Directed mobility induces spatial heterogeneity and epidemic hotspot migration in multiplex networks

Regional epidemics often show asynchronous peaks and shifting hotspots, motivating models that distinguish mobility among epidemiological subpopulations. We develop a multiplex reaction–diffusion epidemic model with directed migration. Linear stability and network spectral analyses characterize stationary and oscillatory Turing instabilities, verified on composite circle, torus, and Klein-bottle networks. Two-parameter analysis reveals pronounced asymmetry: within the parameter ranges studied, increasing susceptible migration σ 1 enlarges the Turing-unstable region, whereas increasing infected migration σ 2 suppresses spatial instability. Transport also changes the instability type. Diffusion alone mainly produces stationary Turing modes, whereas directed migration gives rise to complex modal spectra that can induce oscillatory spatial instability, manifested as oscillatory patterns in nonlinear simulations. Simulations on composite circle, composite torus, Klein-bottle networks, and heterogeneous multiplex networks further show that population-specific transport and network topology jointly regulate pattern morphology and spatial hotspot reorganization. Finally, U.S. COVID-19 data are used for a qualitative phenomenological comparison, showing macroscopically analogous asynchronous regional peaks and reorganization of high-incidence regions. The framework clarifies how population-specific directed mobility organizes spatial epidemic patterns and provides a theoretical basis for distinguishing the effects of susceptible and infected mobility on spatial instability and pattern selection.