Study of phase transition on a temporal network, and application to epidemiology

In this work we address the problem of assessing the vulnerability of a system of interacting individuals targeted with a disease. The disease is modelled using the SIS compartmental model, for which people can be either susceptible S or infectious I. The transition S→I takes place with probability λ, if a S meets an I; the I→S recovery happens spontaneously with probability μ. We dispose of a data-set regarding the interactions of a bipartite community whose contacts change in time. To encode this dynamics we exploit the formalism of temporal networks. We have focused on the so called epidemic threshold (e.t.) in order to give a quantification of the vulnerability of the system. In infinite size systems, the epidemic threshold is the value of diseases's parameters μ, so that for μ >μc the number of infectious people in the stationary state is non zero, while in the other case reaches the disease-free state. We carry out the analysis of the threshold both numerically and analytically. In order to compute the e.t. , we exploit the findings of a recent work on temporal networks coupled with disease dynamics. This work demonstrates that, when it comes to the determination of the epidemic threshold on a temporal network, the microscopic stochastic simulation of a disease-spreading can be replaced with a simpler and faster computation of the spectral radius of a suitable matrix, the so called Infection Propagator. This matrix encodes the coupled dynamics of disease and network in a finite period of time. Using this approach, we evaluate the phase-transition diagram threshold μc(λ) in various portion of the temporal data set. As this system regards woman-man interactions relative to sexual contacts, we also allow for men and women to have different λ, as medical studies show. We find that the impact of the two categories on the threshold is symmetric, and that the present system is largely non vulnerable with respect to the most common STDs. We then address the question of which is the main temporal or topological feature that drives the disease on the network. We brake some properties of the network in order to see their impact on the e.t., and we conclude that the aggregated network is the structure one needs to preserve in order to capture the dynamic of the disease. To tackle the problem analytically, we test on our network three analytical models that allow to write an explicit expression for the e.t. , generalising them to a bipartite structure. Only the time averaged approximation of the temporal network works fine. Lastly, we try to ameliorate this approximation introducing a stochastic correction. We write a N- dimensional Langevin equation for the evolution of the state of probability of infection per each node, p(t), when close to the disease-free state, i.e. p(t)~0. The deterministic part of the equation is the same that describes the time-average regime. The stochastic part includes a Wiener noise-matrix that aims to reproduce the fluctuations of the temporal contact matrix of the network respect to the annealed matrix. The resulting equation is a N-dimensional geometric Brownian motion. By extending the findings for the asymptotic state of a one dimensional GBM, we find an epidemic threshold very close to the actual one. For the future, it would be interesting to delve more into this stochastic approach and test its validity on other networks and on synthetic models; and maybe on other types of noises , such as the Poissonian or shot noise, that resembles more to the real behaviour of human contact dynamics.