Improving Stability Conditions for Equilibria of SIR Epidemic Model with Delay under Stochastic Perturbations

Leonid Shaikhet

doi:10.3390/math8081302

Abstract

So called SIR epidemic model with distributed delay and stochastic perturbations is considered. It is shown, that the known sufficient conditions of stability in probability of the equilibria of this model, formulated immediately in the terms of the system parameters, can be improved by virtue of the method of Lyapunov functionals construction and the method of Linear Matrix Inequalities (LMIs). It is also shown, that stability can be investigated immediately via numerical simulation of a solution of the considered model.

Highlights

Investigation of different versions of the mathematical model of the spread of infections diseases, so called, SIR epidemic models, has a long history, and until now these models are very actual and are very popular in research
Lyapunov functional V = V (zt, G(t)) is used, that satisfies to the conditions (17) of the Theorem 1, which is a classical theorem of the type of Lyapunov-Krasovskii
It is seen that the effect of stochastic perturbations is stronger than in Figure 5, but all trajectories with the same initial conditions converge to the equilibrium E0 = = (5, 0, 0) again

Summary

Introduction

Investigation of different versions of the mathematical model of the spread of infections diseases, so called, SIR epidemic models, has a long history, and until now these models are very actual and are very popular in research (see, for instance, [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]). During its history of development, SIR epidemic models were considered both with constant and distributed delay, both in a deterministic and in a stochastic version. We will consider the SIR epidemic model in the form of the following system of differential equations with distributed delay [2,31]. From (4) it follows that E+ is a positive equilibrium of the system (1) by the condition (5). In [31] the following simple sufficient conditions for stability in probability of the equilibria E0 and E+ are obtained that are formulated immediately in the terms of the system (6) parameters. Note that the second condition (8) contradicts with (5) It means that by the conditions (8) the system (6) does not have the positive equilibrium E+. Inequalities (LMIs) [33] that are essentially less conservative than the conditions (8)–(10)

Some Auxiliary Definitions and Statements

Stability of Equilibria

The First Stability Condition

The Third Stability Condition

Numerical Simulations

Conclusions