Differential equations and dynamical systems with applications in demography, epidemiology and economics

Abstract

A theoretical overview of differential equations and dynamical systems is presented in the first four chapters of the dissertation. The theory is developed systematically, starting with linear and non-linear differential equations in one and two dimensions. Techniques such as proving the existence of closed orbits, bifurcation theory, chaos and iterated maps are dealt with in this section. A brief survey of population models is given in Chapter 5, followed by an in-depth analysis of the Leslie matrix model which is frequently used to describe the dynamics of an age-structured population. A two-sex model is used to project the population of the Maltese islands for the year 2024, taking into consideration the huge impact of the COVID-19 pandemic on migration. Two different scenarios are presented where the model predicts a total population of 529,846 people when considering ‘low’ migration and 580,955 people when considering ‘high’ migration. An overview of classical compartmental models is given in Chapter 6. This idea is extended to multi-site compartments with travelling patterns. An aggregated plot of infected individuals is both modelled and predicted given that the sites are connected via random trees. COVID-19 datasets are used to estimate model parameters applying data fitting. The extended Kalman filter is also implemented to estimate both states and parameter values. The last chapter is dedicated to two economic models which will be examined from a mathematical perspective: The Goodwin model and a model for the relationship between unemployment and inflation.