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To forecast the time dynamics of an epidemic, we propose a discretestochastic model that unifies and generalizes previous approaches to thesubject. Viewing a given population of individuals or groups of individualswith given health state attributes as living in and moving between the nodes ofa graph, we use Monte-Carlo Markov Chain techniques to simulate the movementsand health state changes of the individuals according to given probabilities ofstay that have been preassigned to each of the nodes. We utilize this model toeither capture and predict the future geographic evolution of an epidemic intime, or the evolution of an epidemic inside a heterogeneous population whichis divided into homogeneous sub-populations, or, more generally, its evolutionin a combination or superposition of the previous two contexts. We also provethat when the size of the population increases and a natural hypothesis issatisfied, the stochastic process associated to our model converges to adeterministic process. Indeed, when the length of the time step used in thediscrete model converges to zero, in the limit this deterministic process isdriven by a differential equation yielding the evolution of the expectationvalue of the number of infected as a function of time. In the second part ofthe paper, we apply our model to study the evolution of the Covid-19 epidemic.We deduce a decomposition of the function yielding the number of infectiousindividuals into "wavelets", which allows to trace in time the expectationvalue for the number of infections inside each sub-population. Within thisframework, we also discuss possible causes for the occurrence of multipleepidemiological waves.

Unified modelling of epidemics by coupled dynamics via Monte-Carlo Markov Chain algorithms