Session 6: Stochastic Models in Epidemiology ORGANIZER: Charles Mode
	DESTOBIO 2000 August 23-27, 2000 West Lafayette, Indiana, USA

The papers in this session will focus on stochastic models of infectious diseases, including HIV/AIDS and other infectious diseases. Several of the papers will employ computer intensive methods, including Monte Carlo simulations based on rigorous mathematical formulations.

LIST OF SPEAKERS (in alphabetical order)

Robert Gallop, University of Pennsylvania, Philadelphia, PA: "Determination of Threshold Conditions for a Non-Linear Stochastic Partnership Model for Heterosexually Transmitted Diseases with Stages"

ABSTRACT: To properly understand the modeling structure of an epidemic, it is imperative to understand and capture the characteristics impacting the epidemic's behavior in the population. \ A stochastic model with sufficient parameters describing the behavior of the epidemic was used. \ By embedding non-linear difference equations in the stochastic process in discrete time, a more thorough understanding of the epidemic was achieved. \ To visually enhance the investigation of the epidemic's behavior, comparison of trajectories of the deterministic model and those computed from the samples Monte Carlo realizations were made. \ In this modeling of the heterosexual disease, it is important for the mathematical/statistical structure to accommodate sexual and other contacts among members of the population. \ In this structure, biased partnership selection may arise. \ Threshold conditions are sensitive functions of this and other parameters in the model. \ To derive threshold conditions, non-linear differential equations were derived from the non-linear difference equations. \ Threshold conditions are sensitive functions of this and other parameters in the model. \ To derive threshold conditions, non-linear differential equations were derived from the non-linear difference equations. \ Threshold conditions were determined by investigation of the stability of the Jacobian matrix for the embedded system of non-linear differential equations. \ Threshold conditions for the model were formulated and the sensitivity of these conditions were analyzed under slight deviations of the parameter space. \ Provided are examples of this methodology applied to the HIV/AIDS epidemic in the heterosexual community. \ Comparison of the behavior of the two modeling structures, stochastic and deterministic, with respect to the threshold conditions are also investigated.

Jake Kesinger, Texas Tech University, Lubbock, TX: "Comparison of Some Discrete-Time Deterministic and Stochastic Epidemic Models"

ABSTRACT: This presentation is divided into two parts. In the first part of the presentation, deterministic and stochastic, discrete-time, SIS and SIR models are analyzed and compared. The stochastic models are Markov chains. Models with constant population size and general force of infection are analyzed, then a more general SIS model with variable population size is analyzed.
In the second part of the presentation, discrete-time models for the evolution of infectious diseases in plant pathosystems are analyzed. Deterministic and stochastic models based on the gene-for-gene hypothesis are developed. The evolution of plant resistance and pathogen virulence is studied.

Steve Marion, University of British Columbia, Vancouver, Canada: "Recursive Computation of the Final Size Distribution for Epidemics in Heterogeneous Populations"

ABSTRACT: An efficient and well-behaved algorithm for the recursive calculation of the size distribution is available for the closed SIR (susceptible-infectious-removed) model. It is known that latency between infection and infectivity does not alter the final distribution. Hence if we restrict attention to the final distribution, the same algorithm applies to the closed SLIR model (susceptible, latent, infectious removed). However, several strong restrictions remain: no new susceptibles, no loss of susceptibles except through infection, exponentially distributed waiting times between events, and homogeneous mixing (all susceptibles are at the same risk of infection from a given infectious person). We are able to calculate the final size distribution using modifications of the same algorithm for models that violate one or more of these restrictions. Our approach is to consider "size equivalent" models, models different from the model of interest but which have the same final distribution (independence of latency being one example). We also start from a much more general model that tracks individual level events and allows variation in susceptibility, infectivity and in mixing. Using exchangeability in place of homogeneous mixing, and size equivalence, we are able to calculate the size distribution of many models of practical interest. We are also able to calculate approximations to the transient distribution.

Candace K. Sleeman, Drexel University, Philadelphia, PA: "Computer Implementation of Stochastic Partnership Models for HIV/AIDS"

ABSTRACT: Models accommodating partnerships and heterogeneity with respect to behavioral risk classes were implemented and used to study the evolution of epidemics in various populations. For sexually transmitted diseases with multiple stages like HIV/AIDS, the selection of sexual partners according to disease stage was considered. Computer intensive experimentation was the goal, with a more complete use of latent risk functions and competing risks governing transitions to the infected state than in earlier models. The mathematical structure was used to make connections between the stochastic process and a system of nonlinear differential equations embedded in the process, thus enabling a search for threshold conditions for the stochastic process.

Wai-Yuan Tan, University of Memphis, Memphis, TN: "Stochastic Models of HIV Pathenogenesis under HAART and Development of Drug Resistance"

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