Stability and Bifurcation Analysis of Epidemic Control with Information Delays in Behavioral SIRS Models

This study investigates the profound impact of time delays in information dissemination on infectious disease dynamics, revealing how delayed reporting can dramatically alter epidemic patterns. When news about disease outbreaks, risks, and mortality rates propagates slowly through media and public networks, the resulting behavioral changes among susceptible individuals occur too late to effectively control transmission. Our research examines an SIRS epidemic model incorporating human behavioral responses to disease awareness, where susceptible individuals adopt protective measures upon receiving information. The time delay parameter is investigated, capturing the lag between infection and public awareness. Using the method of multiple scales, we derive normal forms for Hopf and several higher-order bifurcations (Bautin and double-Hopf), allowing us to precisely classify how and when the system transitions between different dynamic states. This approach reveals subtle but critical thresholds where small changes in reporting delays can trigger dramatic shifts in disease patterns. The results demonstrate that delayed information creates remarkably complex dynamics absent in delay-free systems. As reporting delays increase, stability regions are segmented, creating intricate patterns where stable equilibrium, periodic oscillations, and quasi-periodic behaviors intertwine. At certain critical delay values, the system exhibits double-Hopf bifurcations. Furthermore, the global stability of the system’s equilibrium points is investigated in both the presence and absence of delay. We created comprehensive two-parameter bifurcation diagrams using Lyapunov exponents and Poincaré sections to analyze how the system’s dynamics change. These diagrams show how factors like information decay, population growth, recovery rate, and information saturation, when combined with time delay, influence the path of an epidemic. This offers practical guidance for public health officials by identifying the specific parameter zones where swift information spread is critical to stop recurrent outbreaks.