The Covid-19 pandemic outbreak was followed by an huge amount of modeling studies in order to rapidly gain insights to implement the best public health policies. However, most of those compartmental models used a classical ordinary differential equations (ODEs) system based formalism that came with the tacit assumption the time spent in each compartment does not depend of the time already spent in it. To overcome this "memoryless" issue, a widely used workaround is to artificially increase and chain the number of compartments of an unique reality (e.g. many compartments for infected individuals). It allows for a greater heterogeneity and thus be closer to the observed situation, at the cost of rendering the whole model more difficult to apprehend and parametrize. We propose here an alternative formalism based on a partial differential equations (PDEs) system instead of ordinary differential equations, which provides naturally a memory structure for each compartment, and thus allows to keep a restrained number of compartments. We use such a model applied to the French situation, accounting for vaccinal and natural immunity. The results seem to indicate that the vaccination rate is not enough to ensure the end of the epidemic, but, above all, highlight a huge uncertainty attributable to the age-structured contact matrix.