Multiscale System for Environmentally-Driven Infectious Disease with Threshold Control Strategy

A multiscale system for environmentally-driven infectious disease is proposed, in which control measures at three different scales are implemented when the number of infected hosts exceeds a certain threshold. Our coupled model successfully describes the feedback mechanisms of between-host dynamics on within-host dynamics by employing one-scale variable guided enhancement of interventions on other scales. The modeling approach provides a novel idea of how to link the large-scale dynamics to small-scale dynamics. The dynamic behaviors of the multiscale system on two time-scales, i.e. fast system and slow system, are investigated. The slow system is further simplified to a two-dimensional Filippov system. For the Filippov system, we study the dynamics of its two subsystems (i.e. free-system and control-system), the sliding mode dynamics, the boundary equilibrium bifurcations, as well as the global behaviors. We prove that both subsystems may undergo backward bifurcations and the sliding domain exists. Meanwhile, it is possible that the pseudo-equilibrium exists and is globally stable, or the pseudo-equilibrium, the disease-free equilibrium and the real equilibrium are tri-stable, or the pseudo-equilibrium and the real equilibrium are bi-stable, or the pseudo-equilibrium and disease-free equilibrium are bi-stable, which depends on the threshold value and other parameter values. The global stability of the pseudo-equilibrium reveals that we may maintain the number of infected hosts at a previously given value. Moreover, the bi-stability and tri-stability indicate that whether the number of infected individuals tends to zero or a previously given value or other positive values depends on the parameter values and the initial states of the system. These results highlight the challenges in the control of environmentally-driven infectious disease.