Spatial scales of COVID-19 transmission in Mexico.

During outbreaks of emerging infectious diseases, internationally connected cities often experience large and early outbreaks, while rural regions follow after some delay. This hierarchical structure of disease spread is influenced primarily by the multiscale structure of human mobility. However, during the COVID-19 epidemic, public health responses typically did not take into consideration the explicit spatial structure of human mobility when designing nonpharmaceutical interventions (NPIs).

Modelling COVID-19.

As the COVID-19 pandemic continues, mathematical epidemiologists share their views on what models reveal about how the disease has spread, the current state of play and what work still needs to be done.

Crowding and the shape of COVID-19 epidemics.

The coronavirus disease 2019 (COVID-19) pandemic is straining public health systems worldwide, and major non-pharmaceutical interventions have been implemented to slow its spread. During the initial phase of the outbreak, dissemination of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) was primarily determined by human mobility from Wuhan, China. Yet empirical evidence on the effect of key geographic factors on local epidemic transmission is lacking.

Exotic phase transitions of k-cores in clustered networks.

The giant k-core-maximal connected subgraph of a network where each node has at least k neighbors-is important in the study of phase transitions and in applications of network theory. Unlike Erdős-Rényi graphs and other random networks where k-cores emerge discontinuously for k≥3, we show that transitive linking (or triadic closure) leads to 3-cores emerging through single or double phase transitions of both discontinuous and continuous nature. We also develop a k-core calculation that includes clustering and provides insights into how high-level connectivity emerges.

Message-passing approach for recurrent-state epidemic models on networks.

Epidemic processes are common out-of-equilibrium phenomena of broad interdisciplinary interest. Recently, dynamic message-passing (DMP) has been proposed as an efficient algorithm for simulating epidemic models on networks, and in particular for estimating the probability that a given node will become infectious at a particular time. To date, DMP has been applied exclusively to models with one-way state changes, as opposed to models like SIS and SIRS where nodes can return to previously inhabited states.

Message-passing approach for threshold models of behavior in networks.

We study a simple model of how social behaviors, like trends and opinions, propagate in networks where individuals adopt the trend when they are informed by threshold T neighbors who are adopters. Using a dynamic message-passing algorithm, we develop a tractable and computationally efficient method that provides complete time evolution of each individual's probability of adopting the trend or of the frequency of adopters and nonadopters in any arbitrary networks. We validate the method by comparing it with Monte Carlo-based agent simulation in real and synthetic networks and provide an exact analytic scheme for large random networks, where simulation results match well.