Analysis, Estimation, and Validation of Discrete-Time Epidemic Processes

Abstract

Models of spreading processes over nontrivial networks are commonly motivated by modeling and analysis of biological networks, computer networks, and human contact networks. However, learning the spread parameters of such models has not yet been explored in detail, and the models have not been validated by real data. In this paper, we present several different spread models from the literature and explore their relationships to each other; for one of these processes, we present a sufficient condition for asymptotic stability of the healthy equilibrium, show that the condition is necessary and sufficient for uniqueness of the healthy equilibrium, and present necessary and sufficient conditions for estimating the spread parameters. Finally, we employ two real data sets, one from John Snow's seminal work on cholera epidemics in London in the 1850s and the other one from the United States Department of Agriculture, to validate an approximation of a well-studied network-dependent susceptible-infected-susceptible model.

The Viral State Dynamics of the Discrete-Time NIMFA Epidemic Model

“…An extension to more heterogeneous spreading parameters was provided in [7], with the constraint that there is a β i for every node i such that for every node j = i either β ij = β i or β ij = 0. The discrete-time NIMFA model with homogeneous spreading parameters has been studied in [11,10,13]. The discrete-time NIMFA model (2) with fully heterogeneous spreading parameters has been proposed by Paré et al [10], who showed that there is either one stable equilibrium, the healthy state v[k] = 0, or there are two equilibria, the healthy state and a steady-state v ∞ with positive components.…”

“…P5. There is a unique [9,10] In this work, we give answers to the questions Q2 and Q3. In summary, the NIMFA system (2) is a well-behaved and powerful model, which can be applied to a variety of epidemic phenomena due to the full heterogeneity of the spreading parameters W, q.…”

Network Reconstruction and Prediction of Epidemic Outbreaks for General Group-Based Compartmental Epidemic Models

“…We propose a polynomial-time network reconstruction algorithm for the discrete-time NIMFA model based on a basis pursuit formulation. Given only few initial viral state observations, the network reconstruction method allows for an accurate prediction of the further viral state evolution of every node provided that the network is sufficiently sparse.Paré et al [7] analysed the equilibria of the discrete-time NIMFA model (3) and validated the dynamics of real-world epidemics, when the nodes of the network corresponds to groups of individuals, namely either households or counties.Recently, estimation methods were proposed [9, 10, 11] to reconstruct the network from viral state observations of susceptible-infected-susceptible (SIS) epidemic models . The maximum-likelihood SIS network reconstruction problem is NP-hard [12], and the number of required viral state observations n seems [9] to grow (almost) exponentially with respect to the network size N .…”

“…The maximum-likelihood SIS network reconstruction problem is NP-hard [12], and the number of required viral state observations n seems [9] to grow (almost) exponentially with respect to the network size N . For the NIMFA model (3), Paré et al [7] proposed a method to estimate the spreading parameters β T and δ T under the assumption that the adjacency matrix A is known exactly.The network reconstruction method in this work is motivated by two factors. First, the tremendous number of required viral state observations and the NP-hardness seem to render the exact SIS network reconstruction hardly viable, and modelling the viral dynamics by the NIMFA equations (1) may allow for a feasible network reconstruction problem.…”

“…Paré et al [7] analysed the equilibria of the discrete-time NIMFA model (3) and validated the dynamics of real-world epidemics, when the nodes of the network corresponds to groups of individuals, namely either households or counties.…”