Research

My work in this area began with stability analysis of a chemotherapy model with delays and with hepatitis B virus infection models in autonomous, nonautonomous, and stochastic settings. More recent projects couple disease dynamics to human behavior: a model in which vaccination behavior, COVID-19 transmission, and economic outcomes influence one another, a cholera model with sanitation controls, and a study of how adaptive risk awareness and fatigue shape the long term coexistence of epidemics and awareness.

A parallel line extends stability theory itself. With G. Farah I have established adaptive control and Mittag-Leffler stability results for Caputo fractional systems with state dependent delays, so that memory dependent dynamics can be analyzed with the same rigor as classical ones. Across these projects the tools are consistent: threshold quantities such as the basic reproduction number, Lyapunov based stability arguments, bifurcation analysis, and optimal control.

A hybrid model keeps the structure of a mechanistic model, $$\frac{dx}{dt}=f(x,\theta )+g(x;\phi )$$ and lets a flexible component g, typically a neural network, absorb what the mechanism f cannot express. Neural and universal differential equations are the general form. The approach retains interpretability where mechanism is trusted and adds flexibility where it is not, but it raises a question that must be answered before any fitting: which parameters can be recovered from the observed outputs at all?

Structural identifiability answers that question symbolically, from the model equations alone. I have written a tutorial on carrying out this analysis in Julia, and I treat identifiability as the first step of every modeling project. The ongoing goal is a theory of identifiable hybrid models: conditions under which a mechanistic model augmented with a learned component remains identifiable, and training procedures that enforce them.

A clinical prediction model is useful only if it is calibrated, so that predicted probabilities match observed frequencies, and interpretable, so that its reasoning can be inspected. With undergraduate researcher M. Johnson and J. Elrefaei I developed a calibrated and interpretable model for ICU mortality from the first twenty four hours of clinical data, and with clinical collaborators an interpretable prognosis model for mycetoma from routine measurements. In statistical methodology, I developed a method for estimating circular statistics, such as the phase of an oscillation, when each measurement carries a systematic bias.

An emerging direction treats the training of machine learning and deep learning models as a problem in dynamical systems and control. Gradient descent discretizes a gradient flow, momentum methods are second order dynamics, and stochastic gradients inject structured noise, so the stability, delay, and perturbation theory I use for biological systems applies to the algorithms that fit models as well. I am pursuing step size and adaptive schemes with stability guarantees, optimizers whose effect on robustness and calibration can be characterized, and training methods for hybrid and neural differential equation models that respect identifiability constraints. This direction is in progress and has not yet produced publications.