On Deterministic and Stochastic Multiple Pathogen Epidemic Models

In this paper, we consider a stochastic epidemic model with two pathogens. In order to analyze the coexistence of two pathogens, we compute numerically the expectation time until extinction (the mean persistence time), which satisfies a stationary partial differential equation with degenerate variable coefficients, related to backward Kolmogorov equation. I use the finite element method in order to solve this equation, and we implement it in FreeFem++. The main conclusion of this paper is that the deterministic and stochastic epidemic models differ considerably in predicting coexistence of the two diseases and in the extinction outcome of one of them. Now, the main challenge would be to find an explanation for this result.

The nonlocal boundary value problem for one-dimensional backward Kolmogorov equation and associated semigroup

This paper is devoted to a partial differential equation approach to the problem of construction of Feller semigroups associated with one-dimensional diffusion processes with boundary conditions in theory of stochastic processes. In this paper we investigate the boundary-value problem for a one-dimensional linear parabolic equation of the second order (backward Kolmogorov equation) in curvilinear bounded domain with one of the variants of nonlocal Feller-Wentzell boundary condition. We restrict our attention to the case when the boundary condition has only one term and it is of the integral type. The classical solution of the last problem is obtained by the boundary integral equation method with the use of the fundamental solution of backward Kolmogorov equation and the associated parabolic potentials. This solution is used to construct the Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle leaves the boundary of the domain by jumps.

Singular perturbations in noisy dynamical systems - CORRIGENDUM

On the seventh line of the fifth page of the article the Backward Kolmogorov equation should read pt = Lp, and not ps = Lp.The author apologises for the error.

DISPERSIVE GROUNDWATER FLOW AND POLLUTION

By solving asymptotically the Dirichlet problem for the backward Kolmogorov equation describing the random walk of a particle in a dispersive flow, it is computed at what rate contaminated particles cross the boundary of a protected zone. The method also yields an estimate of the expected arrival time.

Solution to the Backward-Kolmogorov Equation for a Nonstationary Oscillation Problem

The solution of a backward-Kolmogorov equation is presented. This equation is associated with a Markov approximation of the response amplitude of a lightly damped linear oscillator driven by an evolutionary random excitation.