Fractional Differential Equations

In this paper we extend the classical susceptible-infected-susceptible epidemic model from a deterministic framework to a stochastic one and formulate it as a stochastic differential equation (SDE) for the number of infectious individuals $I(t)$. We then prove that this SDE has a unique global positive solution $I(t)$ and establish conditions for extinction and persistence of $I(t)$. We discuss perturbation by stochastic noise. In the case of persistence we show the existence of a stationary distribution and derive expressions for its mean and variance. The results are illustrated by computer simulations, including two examples based on real-life diseases.

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A Stochastic Differential Equation SIS Epidemic Model

The threshold of a stochastic SIS epidemic model with vaccination

Threshold behaviour of a stochastic SIR model

Many stochastic differential equations (SDEs) in the literature have a superlinearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately, momentsof the computationally efficient Euler-Maruyama approximation method diverge for these SDEs in finite time. This article develops a general theory for studying integrability properties such as moment bounds for discrete-time stochastic processes. Using this approach, we establish moment bounds for fully and partially drift-implicit Euler methods and for a class of new explicit approximation methods which require only a few more arithmetical operations than the Euler-Maruyama method. These moment bounds are then used to prove strong convergence of the proposed schemes. Finally, we illustrate our results for several SDEs from finance, physics, biology and chemistry.

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Numerical Approximations of Stochastic Differential Equations With Non-globally Lipschitz Continuous Coefficients

We propose a stochastic SIR model with or without distributed time delay and we study the stability of disease-free equilibrium. The numerical simulation of the stochastic SIR model shows that the introduction of noise modifies the threshold of system for an epidemic to occur and the threshold stochastic value is found.

Stability of a stochastic SIR system

Existence and asymptotic properties for quasilinear elliptic equations with gradient dependence

We propose a stochastic disease model where vaccination is included and such that the immunity is permanent. The existence, uniqueness, and positivity of the solution and the stability of the disease-free equilibrium are studied.

SIVR epidemic model with stochastic perturbation

Global solution of bi-dimensional stochastic equation for a viscous gas

Abstract In this paper, by using variational methods and critical point theorems, we prove the existence and multiplicity of solutions for boundary value problem for fractional order differential equations where Riemann-Liouville fractional derivatives and Caputo fractional derivatives are used. Our results extend the second order boundary value problem to the non integer case. Moreover, some conditions to determinate nonnegative solutions are presented and examples are given to illustrate our results.

Elisabetta Tornatore

Papers

Stability of a stochastic SIR system

Existence and asymptotic properties for quasilinear elliptic equations with gradient dependence

SIVR epidemic model with stochastic perturbation

Global solution of bi-dimensional stochastic equation for a viscous gas

On the existence and multiplicity of solutions for Dirichlet's problem for fractional differential equations