubation period of acquired immune deficiency syndrome (AIDS), on the magnitude and duration of the epidemic and on the level of the endemic equilibrium state. Predicted temporal trends in the incidence of AIDS are shown to be particularly sensitive to changes in the intensities and durations of the stages of infectiousness. Most of the paper addresses the influence of changes in sexual behaviour on the magnitude and duration of the epidemic. Numerical simulations show that the manner in which behavioural changes occur and who is influenced by such changes (i.e. infecteds or susceptibles, the sexually active population or new recruits to this population) have a major impact on the future timecourse of the epidemic. The greatest reduction in the incidence of AIDS over the coming decades is induced by changes in the rate of sexual-partner change among the sexually active population, particularly those currently infected. The time periods at which changes in behaviour occur, in relation to the starting point of the epidemic (assumed to be 1979), are also of particular significance to the future pattern of the incidence of disease and infection. Changes in behaviour early on in the timecourse of the epidemic have a much greater impact than equivalent changes at latter time points. On the basis of limited data on the pattern of change in sexual behaviour among the male homosexual community in the U.K., numerical studies of model behaviour tentatively suggest that the epidemic is at, or near to, a period of peak incidence of the disease AIDS. Analyses suggest that, following the peak in incidence, there will be a period of slow decline over many decades provided recent changes in behaviour are maintained in the coming years.(ABSTRACT TRUNCATED AT 400 WORDS)

Variable infectiousness in HIV transmission models.

Two different approaches to the encapsulation of temporal variation in the infectiousness of HIV-infected persons, and variability in the incubation period of the disease AIDS, in simple homogeneous mixing models of viral transmission in male homosexual communities are described. The first approach is based on the division of the infected population into a series of subclasses with differing levels of infectivity and different durations of occupancy. The second approach is more mechanistic in character and is based on an attempt to relate changes in viral abundance within an infected person to the likelihood that the disease AIDS develops. Variable incubation is induced by variation in the rate of change of viral abundance in the infected population. Numerical projections of changes in the incidence of AIDS through time, generated from both types of model, are compared with projections based on the assumption of constant infectivity throughout the incubation period of AIDS. Model formulation highlights areas in which more detailed quantitative epidemiological studies are required. Methods of parameter estimation and future research needs are discussed.

Distributed incubation and infectious periods in models of the transmission dynamics of the human immunodeficiency virus (HIV).

Distributions describing variation in the incubation and infectious periods of the human immunodeficiency virus (HIV) are derived from a series of risk or hazard functions. Four possible forms of the probability density function are considered, namely, exponential, Weibull, Erlang/gamma, and rectangular, and the properties and underlying risk functions are compared and contrasted. Models of the transmission dynamics of the virus, encapsulating different assumptions concerning the distributed incubation and infectious periods, are analysed, and their properties compared by steady-state and local-stability analyses and numerical methods.

Heterogeneous sexual activity models of HIV transmission in male homosexual populations.

A proportionate mixing one-sex model of sexual transmission of HIV is described, in which sexual activity (new partners per unit time) is defined as a continuous variable in a set of integro-partial-differential equations. A discrete activity-class approximation is developed by matching equilibrium state and rate variables as closely as possible with the continuous-variable model, and consists only of ordinary differential equations. Activity-class boundaries are arbitrary, and each class is characterized by a single level of activity. If there are N classes, the level of activity in N - 1 of them is such that the steady-state susceptible class sub-population is equal to the population in the equivalent section of the continuous model. The activity level for the remaining class is chosen such that the condition for endemicity of the infection in the approximation is equal to that for the equivalent continuous-variable model; this minimizes errors in the steady-state population. The relationship between the discrete and continuous-variable models is explored, via numerical and analytical studies, in order to evaluate the accuracy of the approximation.

Is it possible to predict the minimum size of the acquired immunodeficiency syndrome (AIDS) epidemic in the United Kingdom?

A mathematical model of the dynamics of transmission of human immunodeficiency virus within the male homosexual population in the United Kingdom demonstrates that even the minimum size of the acquired immunodeficiency syndrome (AIDS) epidemic in the United Kingdom (based upon the assumption that all transmission ceased at the end of 1986) is difficult to predict. Model predictions are particularly sensitive to assumptions about the distributed incubation period of the disease, differences in frequency and patterns of sexual activity, and the proportion of infected people in whom AIDS later develops. More accurate predictions will depend on the collection of data on the incubation period of the disease, the infectiousness of infected persons, and on the numbers of new sexual partners of each sex per person and the duration of each partnership.

Stage-structure models of populations with distinct growth and development processes.

We extend the repertoire of stage-structure models which can be described in terms of delay-differential equations, by analysing models where the processes of growth and development within a stage are distinct. This permits the use of delay-differential equation models in situations where both population numbers and total biomass are dynamically significant.

The dynamics of population models with distributed maturation periods.

An integro-differential equation for the dynamics of a subpopulation of adults in a closed system where only the adults compete and where there is a distribution of maturation periods is described. We show how the careful choice of a general weighting function based on the gamma distribution with a shift in origin enables us to characterize adequately some observed maturation-period distributions, and also makes local stability and numerical analyses straightforward. Using these results we examine the progression in the behavior of the distributed-delay model as the distribution is narrowed toward the limit of a discrete delay. We conclude that while local stability properties approach those of the limiting equation very rapidly, the persistent fluctuation behavior converges more slowly, with the dominant period and maximum amplitude being least affected by the details of the distribution, and the fine structure of solutions being most sensitive. Finally, we examine the consequences for population modeling, and using several examples of insect populations, conclude that although quite often a full maturation-period distribution should be incorporated in a given model, in many cases a discrete-delay approximation will suffice.