Multi-scale measures of time-varying epidemic spread on human mobility networks

Human movement drives the spatial spread and persistence of many infectious diseases, yet existing theory and real-time operational tools for inferring the instantaneous reproduction number R(t) often assume static and/or homogeneously mixing populations and cannot describe how individuals generate and acquire infections heterogeneously based on their movement patterns within a day. Renewal equations underpin many such popular estimators of R(t), and here, we develop a network-based modelling framework from which we derive new mechanism-led renewal equations and control indicators for outbreaks of infectious diseases. These equations directly integrate within-day human movement to rigorously define a family of instantaneous reproduction numbers; inward, outward, and type R(t) for individual locations, R(t) between locations, R(t) at meeting locations, and R(t) for the entire mobility network. These quantities correct for the unsuitability of existing location-specific R(t) estimators that operate in closed, static populations. Applying our framework to epidemics on diverse types of networks alongside mobile phone data, we demonstrate how our new framework's outputs provide new, multi-scale control indicators at the network, location, and transmission corridor scales, and can be used to design targeted disease control interventions including the strength, type, and length of intervention required across space and time. We capture the biasing effects of different existing ways to measure location-specific and network-level transmission potential without capturing within-day human movements. This generalisable framework redefines reproduction numbers in real-world outbreaks that are shaped by individuals moving across connected locations, enabling more spatially and temporally precise interventions.