Stability analysis of a time-delayed SIQR epidemic model with nonlinear transmission and control parameters.

We introduce an SIQR epidemic model that integrates nonlinear transmission and two discrete time delays corresponding to incubation and treatment durations. The model seeks to encapsulate essential dynamical characteristics of epidemic advancement while being suitable for thorough stability examination. The local asymptotic stability of the disease-free and endemic equilibria is examined by Lyapunov-Krasovskii functionals and delay-dependent linear matrix inequalities (LMIs), which mitigate conservatism in the established stability constraints compared to time-invariant criteria. In the absence of delays, we obtain the classical Routh-Hurwitz criteria for local stability, and for positive delays, we formulate characteristic equations whose roots are examined to identify Hopf bifurcations. Transversality requirements are checked to ensure the presence of Hopf bifurcations and to determine key delay thresholds beyond which persistent oscillations form. We propose a delay-dependent feedback control rule that adaptively modifies transmission and quarantine rates; necessary conditions for stability under this control are provided in LMI form and converted into explicit, interpretable constraints on permissible delays and minimum control intensity. The model is augmented to incorporate basic demographic turnover and multi-stage infection delays to provide subgroup-specific treatment representations. Numerical simulations and bifurcation diagrams demonstrate and validate the theoretical findings, indicating how augmented delays or intensified nonlinear transmission can alter stability thresholds and provoke recurring outbreaks. Our findings quantify (i) essential delay durations that undermine stability and (ii) the control effort necessary to reestablish equilibrium, results that can be articulated as definitive decision thresholds for conversion into policy-relevant outputs.