Analysis of a spatially inhomogeneous stochastic partial differential equation epidemic model

Abstract

This work proposes and analyzes a family of spatially inhomogeneous epidemic models. This is our first effort to use stochastic partial differential equations (SPDEs) to model epidemic dynamics with spatial variations and environmental noise. After setting up the problem, the existence and uniqueness of solutions of the underlying SPDEs are examined. Then, definitions of permanence and extinction are given, and certain sufficient conditions are provided for permanence and extinction. Our hope is that this paper will open up windows for investigation of epidemic models from a new angle.

Stochastic Partial Differential Equation SIS Epidemic Models: Modeling and Analysis

“…By [33,Lemma 3.2], we obtain the positivity of S * (t), I * (t). As a consequence, (2.5) has a unique positive mild solution in L p (Ω; C([0, T ], E)) with p being sufficiently large.…”

“…Moreover, this solution depends continuously on the initial value.Proof. The Theorem can be proved in the spirit of that of[33, Theorem 3.1]. For convenience and completeness, we present a sketch of the proof.…”

Threshold dynamics for a class of stochastic SIRS epidemic models with nonlinear incidence and Markovian switching

“…where α 1 , α 2 , α 3 are constants. Notice that the incidence above combines many general forms existing in the literature (see, [4,5,11,27,34]). Then, we let the parameter values in model (1.2) as follows:…”

Stochastic partial differential equation models for spatially dependent predator-prey equations

“…Because we can not apply Itô's formula to the mild solution as usual, it is very difficult to calculate and estimate. Following our idea in [25], we approximate the mild solution (U (t), V (t)) of (2.4) by a sequence of strong solutions (see [11] for more details about strong solutions, weak solutions, and mild solutions). Consider the following equation…”

“…That is, the consideration of extinction or permanence. Since the mild solution is used, let us modify some definitions in [25] as follows. and that is said to be permanent in the mean if there exist a positive number δ, is independent of initial conditions of population, such that…”