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Maximal Sensitive Dependence and the Optimal Path to Epidemic Extinction

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Abstract

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.

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Authors and Affiliations

  1. Nonlinear Systems Dynamics Section, Plasma Physics Division, Code 6792, US Naval Research Laboratory, Washington, DC, 20375, USA

    Eric Forgoston & Ira B. Schwartz

  2. Department of Applied Science, The College of William & Mary, P.O. Box 8795, Williamsburg, VA, 23187-8795, USA

    Simone Bianco & Leah B. Shaw

Authors
  1. Eric Forgoston
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  2. Simone Bianco
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  3. Leah B. Shaw
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  4. Ira B. Schwartz
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Correspondence to Eric Forgoston.

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Forgoston, E., Bianco, S., Shaw, L.B. et al. Maximal Sensitive Dependence and the Optimal Path to Epidemic Extinction. Bull Math Biol 73, 495–514 (2011). https://doi.org/10.1007/s11538-010-9537-0

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