Dynamical Behavior of a Stochastic SIRC Model for Influenza A

Tongqian Zhang,Tingting Ding,Yi Song,Ning Gao

doi:10.3390/sym12050745

Tongqian Zhang, Tingting Ding + Show 2 more

Open Access

https://doi.org/10.3390/sym12050745

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Journal: Symmetry	Publication Date: May 5, 2020
Citations: 5	License type: CC BY 4.0

Affiliation: Shandong University of Science and Technology

Abstract

In this paper, a stochastic SIRC epidemic model for Influenza A is proposed and investigated. First, we prove that the system exists a unique global positive solution. Second, the extinction of the disease is explored and the sufficient conditions for extinction of the disease are derived. And then the existence of a unique ergodic stationary distribution of the positive solutions for the system is discussed by constructing stochastic Lyapunov function. Furthermore, numerical simulations are employed to illustrate the theoretical results. Finally, we give some further discussions about the system.

Highlights

Influenza is an infectious disease characterized by acute respiratory infection caused by the RNA influenza virus of the mucinous virus family [1,2]
Let σi = 0.1(i = 1, 3, 4), σ2 = 0.9, direct calculation shows R1 = 0.8376 < 1, it follows from Theorem 2, the disease will go to extinction eventually
We proposed and analyzed a SIRC epidemic model for Influenza A with stochastic perturbation

Summary

Introduction

Influenza is an infectious disease characterized by acute respiratory infection caused by the RNA influenza virus of the mucinous virus family [1,2]. According to the degree of antigenic variation, it can be divided into antigenic drift and antigenic shift The former is caused by point mutation and can cause seasonal influenza epidemic. Considering the cross-immunity phenomenon during the spread of influenza A [21,22,23], Casagrandi et al [24] proposed an SIRC model as follows, Ṡ(t) = μ(1 − S(t)) − βS(t) I (t) + γC (t),. According to the way of considering different stochastic disturbances, many scholars have established a variety of stochastic infectious disease models, and investigated the asymptotic behavior of the system. According to the study in [24], cross-immunity causes complex transmission of infectious diseases.

Preliminaries

Global Existence and Uniqueness of the Positive Solution

Extinction of the Disease

Existence of Ergodic Stationary Distribution

Numerical Simulations

Findings

Discussion