Epidemic Models with Time-Varying Transmission and Intervention Effects: Stability Analysis and Threshold Dynamics

We develop and analyse a deterministic SEAIHRV model with explicitly time-varying control functions representing transmission, vaccination, and hospitalization. We introduce an effective reproduction profile Rt, which governs the growth or decay of infection. We establish the existence, positivity, and boundedness of solutions for the non-autonomous system, and we show that the sign of the asymptotic logarithmic average of Rt determines whether the disease is uniformly eliminated or persists. In the autonomous limit, we recover the basic reproduction number R0 as the spectral radius of the next-generation operator, while in periodically forced regimes, we define a periodic reproduction number Rper via Floquet theory, whose sign governs the global stability of disease-free versus endemic T-periodic states. We link adaptive and temporally structured interventions directly to R0, Rper, and the principal Lyapunov exponent of the linearized infectious cocycle. Using NSFD–Patankar simulations, fully consistent with our analytical framework, we illustrate how constant, periodic, abruptly switching, and feedback-controlled interventions shape Rt trajectories, epidemic peaks, duration, and cumulative burden. Together, our results provide a mathematically rigorous and epidemiologically interpretable foundation for understanding and controlling epidemics in dynamically changing environments, and we show that this framework can be readily extended to optimal control formulations, stochastic perturbations, and network-structured epidemic models.