Stability of Spreading Processes in Networks with Non-Markovian Transmission and Recovery

Although viral spreading processes taking place in networks are commonly analyzed using Markovian models in which both the transmission and the recovery times follow exponential distributions, empirical studies show that, in most real scenarios, the distribution of these times are far from exponential. To overcome this limitation, we first introduce a generalized spreading model that allows transmission and recovery times to follow arbitrary distributions within a given accuracy. In this context, we derive conditions for the generalized spreading model to converge exponentially fast towards the infection-free equilibrium with a given rate (in other words, to eradicate the viral spread) without relying on mean-field approximations. Based on our results, we illustrate how the particular shape of the transmission/recovery distribution heavily influences the convergence rate. Therefore, modeling non-exponential transmission/recovery times observed in realistic spreading processes via exponential distributions can induce significant errors in the stability analysis of the spreading dynamics.