Stochastic Partial Differential Equation SIS Epidemic Models: Modeling and Analysis

Abstract

The study on epidemic models plays an important role in mathematical biology and mathematical epidemiology. There has been much effort devoted to epidemic models using ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs). Much study has been carried out and substantial progress has been made. In contrast to the development, this work presents an effort from a different angle, namely, modeling and analysis using stochastic partial differential equations (SPDEs). Specifically, we consider dynamic systems featuring SIS (Susceptible-Infected-Susceptible) epidemic models. Our emphasis is on spatial dependent variations and environmental noise. First, a new epidemic model is proposed. Then existence and uniqueness of solutions of the underlying SPDEs are examined. In addition, stochastic partial differential equation models with Markov switching are examined. Our analysis is based on the use of mild solution. Our hope is that this paper will open up windows for investigation of epidemic models from a new angle.

Stochastic Lotka-Volterra Competitive Reaction-Diffusion Systems Perturbed by Space-Time White Noise: Modeling and Analysis

“…In fact, in [43], we used the following Lemma, whose proof is in [43,Lemma 4.2] and obtained some results of probability one estimates for SIS epidemic model.…”

“…There is also another approach to overcome the second difficulty (being lack of tools regarding change variable), which is introduced in our early works in [39,43,44]. The idea is to approximate the mild solution by a sequence of strong solutions (e.g., the solutions corresponding to the stochastic differential equation driving by finite dimensional noise) and then, we work on these strong solutions, for which the classical Itô's formula is valid.…”

“…The easiest case is to use the finite-trace Q-Wiener process and we refer this case as "nuclear case". Such cases were also considered in some our works in [39,43,44] for epidemic model and predator-prey models. In fact, we considered the "nuclear case" in order to simplify the arguments and help us in investigating the longtime property (sufficient conditions for persistence and extinction).…”

“…In fact, we considered the "nuclear case" in order to simplify the arguments and help us in investigating the longtime property (sufficient conditions for persistence and extinction). The "nuclear case" is more advantageous for approximating mild solutions by sequence of strong solutions and in estimating some quantity like "ln (• • • )"; see [39,43,44] for the details. However, the "finite trace" assumption is too strong and unnecessary in some problems.…”

Stability analysis andB ifurcation for an Bacterial Meningitis S preadingwith S tage S tructure : Mathematical M odeling

“…Mathematical modeling of infectious diseases, over the last century, has tackled this topic with various approaches and assumptions. Among other models, epidemics have been modelled via a system of ordinary differential equations (ODEs) [1,2,3,4,5,6,7], partial differential equations (PDEs) [8,9,10,11], delay differential equations (DDEs) [12,13,14], stochastic differential equations (SDEs) [15,16,17,18] and networks [19,20,21,22,23].…”