Recent network research has focused on the cascading failures in a system of interdependent networks and the necessary preconditions for system collapse. An important question that has not been addressed is how to repair a failing system before it suffers total breakdown. Here we introduce a recovery strategy for nodes and develop an analytic and numerical framework for studying the concurrent failure and recovery of a system of interdependent networks based on an efficient and practically reasonable strategy. Our strategy consists of repairing a fraction of failed nodes, with probability of recovery γ, that are neighbors of the largest connected component of each constituent network. We find that, for a given initial failure of a fraction 1 − p of nodes, there is a critical probability of recovery above which the cascade is halted and the system fully restores to its initial state and below which the system abruptly collapses. As a consequence we find in the plane γ − p of the phase diagram three distinct phases. A phase in which the system never collapses without being restored, another phase in which the recovery strategy avoids the breakdown, and a phase in which even the repairing process cannot prevent system collapse.

In this work we propose and investigate a new strategy of vaccination, which we call "dynamic vaccination". In our model, susceptible people become aware that one or more of their contacts are infected, and thereby get vaccinated with probability ω, before having physical contact with any infected patient. Then, the non-vaccinated individuals will be infected with probability β.We apply the strategy to the SIR epidemic model in a multiplex network composed by two networks, where a fraction q of the nodes acts in both networks. We map this model of dynamic vaccination into bond percolation model, and use the generating functions framework to predict theoretically the behavior of the relevant magnitudes of the system at the steady state. We find a perfect agreement between the solutions of the theoretical equations and the results of stochastic simulations. In addition, we find an interesting phase diagram in the plane β − ω, which is composed by an epidemic and a non-epidemic phases, separated by a critical threshold line β c , which depends on q. As q decreases, β c increases, i.e., as the overlap decreases, the system is more disconnected, therefore more virulent diseases are needed to spread epidemics. Surprisingly we find that, for all values of q, a region in the diagram where the vaccination is so efficient that, regardless of the virulence of the disease, it never becomes an epidemic. We compare our strategy with random immunization and find that using the same amount of vaccines for both scenarios, we obtain that the spread of the disease is much lower in the case of dynamic vaccination when compared to random immunization. Furthermore, we also compare our strategy with targeted immunization and we find that, depending on ω, dynamic vaccination will perform significantly better, and in some cases will stop the disease before it becomes an epidemic. * lgalvere@mdp.edu.ar