AIDS. 3. Modeling contact structure

SPREAD

237

OF EPIDEMICS

represent a wide range of hypotheses regarding the sociological dynamics of mixing and observe the impact on disease outcomes. Two examples based on sexually transmitted AIDS are presented. The first assumes that people target partners on the basis of unchanging characteristics (gender and sexual preference), and the second assumes that people target on the basis of changing characteristics (age). Very different patterns of disease diffusion are shown to result. In the first case, transmission from the seeded group to other populations can be difficult or impossible; in the second case, it is virtually certain.

TOWARD STEPHEN (with Carlos

A UNIFIED BLYTHE,

THEORY

Strathclyde

Castillo-Chavez,

OF PAIR FORMATION

University,

Jeff Palmer,

Glasgow,

and Mingyang

Scotland

Cheng)

Sexually transmitted diseases such as gonorrhea, syphilis, herpes, and AIDS are driven and maintained by epidemiological and sociological factors that are not completely understood. Despite the fact that the processes of pair formation (or social/sexual mixing) and dissolution play a crucial role in disease dynamics, their incorporation into epidemiological models is quite recent. In this paper we present a unified approach to pair formation for a population with an arbitrary number of “sexes.” A new derivation is provided of the mixing formula of Blythe, Busenberg, and Castillo-Chavez, and special cases such as two- and one-sex mixing are discussed. We further illustrate how some of the mixing formulas that have appeared in the literature fit into our framework. Finally, we outline the results of some stochastic simulations and compare the averages of the simplest of these to deterministic solutions-proportionate mixing and Ross’s solutions. Castillo-Chavez, C. and S. P. Blythe, Mixing framework for social/sexual behavior, Lecture Notes Biomath. 83:275-288 (1989). Busenberg, S. and C. Castillo-Chavez, Interaction, pair formation and force of infection terms in sexually transmitted diseases, Lecture Notes Biomath. 83:289-300 (1989).

ON THE COMPUTATION OF THE BASIC REPRODUCTION RATIO R, FOR A CLASS OF INFECTIOUS DISEASE MODELS WITH PAIR FORMATION HANS HEESTERBEEK, Centre for Mathematics Amsterdam, Netherlands (with Odo Diekmann and Klaus Dietz)

and Computer

Science,

Diekmann et al. (1990) shows that the mathematical way to “define” R, is as the spectral radius of a certain operator that describes the transmis-

238

RESEARCH

WORKSHOP

sion dynamics of a disease as a discrete process on subsequent generations of infected individuals. We show what this operator looks like if we take pair formation and separation into account and if we recognize a finite but arbitrary number of possible disease states that an infected individual “moves through” in natural order. The disease state of an individual determines the individual’s current infectivity level. For the pair formation process we follow the simplest model described in Dietz and Hadeler (1988). We describe the model assumptions, explain the ingredients necessary to determine R,, and give the algorithm for its calculation. The operator is an II x n matrix, if we disregard heterogeneity in susceptibility, where II is the number of disease states with positive infectivity. R, is then the dominant eigenvalue of this matrix. These results can be generalized to include heterogeneity in susceptibility. We indicate how, by the right limit procedure, one can “eliminate” the pair formation in our model and in this way recover expressions for R, for non-pair-formation multistage models that were already known. Diekmann,

O., J. A. P. Heesterbeek,

and J. A. J. Metz,

On the definition

calculation of the basic reproduction ratio R, in models for infectious heterogeneous populations,” J. Math. Bid. 28:365-382 (1990). Dietz, K. and K. P. Hadeler, Epidemiological models for sexually transmitted Math. Bid. 26:1-25 (1988).

A STOCHASTIC MODEL WITH PAIR FORMATION CHARLES

MODE,

Drexel

and

the

diseases

in

diseases,

J.

OF AN AIDS EPIDEMIC AND DISSOLUTION

University,

Philadelphia,

Pennsylvania

A non-age-dependent model describing the evolution of a bisexual population is developed and is 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 contact 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. But 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