Dynamics of an epidemic metapopulation system with heterogeneous threshold control and implications for threshold policy design

Threshold control is an essential method for the targeted management of infectious diseases. Consequently, numerous non-smooth dynamic models incorporating state-dependent feedback control have been proposed and thoroughly analyzed. However, most existing studies introduce threshold policies based on homogeneous population models. To fill this gap, this study investigates the impact of population heterogeneity on the design of threshold policies. We developed a Filippov system based on an SIS-type metapopulation model, considering that interventions are triggered when the linear combination of infectious individuals in each group exceeds a critical threshold. Using a structured population with two groups as a case study, we theoretically investigated the existence of sliding regions, the existence and non-existence of pseudo-equilibria, and further analyzed the local and global stability of both pseudo-equilibria and regular equilibria. Additionally, we demonstrated the existence of boundary-node bifurcation in the proposed system as the threshold conditions vary. Furthermore, we showed that the total number of infectious individuals across all groups at the pseudo-equilibrium decreases monotonically as the weight assigned to the infections of one group for designing the threshold condition increases. This suggests that to minimize total infections during the epidemic for a fixed threshold, it is more effective to target the infectious population of a single group-often the group at higher risk of infection-to initialize and stop control measures than to consider combinations of infections across all groups. Moreover, for one fixed group, the monotonicity of the total infections at the pseudo-equilibrium can switch, which is governed by a critical value. Therefore, the selection of the target group to determine the threshold policy depends on the potential control strength and the local characteristics of the population groups.