Graph-based model predicts pandemic waves

Researchers from Skoltech, the University of Arizona and Linköping University in Sweden have created a graphical model describing the spreading patterns of infection outbreaks between different localities. The model helps calculate the probability of a selected group of areas being hit by a local outbreak and map out a minimum set of prevention measures. The paper was published in Scientific Reports.

Predicting the COVID-19 spread patterns remains an unresolved issue. The recent series of hard-to-predict COVID-19 outbreaks underscores the importance of modelling and prediction. However, the modelling encounters two stumbling blocks: first, it involves calibration of an enormous number of parameters based on data and second, models with too much detail and carefully selected parameters turn out to be impractical for predicting the spread patterns selecting the prevention strategies. That said, for the modelling to be effective, the model should not be crammed with detail not to complicate calculations but be detailed enough to predict the spread dynamics and work out the containment strategies.
One way to model the spread of outbreaks is probabilistic graphical models. Each vertex represents a populated area, and the ribs show the probability of a node getting the infection from the neighbouring node. For example, the greater the population flow between areas, the higher the chances of the outbreak moving from one area to another.
In their recent study, the researchers turned for the first time to probabilistic graphical models to solve a two-level task and answer two questions: how does an outbreak spread from a vertex, and what are the minimum steps that would prevent the infection from sprawling all over the graph?
“Our team of experts in applied mathematics that makes part of a larger multidisciplinary group has been modelling the pandemic for over a year. We use models from physics to pin down the most likely post-infection state of the system and prevent further spread. Since any restriction has cost implications, we wish to achieve the utmost effectiveness of containment with a minimum of interventions,” the first author of the paper, Skoltech PhD program graduate Mikhail Krechetov, says.
In their study, the researchers used the mathematical apparatus of statistical physics, namely, the Ising model that describes the material magnetization process. In the team’s model, the graph nodes represented city districts and the ribs the traffic between them. Also, the infection probability calculations considered the prevention factors, such as masks, social distancing, vaccination coverage, and the like.

The team used real data from Seattle (WA, U.S.A.) and mobile tracking data on population flows between areas to bring the model as close to reality as possible.
First, the researchers checked which nodes would be the next to catch the infection from the infected node and then figured out the minimum number of steps needed to curb the spread. They showed that the model helps define transportation restrictions in the case of an outbreak in a specific area and ensures other areas do not become infected. Moreover, graphical models proved to help solve the same task for any number of affected locations.
“Our model is universal as regards its graph topology, and the method works well even for very dense graphs describing cities with massive inter-district population flows. However, the spread patterns and restrictive strategies in dense graphs differ from those in sparse graphs with less intensive migrations. An interesting mathematical implication for us is how the choice of the function for cost estimations influences the solution’s structure: one case requires pointwise but more significant traffic adjustments. At the same time, the other needs small but evenly distributed changes,” Mikhail adds.
This research is a part of a larger study carried out by a research team from the University of Arizona. The researchers plan to combine individual-level agent-based and area-level graphical models seamlessly for more accurate predictions. The authors assume that after further refinement and calibration, their method will provide practical advice on adjusting traffic to prevent the rampant spread of the infection.

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