Given several related critical infrastructure (CI) networks, such as power grid, transportation, and water systems, one crucial question emerges: how to model the propagation of failed facilities and predict their spread over time to the whole system? Given digital surveillance data, can we predict the impact of Influenza-Like Illness (ILI), including the percentage of outpatient doctors visits, the season duration, and peak? These two questions are related to modeling and predicting the incidence of different types of contagions. In the case of CI, the contagions are the failures of facilities. In the case of flu spread, the contagions are the infective ILI. In this thesis, in the case of CI, we give a novel model of failure cascades and use it to identify key facilities in an optimization-based approach, called HotSpots. In the case of flu spread, we develop a deep neural network, EpiDeep, to predict multiple key epidemiology metrics. In both of these applications, we use the dynamics of propagation to develop better approaches. By collaborating with Oak Ridge National Laboratory (ORNL) and working on the real CI networks provided by them, we find that HotSpots helps solve what-if scenarios. By using the digital surveillance data reported by the Centers for Disease Control and Prevention (CDC), we carry on experiments and find that EpiDeep is better than non-trivial baselines and outperforms them by up to 40%. We believe the generality of our approaches, and it can be applied to other propagation-based scenarios in infrastructure and epidemiology. / M.S. / Critical Infrastructure Systems (CIS), including the power grid, transportation, and gas systems, are essential to national security, economy, and political stability. Moreover, they are interconnected and are vulnerable to potential failures. The previous event, like 2012 Hurricane Sandy, showed how these interdependencies can lead to catastrophic disasters among the whole systems. Therefore, one crucial question emerges: Given several related CIS networks: how to model the propagation of failed facilities and predict their spread over time to the whole system? Similarly, in the case of seasonal influenza, it always remains a significant health issue for many people in every country. The time-series of the weighted Influenza-like Illness (wILI) data are provided to researchers by the US Center for Disease Control and Prevention (CDC), and researchers use them to predict several key epidemiological metrics. The question, in this case, is: Given the wILI time-series, can we predict the impact of Influenza-Like Illness (ILI) accurately and efficiently? Both of these questions are related to modeling and predicting the incidence of different types of contagions. Contagions are any infective trend which can spread inside a network, including failures of facilities, illness of human, and popular news. In the case of CIS, the contagions are the failures of facilities. In the case of flu spread, the contagions are the infective ILI. In this thesis, in the case of CI, we present a novel model of failure cascades and use it to identify critical facilities in an optimization-based approach. In the case of flu spread, we develop a deep neural network to predict multiple key epidemiology metrics. In both of these applications, we use the dynamics of propagation to create better approaches. By collaborating with ORNL and working on the real CI networks provided by them, we find that F-CAS captures the dynamics of the interconnected CI networks. In the experiments using the wILI data from CDC, we find that EpiDeep is better than non-trivial baselines and outperforms them by up to 40%. We believe the generality of our approaches, and it can be applied to other propagation-based scenarios in infrastructure and epidemiology.