A Strict Smooth Lyapunov Function and Input-to-State Stability of SIR Model

This paper shows that the SIR model of infectious diseases is input-to-state stable (ISS) with respect to the net flow of susceptible individuals into the region in both disease-free and epidemic situations. The key development is the construction of a continuously differentiable strict Lyapunov function. Firstly, for asymptotic stability, this paper clarifies that a continuously differentiable Lyapunov function can deduce the stability on the whole state space even if its derivative is not strictly negative. Secondly, it is demonstrated that a new idea of modifying the Lyapunov function to render its derivative strictly negative allows one to detect an appropriate margin to prove ISS of the SIR model. The accomplishment is based on the pursuit of a Lyapunov function which is not entirely separable into components. It contrasts with previously studied and popular Lyapunov functions which are incapable of assessing robustness properties such as ISS.