Detectability limits of epidemic sources in networks

The detection of an epidemic source or the patient zero is an important practical problem that can help in developing the epidemic control strategies. In this paper, we study the statistical inference problem of detecting the source of epidemics from a snapshot of a contagion spreading process at some time on an arbitrary network structure. By using exact analytic calculations and Monte Carlo simulations, we demonstrate the detectability limits for the SIR model, which primarily depend on the spreading process characteristics. We introduce an efficient Bayesian Monte Carlo source probability estimator and compare its performance against state-of-the-art approaches. Finally, we demonstrate the applicability of the approach in a realistic setting of an epidemic spreading over an empirical temporal network of sexual interactions. Introduction The majority of biological, technological, social and information systems structures can be represented as a complex network [1, 2, 3]. The most prevalent type of dynamic processes of public interest characteristic for the real-life complex networks are contagion processes [4]. Different mathematical methods have been used to study the epidemic spreading on complex networks including the bond percolation method [7, 8], the mean-field approach [9, 10], the reaction-diffusion processes [11, 12], pair and the master equation approximations [13] as well as models with the complex compartmental structure along with population mobility dynamics [14]. Epidemiologists detect the epidemic source or the patient-zero either by analysing the temporal genetic evolution of virus strains [5] or try to do a contact backtracking [6] from the available observed data. However, in cases where the information on the times of contact is unknown, or incomplete, the backtracking method is no longer adequate. This 1 ar X iv :1 40 6. 29 09 v1 [ cs .S I] 1 1 Ju n 20 14 becomes especially hard if the only data available is a static snapshot of a epidemic process at some time. Even in cases when we do have some information on the times of contact, the longer the recovery time and subtler the symptoms the harder it becomes to establish the proper ordering of the transmissions that have occurred. Due to its practical aspects and theoretical importance, the epidemic source detection problem on contact networks has recently gained a lot of attention in complex network science community. This has led to the development of many different source detection estimators for static networks, which vary in their assumptions on the network structure and the spreading process models [15, 16, 17, 18, 19, 20, 21, 22, 23]. For the source detection with the SI model the following interesting results have been obtained. Zaman et. al. developed a rumor centrality measure, which is the maximum likelihood estimator for regular trees under the SI model [15]. Dong et. al. also studied the problem of rooting the rumor source with the SI model and demonstrated the asymptotic source detection probability on regular tree-type networks [16]. Comin et. al. compared the different centrality measures e.g. the degree, the betweenness, the closeness and the eigenvector centrality as the source detection estimators [22]. Wang et. al. addressed the problem of source estimation from multiple observations under the SI model [17]. Pinto et. al. used the SI model and assumed that the direction and the times of the infection are known exactly, and solved diffusion tree problem using breadth first search from sparsely placed observers [19]. In the case of the SIR model there are two different approaches. Zhu et. al. adopted the SIR model and proposed a sample path counting approach for the source detection [18]. They proved that the source node on infinite trees minimizes the maximum distance (Jordan centrality) to the infected nodes. Lokhov et. al. used a dynamic message-passing algorithm (DMP) for the SIR model to estimate the probability that a given node produces the observed snapshot. They use a mean-field-like approximation (independence approximation) and an assumption of a tree-like contact network to compute the marginal probabilities [20]. The main contributions of the paper are the following: (i) given the non-uniqueness of finding a single epidemic source of the SIR realization on general networks, we turn the problem to finding a source probability distribution, which is a well-posed problem; (ii) we develop the analytic combinatoric and the direct Monte-Carlo approaches for determining theoretical source probability distribution and produce the benchmark solutions on the 4-connected lattice; (iii) we measure the source detectability by using the normalized Shannon entropy of the estimated source probability distribution for each of the source detection problems and observe the existence of the highly detectable and the highly undetectable regimes; (iv) using the above insights, we construct the Soft Margin epidemic source detection estimator for the arbitrary networks (static and temporal) and show that it is robust and more accurate than the state-ofthe-art approaches and much faster than the analytic combinatoric or the direct Monte-Carlo approach; (v) by using the simulations of the sexually transmitted disease (STD) model on a realistic time interval of 200 days on an empirical temporal network of sexual contacts (see the network visualization in Figure 4, plot C) we demonstrate the robustness to the uncertainty in the epidemic starting time, the network interaction orderings and in incompleteness of observations. Although we use the SIR model of epidemic spreading, our algorithms are easily applicable to other compartmental models, e.g. SI and SEIR and all other compartmental models where the states cannot be recurrent. 1 Detectability limits The main goals of this work are to better understand the nature of the epidemic source detection problem in networks, characterize its complexity, and develop efficient algorithms for estimating source probability distribution. Next, we introduce the terminology and formalize the problem. In a general case, the contactnetwork during an epidemic process can be temporal