  In the context of epidemic spreading, many intricate dynamical patterns can
emerge due to the cooperation of different types of pathogens or the
interaction between the disease spread and other failure propagation mechanism.
To unravel such patterns, simulation frameworks are usually adopted, but they
are computationally demanding on big networks and subject to large statistical
uncertainty. Here, we study the two-layer spreading processes on
unidirectionally dependent networks, where the spreading infection of diseases
or malware in one layer can trigger cascading failures in another layer and
lead to secondary disasters, e.g., disrupting public services, supply chains,
or power distribution. We utilize a dynamic message-passing method to devise
efficient algorithms for inferring the system states, which allows one to
investigate systematically the nature of complex intertwined spreading
processes and evaluate their impact. Based on such dynamic message-passing
framework and optimal control, we further develop an effective optimization
algorithm for mitigating network failures.
