In this paper, we investigate the well-posedness and dynamics of a class of hybrid models, obtained by coupling a system of ordinary differential equations and an agent-based model. These hybrid models intend to integrate the microscopic dynamics of individual behaviors into the macroscopic evolution of various population dynamics models, and can be applied to a great number of complex problems arising in economics, sociology, geography and epidemiology. Here, in particular, we apply our general framework to the current COVID-19 pandemic. We establish, at a theoretical level, sufficient conditions which lead to particular solutions exhibiting irregular oscillations and interpret those particular solutions as pandemic waves. We perform numerical simulations of a set of relevant scenarios which show how the microscopic processes impact the macroscopic dynamics.

In this paper, we analyze the properties of a complex network of predator-prey systems, modeling the ecological dynamics of interacting species living in a fragmented environment. We consider non-identical instances of a Lotka-Volterra model with Holling type II functional response, which undergoes a Hopf bifurcation, and focus on the possible synchronization of distinct local behaviours. We prove an original result for the near-synchronization of non-identical systems, which shows how to and to what extent an extinction dynamic can be driven to a persistence equilibrium. Our theoretical statements are illustrated by appropriate numerical simulations.