Robustness of feedback control for siqr epidemic model under measurement uncertainty

We study a model for susceptible, infected, quarantined, and re-covered (SIQR) populations, in the presence of an infectious disease. The three feedback controls represent isolation, contact regulation, and vaccination. The model contains four time-varying uncertainties. One uncertainty models un-certain immigration. The other three uncertainties represent uncertainties in the measurements of state components that are used in the feedback control. We use the input-to-state stability (ISS) framework. Using a strict Lyapunov function construction method from a recent paper by H. Ito, M. Malisoff, and F. Mazenc whose uncertainties were confined to be uncertain immigration, we provide two ISS results that quantify the robustness of the feedback control, with respect to the four uncertainties. Our first theorem proves ISS of the SIQR error dynamics with all four uncertainties present in the ISS overshoot term. The error variable measures the difference between the current state of the SIQR dynamics and the prescribed endemic equilibrium. Our second theorem proves ISS where the overshoot in the ISS estimate only depends on the immigration uncertainty, and so ensures robust asymptotic stability when no immigration uncertainty is present, under a new condition on the allowable measurement uncertainties. We illustrate the effectiveness of our approach in simulations, using parameter values from the COVID-19 pandemic.