Modeling the dynamics of epidemic spreading on homogenous and heterogeneous networks

Abstract

This paper proposes two modified susceptible-infected-recovered-susceptible models on homogenous and heterogeneous networks, respectively. In the study of the homogenous network model, it is proved that if the basic reproduction number of the model is less than one, then the disease-free equilibrium is locally asymptotically stable and becomes globally asymptotically stable under the condition that the threshold value is less than one. Otherwise, if is more than one, the endemic equilibrium is locally asymptotically stable and becomes globally asymptotically stable under the assumption that the total population will tend to a specific plane. In the study of the heterogeneous network model, this paper discusses the existences of the disease-free equilibrium and endemic equilibrium of the model. It is proved that if the threshold value is less than one, then the disease-free equilibrium is globally asymptotically stable. Otherwise, if is more than one, the system is permanent. A series of numerical experiments are given to illustrate the theoretical results. We also numerically predict the effect of vaccination ratio on the size of HBV-infected mainland Chinese population.

Keywords:

• SIRS model
• homogenous network
• heterogeneous network
• local stability
• global stability

AMS Subject Classifications:

• 92B05
• 91D30

Acknowledgements

The authors would like to thank the anonymous referees for valuable comments. This research is supported by the National Natural Science Foundation of China under Grant Nos 61074192, and the NSF grants DMS-0436341 and DMS-0920744 of US.

Notes

This research is supported by the National Natural Science Foundation of China [grant number 61074192]; the NSF [grant number DMS-0436341], [grant number DMS-0920744] of US.