Modeling and Analysis of Epidemic Diffusion within Small‐World Network

Ming Liu,Yihong Xiao

doi:10.1155/2012/841531

Abstract

To depict the rule of epidemic diffusion, two different models, the Susceptible‐Exposure‐Infected‐Recovered‐Susceptible (SEIRS) model and the Susceptible‐Exposure‐Infected‐Quarantine‐Recovered‐Susceptible (SEIQRS) model, are proposed and analyzed within small‐world network in this paper. Firstly, the epidemic diffusion models are constructed with mean‐filed theory, and condition for the occurrence of disease diffusion is explored. Then, the existence and global stability of the disease‐free equilibrium and the endemic equilibrium for these two complex epidemic systems are proved by differential equations knowledge and Routh‐Hurwiz theory. At last, a numerical example which includes key parameters analysis and critical topic discussion is presented to test how well the proposed two models may be applied in practice. These works may provide some guidelines for decision makers when coping with epidemic diffusion controlling problems.

Highlights

Disastrous epidemics such as SARS and H1N1 can significantly impact people’s life
It is widely recognized that a large-scale epidemic diffusion can conceivably cause many deaths and more people of permanent sequelae, which presents a severe challenge to a local or regional health-care systems
Many recent research efforts have been devoted to understanding the prevention and control of epidemics, such as those of Wein et al 1, Wein et al 2, Craft et al 3, Kaplan et al 4, 5, and Matsuura et al 6

Summary

Introduction

Disastrous epidemics such as SARS and H1N1 can significantly impact people’s life. The outbreak of infections in Europe last year is another recent example. Epidemic diffusion models they adopted are based on traditional compartment models, while complex topological structure of social contact network has not been considered. As it is well known, a class of network with a topology interpolating between that of lattices and random graphs is proposed by Watts and Strogatz 8. Xu et al presented a modified SIS model based on complex networks, small world and scale free, to study the spread of an epidemic by considering the effect of time delay. Two different models SEIRS, SEIQRS based on small-world network are formulated for the spread of infectious diseases.

Basic Introduction

Analysis of the SEIRS Model

Numerical Experiment

A Critical Topic Discussion

Conclusions