Mathematical Biology

Mathematical modeling, when combined with diverse epidemiological datasets, provides valuable insights for understanding and controlling infectious diseases. In this talk, I will present a series of case studies demonstrating how the synergy between modeling and serological, case-based, and wastewater surveillance data can enhance disease monitoring and inform public health strategies.

Serological data measures biomarkers of infection or vaccination, offering direct estimates of population immunity. This approach complements case data by providing a broader understanding of disease epidemiology. Wastewater surveillance, which detects pathogen genomes in sewage, captures infections across the entire population, including symptomatic, asymptomatic, and pre-/post-symptomatic individuals. This approach complements traditional case reporting by providing a broader, community-wide perspective on disease transmission.

In the first case study, I will discuss how we utilized serological data and a static cohort model to quantify the relative contributions of natural infection, routine vaccination, and supplementary immunization activities (SIAs) to measles seroconversion in Kenyan children. In the second case study we developed a simple static model combined with serological data to evaluate the effectiveness of SIAs in reducing the risk of a measles outbreak in the post-pandemic period. Finally, I will introduce my current work on wastewater-based epidemic modeling for mpox surveillance in British Columbia, demonstrating how wastewater and case data can be integrated within a dynamic transmission model to predict future scenarios of mpox outbreak.

These case studies illustrate the power of mathematical modeling in integrating multiple data sources to inform public health strategies and improve infectious disease control efforts.

Individual and collective cell polarity has fascinated mathematical modelers for a long time. Recently, a more subtle type of symmetry breaking started to attract attention of experimentalists and theorists alike - emergence of chirality in single cells and in cell groups. I will describe a joint project with Bershadsky/Tee lab to understand collective cell chirality on adhesive islands. From the initial microscopy data, two potential models emerged: in one, cells elongate and slowly rotate, and neighboring cells align with each other. When the collective rotation is stopped by the island boundaries, chirality emerges. In an alternative model, cells become chiral due to stress fibers turns inside the cells on the boundary, and then the polarity pattern propagates inward into the cellular groups. We used agent-based modeling to simulate these two hypotheses. The models make many predictions, and I will show how we discriminated between the models by comparing the data to these predictions.

Cell movement requires long-range coordination of the cytoskeletal machinery that organizes cell morphogenesis. We have found that reciprocal interactions between biochemical signals and physical forces enable this long-range signal integration. Through a combination of optogenetic inputs, mechanical measurements, and mathematical modeling, we resolve a recent controversy regarding the role of membrane tension propagation in this process and reveal the requirements for long-range transmission of tension in cells. Most cells don't move in isolation-- they collectively migrate by sharing information similar to the flocking of birds, the schooling of fish, and the swarming of ants. We reveal a novel active signal relay system that rapidly and robustly ensures the proper level of immune cell recruitment to sites of injury and infection.

The Moran process models the evolutionary dynamics between two competing types in a population, traditionally assuming a fixed population size. We investigate an extension to this process which adds ecological aspects through variable population sizes. For the original Moran process, birth and death events are correlated to maintain a constant population size. Here we decouple the two events and derive the stochastic differential equation that represents the dynamics in a well-mixed population and captures its behaviour as the population size becomes arbitrarily large. Our analysis explores the impact of this decoupling on two key determinants of the evolutionary process: fixation probabilities and fixation times. In evolutionary graph theory, these statistics depend significantly on the population structure, such that structures have been identified that act as ‘amplifiers’ of selection while others are ‘suppressors’ of selection. However, these features are crucially dependent on the sequence of events, such as birth-death vs death-birth – a seemingly small change with significant consequences. In our extension of the Moran process this distinction is no longer necessary or possible. We determine the fixation probabilities and times for the well-mixed population, regular graphs as well as amplifiers and suppressors, and compare them to the original Moran process.

Skeletal muscle is composed of cells collectively referred to as fibers, which themselves contain contractile proteins arranged longtitudinally into sarcomeres. These latter respond to signals from the nervous system, and contract; unlike cardiac muscle, skeletal muscles can respond to voluntary control. Muscles react to mechanical forces - they contain connective tissue and fluid, and are linked via tendons to the skeletal system - but they also are capable of activation via stimulation (and hence, contraction) of the sacromeres. The restorative along-fibre force introduce strong mechanical anisotropy, and depend on departures from a characteristic length of the sarcomeres; diseases such as cerebral palsy cause this characteristic length to change, thereby impacting muscle force. In the 1910s, A.V. Hill [1] posited a mathematical description of skeletal muscles which approximated muscle as a 1-dimensional nonlinear and massless spring. This has been a remarkably successful model, and remains in wide use. Yet skeletal muscle is three dimensional, has mass, and a fairly complicated structure. Are these features important? What insights are gained if we include some of this complexity in our models? Many mathematical questions of interest in skeletal muscle mechanics arise: how to model this system, how to discretize it, and what theoretical properties does it have? In this talk, we survey recent work on the modeling, parameter estimation, simulation and validation of a fully 3-D continuum elasticity approach for skeletal muscle dynamics. This is joint work based on a long-standing collaboration with James Wakeling (Dept. of Biomedical Physiology and Kinesiology, SFU).