Periodic Orbits and Global Stability for a Discontinuous SIR Model with Delayed Control

Abstract

AbstractWe propose and analyse a mathematical model for infectious disease dynamics with a discontinuous control function, where the control is activated with some time lag after the density of the infected population reaches a threshold. The model is mathematically formulated as a delayed relay system, and the dynamics is determined by the switching between two vector fields (the so-called free and control systems) with a time delay with respect to a switching manifold. First we establish the usual threshold dynamics: when the basic reproduction number $$\,{\mathcal {R}}_0\le 1$$ R 0 ≤ 1 , then the disease will be eradicated, while for $$\,{\mathcal {R}}_0>1$$ R 0 > 1 the disease persists in the population. Then, for $$\,{\mathcal {R}}_0>1$$ R 0 > 1 , we divide the parameter domain into three regions, and prove results about the global dynamics of the switching system for each case: we find conditions for the global convergence to the endemic equilibrium of the free system, for the global convergence to the endemic equilibrium of the control system, and for the existence of periodic solutions that oscillate between the two sides of the switching manifold. The proof of the latter result is based on the construction of a suitable return map on a subset of the infinite dimensional phase space. Our results provide insight into disease management, by exploring the effect of the interplay of the control efficacy, the triggering threshold and the delay in implementation.