For decades, scientists have relied on mathematical models to understand and predict how infections move through populations and impact communities.
But a central mathematical question has persisted: Do these models predict that the number of new cases will fluctuate endlessly or stabilize at a constant level?
Computer simulations of epidemic models frequently show a rise and fall of case numbers in waves that diminish gradually, but attempts to determine what the models predict in the long term have succeeded only on a case-by-case basis.
Researchers at McMaster University and Wilfrid Laurier University have proven that for countless epidemic models — including thousands used previously to study diseases like influenza, smallpox, measles, and COVID-19 — infection levels will always approach a steady state, regardless of how an outbreak might begin.
The theorem, published this week in the Proceedings of the National Academy of Sciences (PNAS), applies to an enormous range of models that share a simple property: All individuals who are susceptible to infection are equally susceptible.
“Because extremely complex models are now proved to have the same long-term behaviour as simple models, we can more confidently use simple models to guide public health policy,” explains David Earn, professor in the Department of Mathematics at McMaster, who conducted the work with former postdoctoral fellow Connell McCluskey, now a professor of mathematics at Wilfrid Laurier University.
“A quick check that a model satisfies the assumptions of our new theorem removes the need for the convoluted and time-consuming analysis that has previously been required to determine the long-term behaviour of any specific model,” says McCluskey.
The work also has broader implications, says Earn, in that the new methodology can be applied to other complex systems, such as models of predator-prey relationships or food webs.