8. Alicia Prieto, Math. Concrete Problems, Agent-Based Modeling & the Joy of Learning New Things

More than mathematics itself, Alicia Prieto enjoys learning new things. In approaching mathematics she searches first for concrete problems that interest her. Such an approach has its positives and negatives. For Dr. Prieto a main attraction is how it forces her to constantly learn new things. A potential downside is that it is slow. After all, if you start on a problem where you have to learn a bunch before you can make progress, you will never produce results as quickly as others. Using this approach Dr. Preito has worked on diverse problems ranging from modeling immune response in mathematical biology to recommender systems in data science for student course selection. A major focus for Dr. Prieto has been agent-based modeling. This stochastic approach to modeling systems treats elements of interest as “agents” who have a some set distribution for how they move and/or interact with the surrounding systems. As an example, Dr. Prieto discusses her Ph.D. project in some detail. For this project she was modeling the immune response to a biomedical implant. The system was modeled as a grid (essentially a giant matrix) with several levels (so really several copies of the matrix stacked on each other). Each level represented some aspect of the system. For instance one level would represent the position of certain immune cells (say killer T-cells). The movement of each T-cell is stochastic, meaning at each time step there is a probability of the cell moving in each of the different directions. To make such a model work at each time step the random distribution is sampled for every cell and the cells move based on the sampling. A single run of the model means almost nothing. The point of a stochastic model is to run it many, many times (what counts as many depends on the details of the situation). The hope is that the samples run many times will reflect the range of possible outcomes for the actual biological system. We discuss the idea of picking an appropriate model for the situation, contrasting the physics versus biology. In physics situations the degree of control and certainty over the situation often allow for deterministic models. However, in biological situations the phenomena are fundamentally uncertain and variable. We will never know exactly where all the cells are and the cells will all be unique and prone to moving randomly (though random does not mean without connection to external or internal signals). Stochastic models are often appropriate in such situations as they are more flexible and less rigid, meaning they can more readily be modified to accommodate changes in belief, something that more deterministic differential equation-type models often cannot accommodate. We also talk about the problem of verification in doing any applied work and how Dr. Prieto was able to come full circle in verifying aspects of the model she built for her Ph.D. project. Enjoy!