Final Report Summary - EPIDELAY (Delay differential models and transmission dynamics of infectious diseases) The main goal of this project was to advance the mathematical theory of functional differential equations and apply it to better understand the transmission dynamics of infectious diseases. Time delays arise naturally in epidemiological modelling in many forms, such as the length of the incubation period, the duration of immunity, the time needed to travel to distant places, the maturation period of disease transmitting vectors, the time to initiate an intervention strategy, or a delay in start of the treatment of a patient. We have developed new mathematical tools so that the investigation of mathematical models taking into account such factors became feasible from the modelling, analytical, numerical and simulation points of view. Our mathematical toolbox now allows us to go beyond the case of a single constant delay and treat models with more realistic and complex forms of the delays, such as multiple, distributed, unbounded or state dependent delays. Using our delay differential models and methods, we have not only constructed more accurate and realistic models, but also uncovered and explained new nonlinear phenomena of disease dynamics, and discovered novel bifurcation diagrams. We have obtained several counterintuitive results which can be obtained only by the careful mathematical analysis of nonlinear functional differential equations. We have had new results for some classical model equations where there has been no progress for a long time, and proved a more than two decades old global stability conjecture. We applied our techniques to real life epidemiological problems related to influenza, measles, ebola, malaria, pertussis, vaccination strategies or the emergence of drug-resistance.