I dropped into today’s special session on mathematical biology to hear Yanyu Xiao of the University of Cincinnati speak. She talked about how her group modelled contact patterns in Ontario and used them to study the spread of COVID-19 this past year.
The standard model for predicting infectious disease spread is an agent-based model known as an SIR model. This acronym stands for Susceptible, Infected, and Recovered, referring to the three types of agents. Infected agents will remain so for a period of time until they recover (or die) and become immune, and in the meantime will spread their disease to susceptible agents. From this, you can derive a system of differential equations that describe how the number of susceptible, infected, and recovered people in a population evolves over time. To get the details right, you need data about how infectious the disease is and how often agents come in contact with one another.
Getting that data for a real-world situation like COVID-19 is easier said than done. People interact in tons of different ways, and establishing how likely one person is to infect another requires analyzing all of that complexity. How likely are those people to meet in the grocery store—and how likely is an infection to occur there? What if they meet at a backyard barbecue?
To answer these questions, Xiao explained, researchers distill the different settings in which people meet one another into four main groups: households, workplaces, schools, and community. They also split the population into age groups. For each setting, they can create a “contact matrix” of age groups. So, for example, in the workplace matrix, entry (i, j) represents the number of workplace contacts in age group j that a person in age group i has. I would imagine that this method comes in handy especially when studying a disease like COVID-19, whose behaviour seems to very drastically with age.
Reference contact matrices used in Xiao’s study.
Xiao and her colleagues used survey data from 2006 to create benchmark contact matrices, and used demographic adjustments to estimate the correct matrices for Ontario 2020. As Ontario went through its various shutdown and reopening phases, they modelled the overall contact matrix accordingly. For example: before any businesses or schools had been shut down or physical distancing measures recommended, the overall contact matrix was merely the sum of the matrices from each setting, C = C(Household) + C(Workplace) + C(Community) + C(School). But once schools shut down, C took a different form: C = (1 + p) C(Household) + C(Workplace) + (1 + q) C(Community). This formula reflects the fact that there were no longer any school contacts. But with children and teachers spending more time at home or out in their neighbourhoods, household and community contacts would increase.
Based on this data, Xiao could implement the SIR model to estimate the cumulative number of infections in Ontario, and find the parameters that best fit the data. The model could then be used to evaluate the reopening plan. And with the advent of vaccines, it may be used to analyze distribution strategies.