Research

Summary

My research interests lie at the crossroads of dynamical systems, mathematical modeling, and computational sciences. A common thread in my research is exploring problems that arise in population dynamics, control theory, and epidemiology by using tools of applied analysis and computational methods to characterize these models in order to address relevant biological or epidemiological questions.

Overview

Typically, my research is structured into two key components. The first step is to better understand these biosystems in order to formulate mathematical models that describe these phenomena. Oftentimes, these models are represented by nonlinear dynamical systems. The second step is to mathematically analyze these problems to gain insights and characterize the initial problems. These two undertakings are synergetic: biological questions lead to new challenges in mathematics, and the study of these mathematical problems provide tools in understanding the behavior of natural phenomena. My focus is on three primary areas of research detailed below. But before weaving into details, a word cloud of my research is this image (shout-out to those who recognized the shape of the cloud).

1. Residence times concepts and Lagrangian approaches:  I have developed a new framework in modeling infectious diseases which consists of using environmental risk factors and proportion of times spent in these environment – or patches – as a proxy of contact rates, which are difficult to measure or define for communicable diseases.  These investigations have led to advances in understanding the effects of virtual dispersal,  heterogeneity in groups and patches among others. 
   
2. Global stability for large dynamical systems: Until recently, the global stability of non-trivial equilibria of models describing the evolution of epidemics has been an open problem. The existence and the global stability study of these equilibria become even more challenging for large-scale dynamical systems, which capture more detailed features of diseases. I have contributed to solving these types of problems in different settings, using Lyapunov functions and monotone systems theory. These constributions includes competive exclusion principle, vector-borne diseases and zoonoses.
   
3. States and parameters estimation: The estimation of unknown parameters and some state variables is a notorious problem in epidemiology. Indeed, in many epidemic models, the transmission parameter, the index cases (as a component of the initial conditions of the dynamical system), the number of latent, or the total number of mosquito populations are not known. I have developed methods to estimate parameters and state variables for some epidemic models using tools of control theory.