Extinction of an infectious disease: A large fluctuation in a nonequilibrium system

Abstract

We develop a theory of first passage processes in stochastic non-equilibrium systems of birthdeath type using two closely related epidemiological models as examples. Our method employs the probability generating function technique in conjunction with the eikonal approximation. In this way the problem is reduced to finding the optimal path to extinction: a heteroclinic trajectory of an effective multi-dimensional classical Hamiltonian system. We compute this trajectory and mean extinction time of the disease numerically and uncover a non-monotone, spiral path to extinction of a disease. We also obtain analytical results close to a bifurcation point, where the problem is described by a Hamiltonian previously identified in one-species population models.PACS numbers: 87.23.Cc, 02.50.Ga Statistics of large fluctuations in stochastic nonequilibrium systems has received much recent attention [1]. While the equilibrium fluctuation probability is determined by the Boltzmann distribution, there is no similar general principle away from equilibrium. The underlying reason is the absence of time reversal symmetry between the relaxation and excitation dynamics in out-of-equilibrium systems. As a consequence, the most probable fluctuation path is not determined by the relaxation trajectory of the underlying deterministic system.An important class of stochastic non-equilibrium systems is reaction kinetics, or birth-death, systems [2]. Rather than being caused by external factors, the noise in these systems is intrinsic, as it originates from discreteness of the reacting agents and random character of their interactions. When the typical number of agents is large, the Fokker-Planck (FP) approximation to the master equation, see e.g. Ref.[2], can accurately describe small deviations from the probability distribution maxima. It fails, however, in determining the probability of large fluctuations [3,4,5]. Therefore, developing adequate theoretical tools for dealing with large fluctuations is an important task.One of the areas where the birth-death models have been very successful is mathematical epidemiology, see Refs. [6]. In this Letter we consider two closely related models of spread of disease in a population. Although they have served as standard multi-population epidemiological models, the analysis of large fluctuations in each of these models has not been satisfactory. We will use the two models as prototypical examples of multidimensional stochastic non-equilibrium systems.Observing the dynamics of a disease in a finite population, one notices the remarkable phenomenon of extinction of the disease in a finite time. The expected time to extinction (and the possibility to affect it) is of great practical interest. Here we develop an efficient theoretical approach capable of computing, among other things, this quantity. The approach employs the probability generating function formalism in conjunction with the eikonal approximation. In this way the problem is reduced to the dynamics of an effective classical Hamiltonian sys...

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