# Analysis of an Epidemic System with Two Response Delays in Media Impact Function

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## Abstract

A functional differential model of SEIS-M type with two time delays, representing the response time for mass media to cover the current infection and for individuals’ behavior changes to media coverage, was proposed to examine the delayed media impact on the transmission dynamics of emergent infectious diseases. The threshold dynamics were established in terms of the basic reproduction number \({\mathcal {R}}_{0}\). When there are no time delays, we showed that if the media impact is low, the endemic equilibrium is globally asymptotically stable for \({\mathcal {R}}_{0}>1\), while the endemic equilibrium may become unstable and Hopf bifurcation occurs for some appropriate conditions by taking the level of media impact as bifurcation parameter. With two time delays, we comprehensively investigated the local and global bifurcation by considering the summation of delays as a bifurcation parameter, and theoretically and numerically examined the onset and termination of Hopf bifurcations from the endemic equilibrium. Main results show that either the media described feedback cycle, from infection to the level of mass media and back to disease incidence, or time delays can induce Hopf bifurcation and result in periodic oscillations. The findings indicate that the delayed media impact leads to a richer dynamics that may significantly affect the disease infections.

## Keywords

Media impact Functional differential model Time delays Global Hopf bifurcation## Notes

### Acknowledgements

PS was supported by the China Scholarship Council; YX was supported by the National Natural Science Foundation of China(NSFC, 11631012, 11571273(YX)). The authors would like to thank the referees for many helpful comments, which lead to improvements in Theorems 1–3. The authors would like to thank Prof Xiaoqiang Zhao for his generous help in discussing the theory of the limiting system.

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