During infectious disease outbreaks, media dissemination, and time delays critically influence the transmission dynamics. To capture the coupled interactions among media influence, time delay, and hospitalization resources, we propose a delayed $ \mathit{susceptible} $–$ \mathit{exposed} $–$ \mathit{infected} $–$ \mathit{hospitalized} $–$ \mathit{recovered} $–$ \mathit{media} $ ($ \mathit{SEIHRM} $) epidemic model incorporating media feedback and systematically investigate its dynamic behavior both theoretically and numerically. The model is shown to be well-posed in terms of the existence, uniqueness, positivity, and boundedness of solutions. The basic reproduction number $ R_0 $ is derived, thereby determining the stability of the disease-free equilibria and serving as a threshold for the existence of endemic equilibria. Bifurcation analysis with the time delay $ \tau $ as the bifurcation parameter reveals that once $ \tau $ exceeds a critical threshold $ \tau_0 $, a Hopf bifurcation emerges, giving rise to sustained periodic oscillations and thereby recurrent epidemic outbreaks. Numerical simulations substantiate the theoretical results and reveal the distribution of stable equilibria, periodic oscillations, and unstable dynamics across parameter regimes. The results demonstrate that the coupling between media dissemination and time delay can drive complex oscillatory dynamics via delayed feedback, providing a mechanistic explanation for both endemic persistence and recurrent epidemic outbreaks. This study offers a novel theoretical perspective on information-feedback-driven epidemic dynamics and provides a quantitative basis for delay-based targeted intervention strategies.
Citation: Chan Ma, Yun Zhao, Xunwen Tan, Wenjie Shi. Dynamical analysis and bifurcation study of an epidemic model with media coverage and time delay[J]. Networks and Heterogeneous Media, 2026, 21(5): 1600-1640. doi: 10.3934/nhm.2026061
During infectious disease outbreaks, media dissemination, and time delays critically influence the transmission dynamics. To capture the coupled interactions among media influence, time delay, and hospitalization resources, we propose a delayed $ \mathit{susceptible} $–$ \mathit{exposed} $–$ \mathit{infected} $–$ \mathit{hospitalized} $–$ \mathit{recovered} $–$ \mathit{media} $ ($ \mathit{SEIHRM} $) epidemic model incorporating media feedback and systematically investigate its dynamic behavior both theoretically and numerically. The model is shown to be well-posed in terms of the existence, uniqueness, positivity, and boundedness of solutions. The basic reproduction number $ R_0 $ is derived, thereby determining the stability of the disease-free equilibria and serving as a threshold for the existence of endemic equilibria. Bifurcation analysis with the time delay $ \tau $ as the bifurcation parameter reveals that once $ \tau $ exceeds a critical threshold $ \tau_0 $, a Hopf bifurcation emerges, giving rise to sustained periodic oscillations and thereby recurrent epidemic outbreaks. Numerical simulations substantiate the theoretical results and reveal the distribution of stable equilibria, periodic oscillations, and unstable dynamics across parameter regimes. The results demonstrate that the coupling between media dissemination and time delay can drive complex oscillatory dynamics via delayed feedback, providing a mechanistic explanation for both endemic persistence and recurrent epidemic outbreaks. This study offers a novel theoretical perspective on information-feedback-driven epidemic dynamics and provides a quantitative basis for delay-based targeted intervention strategies.
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Chan Ma, Yun Zhao, Xunwen Tan, Wenjie Shi. Dynamical analysis and bifurcation study of an epidemic model with media coverage and time delay[J]. Networks and Heterogeneous Media, 2026, 21(5): 1600-1640. doi: 10.3934/nhm.2026061
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