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In an epidemic network, lags due to travel time between populations, latent period, and recovery period can significantly change the epidemic behavior and result in successive echoing waves of the spread between various population clusters. Moreover, external shocks to a given population can propagate to other populations within the network, potentially snowballing into waves of resurgent epidemics. The main objective of this study is to investigate the effect of time delay and small shocks/uncertainties on the linear susceptible-infectious-susceptible (SIS) dynamics of epidemic networks. In this regard, the asymptotic stability of this class of networks is first studied, and then its performance loss due to small shocks/uncertainties is evaluated based on the notion of the norm. It is shown that network performance loss is correlated with the structure of the underlying graph, intrinsic time delays, epidemic characteristics, and external shocks. This performance measure is then used to develop an optimal traffic restriction algorithm for network performance enhancement, resulting in reduced infection in the metapopulation. A novel epidemic-based centrality index is also defined to evaluate the impact of every subpopulation on network performance, and its asymptotic behavior is investigated. It is shown that for specific choices of parameters, the output of the epidemic-based centrality index converges to the results obtained by local or eigenvector centralities. Moreover, given that epidemic-based centrality depends on the epidemic properties of the disease, it may yield distinct node rankings as the disease characteristics slowly change over time or as different types of infections spread. This node interlacing phenomenon is not observed in other centralities that rely solely on network structure. This unique characteristic of epidemic-based centrality enables it to adjust to various epidemic features. The derived centrality index is then adopted to improve the network robustness against external shocks on the epidemic network. The numerical results, along with the theoretical expectations, highlight the role of time delay as well as small shocks in investigating the most effective methods of epidemic containment. 

Several sources of delay in an epidemic network might negatively affect the stability and robustness of the entire network. In this paper, a multi-delayed Susceptible-Infectious-Susceptible (SIS) model is applied on a metapopulation network, where the epidemic delays are categorized into local and global delays. While local delays result from intra-population lags such as symptom development duration or recovery period, global delays stem from inter-population lags, e.g., transition duration between subpopulations. The theoretical results for a network of subpopulations with identical linear SIS dynamics and different types of time-delay show that depending on the type of time-delay in the network, different eigenvalues of the underlying graph should be evaluated to obtain the feasible regions of stability. The delay-dependent stability of such epidemic networks has been analytically derived, which eliminates potentially expensive computations required by current algorithms. The effect of time-delay on the H2 norm-based performance of a class of epidemic networks with additive noise inputs and multiple delays is studied and the closed form of their performance measure is derived using the solution of delayed Lyapunov equations. As a case study, the theoretical findings are implemented on a network of United States’ busiest airports.