An analytical framework for the study of epidemic models on activity driven networks

Network theory has greatly contributed to an improved understanding of epidemic processes, offering an empowering framework for the analysis of real-world data, prediction of disease outbreaks, and formulation of containment strategies. However, the current state of knowledge largely relies on time-invariant networks, which are not adequate to capture several key features of a number of infectious diseases. Activity driven networks (ADNs) constitute a promising modelling framework to describe epidemic spreading over time varying networks, but a number of technical and theoretical gaps remain open. Here, we lay the foundations for a novel theory to model general epidemic spreading processes over time-varying, ADNs. Our theory derives a continuous-time model, based on ordinary differential equations (ODEs), which can reproduce the dynamics of any discrete-time epidemic model evolving over an ADN. A rigorous, formal framework is developed, so that a general epidemic process can be systematically mapped, at first, on a Markov jump process, and then, in the thermodynamic limit, on a system of ODEs. The obtained ODEs can be integrated to simulate the system dynamics, instead of using computationally intensive Monte Carlo simulations. An array of mathematical tools for the analysis of the proposed model is offered, together with techniques to approximate and predict the dynamics of the epidemic spreading, from its inception to the endemic equilibrium. The theoretical framework is illustrated step-by-step through the analysis of a susceptible–infected–susceptible process. Once the framework is established, applications to more complex epidemic models are presented, along with numerical results that corroborate the validity of our approach. Our framework is expected to find application in the study of a number of critical phenomena, including behavioural changes due to the infection, unconscious spread of the disease by exposed individuals, or the removal of nodes from the network of contacts.