Compartment models, based on coupled systems of ordinary differential equations, are found everywhere in modern mathematical biology. Some examples include the susceptible-infected-recovered (“SIR”) model in epidemiology, the predator-prey models of animal populations in the wild, and physiological based pharmacokinetic (PBPK) modelling for the interaction and absorption of pharmaceuticals in various tissues in the human body.
In many cases these smooth models represent a sort of “mean-field” or “large population limit" of an underlying stochastic reality. For example, the spread of a disease will essentially be a random process on a discrete population. However, SIR models based on coupled ODEs, that are continuous and deterministic, will closely resemble the progression of a typical epidemic. So, they continue to be widely employed.
But certain stochastic processes need special treatment. For example, a stochastic process with age structure or anomalous waiting times does not necessarily correspond to a first order ODE model. In this project you will explore the basic assumptions of these classical models and situations where they may not be so applicable. You will discover the mathematical theory to handle these cases, including the world of non-local or integral operators and fractional calculus.
This project would suit students interested in biological modelling and scientific computing. Some experience in numerical computation in Python would be useful. The project will be in partnership with other researchers at the Biological Data Science Institute (BDSI).
Angstmann, C.N., Erickson, A.M., Henry, B.I., McGann, A.V., Murray, J.M., Nichols, J.A., 2021. A General Framework for Fractional Order Compartment Models. SIAM Rev. 63, 375–392. https://doi.org/10.1137/21M1398549