Mathematical Models in Life and Social Sciences

Structured population dynamics deals with the time evolution of the population composition with respect to the state of the individuals. Typically, the structuring variables are age (with different interpretations, e.g. chronological or time since infection or time since cell division), size, phenotype, maturity level, space, etc. The simplest structure is given by the age of the individuals since the evolution of age over time proceeds with speed one. Moreover, age-structured population models play a central role since, in some cases, models considering other physiological characteristics can be reduced to them. The course encompasses (describes/analyzes) different types of models: unstructured and structured, continuous and discrete, all of them coming from ecology (highlighting the interactions between individuals and their environment) and epidemiology (e.g. the spread of infectious diseases). The course is divided in 4 parts: an introduction to the abstract setting for the linear case (semigroup approach); early ecological and epidemiological models (ordinary differential equations); continuously structured population models (partial differential equations/integral equations); and matrix population models (discrete structure). Background: basic knowledge of ordinary differential equations and linear algebra.