Nonlinear Dynamics of Epidemics with Behavioral Lags: SIR Modification and Implications for Public Health Policy

The article develops and analyzes a behaviorally sensitive modification of the SIR model with temporal support via convolutions that reflect delayed negative feedback between epidemic load and adherence to preventive measures. The approach is based on the observation that the classical SIR model with constant intensive behavior does not reproduce the adaptive behavior of the population, while the behavioral-informational circuit significantly changes the process trajectory and causes wave dynamics even with limited seasonal forcing. The modified SIR model demonstrates a transition from spiral convergence to a stationary equilibrium (without memory) and to a limit cycle (with convolutions), while the amplitude and period of oscillation monotonically rise with increasing “inertia” of the behavioral circuit (time constants of memory, intensity of fatigue). The paper shows that fluctuations in people’s behavior are translated into fluctuations of economic activity, budget revenues, social expenditures, etc. It emphasizes the need to combine medical-biological, behavioral, and economic parameters in anti-epidemic policy. The practical model value lies in the possibility of scenario forecasting peaks/intervals between waves and testing guided interventions to reduce the peak load on healthcare and optimize costs. The results confirm that the integration of biological, behavioral, and informational subsystems forms a dynamic with feedback loops capable of generating complex regimes, thereby expanding the scope of applicability of SIR models in the socio-economic analysis of epidemics. Numerical analysis is conducted via the 4th-order Runge-Kutta method, which allowed to construct phase portraits and trace trajectories in the vicinity of stationary regimes. The results demonstrate two qualitatively different regimes of dynamics. In the case of an instantaneous behavioral response, the system exhibits a monotonic relaxation to equilibrium with a single peak of infected individuals. With present convolutions, short oscillations of altered and closed memory orbits are observed in the phase portraits. At the same time, the oscillations are caused not by biological parameters of diseases but by social behavior, inertia, and support in the formation of compliance, which emphasizes information policy and behavior management in accordance with medical measures. Given the demonstrated sensitivity of the dynamics to time lags of behavior, a modification of the SIR model with memory is proposed that serves as a convenient framework for rapid scenario assessment and supports decisions in public health.