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Mathematics Colloquium: Dynamic and Control of Epidemic Processes
10:00 a.m. Neill 5W
Abstract: Epidemic processes are characterized by a basic interaction mechanism. Infected individuals interact with not yet infected persons rendering them infected. The individual propensity of getting infected, i.e. a micro quantity depends on macro variables. Such diffusion mechanism plays a crucial role in various different fields like health planning, marketing, social interactions and networks, neighboring behavior, development of social norms and deviant behavior. The inherent nonlinearities generate multiple long-run equilibria and related complex behavior of the optimal trajectories. The thresholds separating the various basins of attraction deliver interesting insight into the 'tipping' behavior of optimal solutions. The purpose of this paper is to present an intertemporal cost-benefit analysis of three kinds of epidemic processes. First a word-of-mouth two-state compartment model is presented, the number of satisfied and not-satisfied customers being the state variables. Both groups spread messages influencing the buying behavior of potential customers in a different manner. The model is validated by data of parties of the student union of the Vienna University. Secondly, we present a larger case study, namely the US cocaine epidemic. At the strategic level, drug policy can be viewed as a resource allocation problem: How should scarce resources be divided among competing drug control programs? Drug problems are inherently dynamic, evolving over time with significant nonlinear feedback effects. Thus, one might expect the optimal mix of drug control interventions to vary over time. The lecture investigates that possibility for a couple of models of drug use with optimal control theory. The third application is localized at the intersection of population dynamics, epidemiology and health management. It deals with the dynamics and the control of the HIV/AIDS pandemic. Governments in developing countries are spending a lot of money on antiretroviral medication programs in hope that the programs curb the spread of HIV/AIDS and reduce the suffering of those infected. This makes the question of determining the optimal mix of prevention and medication ever more pressing. We construct and parameterize a model of the spread of HIV/AIDS in Botswana that allows us to compute the balance of costs and benefits associated with various policy mixes.