Maximal Sensitive Dependence and the Optimal Path to Epidemic Extinction
Extinction of an epidemic or a species is a rare event that occurs due to a large, rare stochastic fluctuation. Although the extinction process is dynamically unstable, it follows an optimal path that maximizes the probability of extinction. We show that the optimal path is also directly related to the finite-time Lyapunov exponents of the underlying dynamical system in that the optimal path displays maximum sensitivity to initial conditions. We consider several stochastic epidemic models, and examine the extinction process in a dynamical systems framework. Using the dynamics of the finite-time Lyapunov exponents as a constructive tool, we demonstrate that the dynamical systems viewpoint of extinction evolves naturally toward the optimal path.
KeywordsStochastic dynamical systems and Lyapunov exponents Optimal path to extinction
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- Anderson, R. M. & May, R. M. (1991). Infectious diseases of humans. Oxford: Oxford University Press. Google Scholar
- Bartlett, M. S. (1961). Stochastic population models in ecology and epidemiology. New York: Wiley. Google Scholar
- Guckenheimer, J. & Holmes, P. (1986). Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Berlin: Springer. Google Scholar
- Schwartz, I. B., Billings, L., Dykman, M., & Landsman, A. (2009). Predicting extinction rates in stochastic epidemic models. J. Stat. Mech.—Theory E, P01005. Google Scholar
- Schwartz, I. B., Forgoston, E., Bianco, S., & Shaw, L. B. (2010). Converging towards the optimal path to extinction. Submitted. Google Scholar
- Shaw, L. B. & Schwartz, I. B. (2010). Enhanced vaccine control of epidemics in adaptive networks. Phys. Rev. E. In press. Google Scholar
- van Kampen, N. G. (2007). Stochastic processes in physics and chemistry. Amsterdam: Elsevier. Google Scholar
- Wentzell, A. (1976). Rough limit theorems on large deviations for Markov stochastic processes, I. Theor. Probab. Appl., 21, 227–242. Google Scholar