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The problem of identifying the source of an epidemic (also called patient zero) given a network of contacts and a set of infected individuals has attracted interest from a broad range of research communities. The successful and timely identification of the source can prevent a lot of harm as the number of possible infection routes can be narrowed down and potentially infected individuals can be isolated. Previous research on this topic often assumes that it is possible to observe the state of a substantial fraction of individuals in the network before attempting to identify the source. We, on the contrary, assume that observing the state of individuals in the network is costly or difficult and, hence, only the state of one or few individuals is initially observed. Moreover, we presume that not only the source is unknown, but also the duration for which the epidemic has evolved. From this more general problem setting a need to query the state of other (so far unobserved) individuals arises. In analogy with active learning, this leads us to formulate the active querying problem. In the active querying problem, we alternate between a source inference step and a querying step. For the source inference step, we rely on existing work but take a Bayesian perspective by putting a prior on the duration of the epidemic. In the querying step, we aim to query the states of individuals that provide the most information about the source of the epidemic, and to this end, we propose strategies inspired by the active learning literature. Our results are strongly in favor of a querying strategy that selects individuals for whom the disagreement between individual predictions, made by all possible sources separately, and a consensus prediction is maximal. Our approach is flexible and, in particular, can be applied to static as well as temporal networks. To demonstrate our approach's practical importance, we experiment with three empirical (temporal) contact networks: a network of pig movements, a network of sexual contacts, and a network of face-to-face contacts between residents of a village in Malawi. The results show that active querying strategies can lead to substantially improved source inference results as compared to baseline heuristics. In fact, querying only a small fraction of nodes in a network is often enough to achieve a source inference performance comparable to a situation where the infection states of all nodes are known. [ABSTRACT FROM AUTHOR]

We consider Y-system functional equations of the form Y n (x + i) Y n (x - i) = ∏ m = 1 N 1 + Y m (x) G nm and the corresponding nonlinear integral equations of the thermodynamic Bethe ansatz. We prove an existence and uniqueness result for solutions of these equations, subject to appropriate conditions on the analytical properties of the Y n , in particular the absence of zeros in a strip around the real axis. The matrix G nm must have non-negative real entries and be irreducible and diagonalisable over R with spectral radius less than 2. This includes the adjacency matrices of finite Dynkin diagrams, but covers much more as we do not require G nm to be integers. Our results specialise to the constant Y-system, proving existence and uniqueness of a strictly positive solution in that case. [ABSTRACT FROM AUTHOR]