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AIDS. 3. Modeling contact structure
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
SPREAD
239
OF EPIDEMICS
prevalence of HIV infection among females ranged from about 10 to 15%, depending in part on the expected duration of partnerships and whether the frequencyor 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.
THE EFFECT OF STRUCTURAL BEHAVIOR CHANGE ON THE SPREAD OF HIV IN A ONE-SEX POPULATION PAOLO
SCALIA-TOMBA,
Statens
Bakteriologiska
Laboratorium,
Stockholm,
A simple deterministic SIR model, with two sexual activity preferred mixing, is extended to represent “structural behavior allowing people to move between activity classes independently status. Some aspects of this model, such as equilibria and local discussed and compared to results for other models.
Sweden
classes and change,” by of infection stability, are
STOCHASTIC MODELING ON RANDOM GRAPHS OF THE SPREAD OF SEXUALLY TRANSMITTED DISEASES PHILIPPE
BLANCHARD,
University
of Bielefeld,
Bielefeld,
West Germany
A model is introduced for the epidemic dynamics of sexually transmitted diseases that incorporates the structure of sexual contacts inside a society as a random graph and reflects the inherent stochasticity of the transmission dynamics using a discrete stochastic process. Analytical results (threshold theorems for models generated by independent matchings) are discussed, and results from several simulations with varied epidemiological parameters, including time-dependent infectivity and certain prevention scenarios, are presented. Moreover, the results are contrasted with those obtained from a simple differential equation model to underline the need for proper inclusion of the contact structure and to discuss qualitative differences in predictions. Blanchard, P., G. F. Bolz, and T. Kruger, Modeling AIDS epidemics or any venereal disease on random graphs, Lecture Notes Biomath. 86:104-117 (1990). Bolz, G. F., P. Blanchard, and T. Kruger, Stochastic modeling on random graphs of the spread of sexually transmitted disease, in Progress in AIDS-Research in the Federal Republic of Germany, M. Schauzu, Ed., MMV-Verlag, Munchen, 1990.
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
SPREAD
239
OF EPIDEMICS
prevalence of HIV infection among females ranged from about 10 to 15%, depending in part on the expected duration of partnerships and whether the frequencyor 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.
THE EFFECT OF STRUCTURAL BEHAVIOR CHANGE ON THE SPREAD OF HIV IN A ONE-SEX POPULATION PAOLO
SCALIA-TOMBA,
Statens
Bakteriologiska
Laboratorium,
Stockholm,
A simple deterministic SIR model, with two sexual activity preferred mixing, is extended to represent “structural behavior allowing people to move between activity classes independently status. Some aspects of this model, such as equilibria and local discussed and compared to results for other models.
Sweden
classes and change,” by of infection stability, are
STOCHASTIC MODELING ON RANDOM GRAPHS OF THE SPREAD OF SEXUALLY TRANSMITTED DISEASES PHILIPPE
BLANCHARD,
University
of Bielefeld,
Bielefeld,
West Germany
A model is introduced for the epidemic dynamics of sexually transmitted diseases that incorporates the structure of sexual contacts inside a society as a random graph and reflects the inherent stochasticity of the transmission dynamics using a discrete stochastic process. Analytical results (threshold theorems for models generated by independent matchings) are discussed, and results from several simulations with varied epidemiological parameters, including time-dependent infectivity and certain prevention scenarios, are presented. Moreover, the results are contrasted with those obtained from a simple differential equation model to underline the need for proper inclusion of the contact structure and to discuss qualitative differences in predictions. Blanchard, P., G. F. Bolz, and T. Kruger, Modeling AIDS epidemics or any venereal disease on random graphs, Lecture Notes Biomath. 86:104-117 (1990). Bolz, G. F., P. Blanchard, and T. Kruger, Stochastic modeling on random graphs of the spread of sexually transmitted disease, in Progress in AIDS-Research in the Federal Republic of Germany, M. Schauzu, Ed., MMV-Verlag, Munchen, 1990.