Grant: Discovery Grants Program - Individual

Models of the spread of infectious diseases have been invaluable in assessing the impact of intervention measures for controlling COVID-19 and other diseases.  Two key questions of interest that associated with a model are: - What happens during an outbreak? - What happens in the long-run? My work is focused on this second question - the long-term behaviour attached to models. In making modelling decisions, we can create both simple and complicated models of the same situation.  The complicated model is almost certainly harder to study.  Often we start by studying a simpler model, and then move on to the more complicated one.  One of the properties that is sometimes shown is that the model has a globally stable equilibrium, meaning that the disease level in population tends to a constant level in the long-run. One of the ways that a model is shown to have a globally stable equilibrium is by using Lyapunov's Direct Method.  To do this, an extra function (called a Lyapunov function) is introduced and used for a sequence of calculations.  If the calculation works out, then we know that model has a globally stable equilibrium, and so the disease prevalence will eventually level out - assuming that the model is accurate. My interest is in determining when the stability calculation for the simpler model implies that the more complicated model is also stable.  The alternative is that the more complicated model (which includes more detail) might exhibit real instabilities that the simpler model failed to detect.  If this is the case, we would like to know.  Similarly, if this is not the case, then we would like to know. We have many examples where simple models with a Lyapunov function have been extended to more complicated models that also have a Lyapunov function - and so both models are shown to be stable.  There is generally a connection between the two Lyapunov functions.  However, this connection has not been properly studied as general phenomenon. Having a better understanding of this connection will provide a better understanding of the stability of complicated models.  That will help with planning disease interventions, because the long-term consequences will be more clear.