Global hopf bifurcation of a delayed equation describing the lag effect of media impact on the spread of infectious disease.

We proposed a delay differential model, associated with the response time for individuals to the current infection, to examine the media impact on the transmission dynamics of infectious diseases. We investigated the global bifurcation by considering the delay as a bifurcation parameter and examined the onset and termination of Hopf bifurcations from a positive equilibrium. Numerical studies to identify ranges of parameters for coexisting multiple periodic solutions are guided by the bifurcation analysis and the Matlab package DDE-BIFTOOL developed by Engelborghs et al. Further, we parameterized the proposed model on the basis of the 2009 A/H1N1 pandemic influenza data in Shaanxi province, China, and estimated the basic reproduction number to be 1.79 [95% CI (1.77-1.80)] and the time delay to be 2.94 days [95% CI (2.56-3.24)]. Our main results indicated that media impact with time delay significantly influenced the transmission dynamics of infectious diseases.