Topological and Geometric approaches in Epidemiology

Abstract

Science has tried to use advanced modelling to understand nature behaviours. Dynamical systems often rise as the first option to forecast epidemic courses from parameter estimation and epidemic curve fitting. Even though such techniques do not always lead to successful results, the answer to this fact might be coupled with the variety of data sets and presence of noise and delays in the time collection, which might interview directly in the model precision. As a well-known parallel data-driven approach, Topological Data Analysis (TDA) emerges as a trend since the last decade as a promising and powerful tool for data science. Currently, infectious diseases have threatening humanity and concerning health system worldwide. Recent Dengue outbreaks have been reported over the past years, concerned with world health care and became our motivation to use and build geometric and topological techniques for a more in-depth, data-driven understanding of this topic. Yet, the damage of the novel Coronavirus disease (COVID-19) is reaching unprecedented scales. Numerous classical epidemiology models are trying to quantify epidemiology metrics. In the work of this thesis, we propose a data-driven, parameter-free, topological and geometric approaches to access the emergence of a pandemic states by studying the Euler characteristics and Ricci curvature discretizations. We first compute this curvatures in toy-models of epidemic time-series, which allows us to create epidemic networks. Those curvatures allow us to detect early warning signs of the emergence of the pandemic. The advantage of our method lies in providing an early geometrical data marker for the pandemic state, regardless of parameter estimation and stochastic modelling. This work opens the possibility of using discrete geometry to study epidemic networks.