Abstract
We 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}}_0le 1, then the disease will be eradicated, while for ,{mathcal {R}}_0>1 the disease persists in the population. Then, for ,{mathcal {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.
Highlights
Switching models have been used recently in the compartmental models of mathematical epidemiology to analyze the impact of control measures on the disease dynamics
Such a sudden change may even be discontinuous, for example due to the implementation or termination of an intervention policy such as vaccination or school closures. Such situations are described by Filippov systems, when the phase space is divided into two parts and the system is given by different vector fields in each of those parts
Delayed relay systems have been applied to an SIS model [5], where explicit periodic solutions were constructed for the case of a delayed reduction in the contact rate after the density of infection in the population passed through a threshold value
Summary
Switching models have been used recently in the compartmental models of mathematical epidemiology to analyze the impact of control measures on the disease dynamics. Such a sudden change may even be discontinuous, for example due to the implementation or termination of an intervention policy such as vaccination or school closures Such situations are described by Filippov systems, when the phase space is divided into two (or more) parts and the system is given by different vector fields in each of those parts. Delayed relay systems have been applied to an SIS model [5], where explicit periodic solutions were constructed for the case of a delayed reduction in the contact rate after the density of infection in the population passed through a threshold value The dynamics of this discontinuous system was different from its continuous counterpart [3,4], showing that it is worthwhile to analyse the dynamics of epidemiological systems with delayed switching. Our results contribute to the development of a systematic way of designing implementable controls that drive the dynamics towards disease control or mitigation
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