Abstract
In 1927, Kermack & McKendrick published the susceptible -- infected -- recovered (SIR) model, based on a system of ordinary differential equations (ODEs), which still forms the basis of many compartmental models in epidemiology. This formalism assumes that residence time in the infectious compartment follows an exponential law. This law is Markovian, meaning that the probability of leaving the compartment does not depend on the time spent in the compartment, which is biologically unrealistic. This aspect was later corrected by chaining together several compartments corresponding to the same epidemiological reality, transforming the underlying exponential distribution into a hypoexponential one.However, as early as 1932, Kermack & McKendrick proposed an alternative based on a system of partial differential equations (PDEs) whose use remained marginal in the decades that followed, despite the non-Markovian aspect. In this thesis, we return to the contribution of non-Markovian properties in epidemiology and evolution, focusing on this PDE-based formalism. Taking SARS-CoV-2 as an example, we look at the modelling of different biological processes. In particular, we'll be focusing on modelling immunity, which is empirically dependent on time since clearance and is therefore non-Markovian. This component is crucial in the medium- and long-term SARS-CoV-2 epidemic. The use of PDEs is particularly interesting for modelling imperfect immunity, where each individual can be reinfected provided the strength of the infection is high enough --- which seems to be the case for SARS-CoV-2.