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One of the popular dynamics on complex networks is the epidemic spreading. An epidemic model describes how infections spread throughout a network. Among the compartmental models used to describe epidemics, the Susceptible-Infected-Susceptible (SIS) model has been widely used. In the SIS model, each node can be susceptible, become infected with a given infection rate, and become again susceptible with a given curing rate. In this paper, we add a new compartment to the classic SIS model to account for human response to epidemic spread. Each individual can be infected, susceptible, or alert. Susceptible individuals can become alert with an alerting rate if infected individuals exist in their neighborhood. An individual in the alert state is less probable to become infected than an individual in the susceptible state; due to a newly adopted cautious behavior. The problem is formulated as a continuous-time Markov process on a general static graph and then modeled into a set of ordinary differential equations using mean field approximation method and the corresponding Kolmogorov forward equations. The model is then studied using results from algebraic graph theory and center manifold theorem. We analytically show that our model exhibits two distinct thresholds in the dynamics of epidemic spread. Below the first threshold, infection dies out exponentially. Beyond the second threshold, infection persists in the steady state. Between the two thresholds, the infection spreads at the first stage but then dies out asymptotically as the result of increased alertness in the network. Finally, simulations are provided to support our findings. Our results suggest that alertness can be considered as a strategy of controlling the epidemics which propose multiple potential areas of applications, from infectious diseases mitigations to malware impact reduction.
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We study the problem of estimating the parameters (i.e., infection rate and recovery rate) governing the spread of epidemics in networks. Such parameters are typically estimated by measuring various characteristics (such as the number of infected and recovered individuals) of the infected populations over time. However, these measurements also incur certain costs, depending on the population being tested and the times at which the tests are administered. We thus formulate the epidemic parameter estimation problem as an optimization problem, where the goal is to either minimize the total cost spent on collecting measurements, or to optimize the parameter estimates while remaining within a measurement budget. We show that these problems are NP-hard to solve in general, and then propose approximation algorithms with performance guarantees. We validate our algorithms using numerical examples.
In the real world, many complex systems interact with other systems. In addition, the intra- or inter-systems for the spread of information about infectious diseases and the transmission of infectious diseases are often not random, but with direction. Hence, in this paper, we build epidemic model based on an interconnected directed network, which can be considered as the generalization of undirected networks and bipartite networks. By using the mean-field approach, we establish the Susceptible-Infectious-Susceptible model on this network. We theoretically analyze the model, and obtain the basic reproduction number, which is also the generalization of the critical number corresponding to undirected or bipartite networks. And we prove the global stability of disease-free and endemic equilibria via the basic reproduction number as a forward bifurcation parameter. We also give a condition for epidemic prevalence only on a single subnetwork. Furthermore, we carry out numerical simulations, and find that the independence between each nodes in- and out-degrees greatly reduce the impact of the networks topological structure on disease spread.
Real epidemic spreading networks often composed of several kinds of networks interconnected with each other, and the interrelated networks have the different topologies and epidemic dynamics. Moreover, most human diseases are derived from animals, and the zoonotic infections always spread on interconnected networks. In this paper, we consider the epidemic spreading on one-way circular-coupled network consist of three interconnected subnetworks. Here, two one-way three-layer circular interactive networks are established by introducing the heterogeneous mean-field approach method, then we get the basic reproduction numbers and prove the global stability of the disease-free equilibrium and endemic equilibrium of the models. Through mathematical analysis and numerical simulations, it is found that the basic reproduction numbers $R_0$ of the two models are dependent on the infection rates, infection periods, average degrees and degree ratios. In the first model, the network structures of the inner contact patterns have a bigger impact on $R_0$ than that of the cross contact patterns. Under the same contact pattern, the internal infection rates have greater influence on $R_0$ than the cross-infection rates. In the second model, the disease prevails in a heterogeneous network has a greater impact on $R_0$ than the disease from a homogeneous network, and the infections among the three subnetworks all play a important role in the propagation process. Numerical examples verify and expand these theoretical results very well.
Temporal networks are widely used to represent a vast diversity of systems, including in particular social interactions, and the spreading processes unfolding on top of them. The identification of structures playing important roles in such processes remains largely an open question, despite recent progresses in the case of static networks. Here, we consider as candidate structures the recently introduced concept of span-cores: the span-cores decompose a temporal network into subgraphs of controlled duration and increasing connectivity, generalizing the core-decomposition of static graphs. To assess the relevance of such structures, we explore the effectiveness of strategies aimed either at containing or maximizing the impact of a spread, based respectively on removing span-cores of high cohesiveness or duration to decrease the epidemic risk, or on seeding the process from such structures. The effectiveness of such strategies is assessed in a variety of empirical data sets and compared to baselines that use only static information on the centrality of nodes and static concepts of coreness, as well as to a baseline based on a temporal centrality measure. Our results show that the most stable and cohesive temporal cores play indeed an important role in epidemic processes on temporal networks, and that their nodes are likely to represent influential spreaders.
The customary perspective to reason about epidemic mitigation in temporal networks hinges on the identification of nodes with specific features or network roles. The ensuing individual-based control strategies, however, are difficult to carry out in practice and ignore important correlations between topological and temporal patterns. Here we adopt a mesoscopic perspective and present a principled framework to identify collective features at multiple scales and rank their importance for epidemic spread. We use tensor decomposition techniques to build an additive representation of a temporal network in terms of mesostructures, such as cohesive clusters and temporally-localized mixing patterns. This representation allows to determine the impact of individual mesostructures on epidemic spread and to assess the effect of targeted interventions that remove chosen structures. We illustrate this approach using high-resolution social network data on face-to-face interactions in a school and show that our method affords the design of effective mesoscale interventions.
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