Siva Athreya

Wastewater surveillance is an effective tool for monitoring the spread of infectious diseases such as COVID-19. In August 2021, a citywide surveillance effort was initiated in Bengaluru to analyze viral loads from 28 sewage treatment plants (STPs). The study found a strong correlation between aggregated viral loads and citywide COVID-19 case counts. However, the lack of localized clinical data limited the ability to assess infection trends at the STP level. In this follow-up study, we incorporate granular clinical data from 198 administrative units in the city. We find similar trends between viral loads at individual STPs and the cases in their catchment areas. A typical confidence interval for the Pearson correlation between clinical cases and wastewater viral loads at an STP is approximately 0.56 – 1.00 , based on the median bounds across the STPs; at the city-level it is 0.67 – 1.00 . However, our analysis shows no reliable indication of a lead or early warning—the viral loads and reported cases rise simultaneously. It is important to note that our study is limited to the first Omicron wave of the pandemic. To quantify lead time, we used correlation and change-point analysis. These results underscore the potential of localized wastewater surveillance for real-time monitoring but highlight its limitation in early outbreak detection.

In this article we prove under suitable assumptions that the marginals of any solution to a relaxed controlled martingale problem on a Polish space [Formula: see text] can be mimicked by a Markovian solution of a Markov-relaxed controlled martingale problem. We also show how such ‘`Markov mimics’’ can be obtained by relative entropy minimization. We provide many examples where the above results can be applied.

We consider two classes of natural stochastic processes on finite unlabeled graphs. These are Euclidean stochastic optimization algorithms on the adjacency matrix of weighted graphs and a modified version of the Metropolis MCMC algorithm on stochastic block models over unweighted graphs. In both cases we show that, as the size of the graph goes to infinity, the random trajectories of the stochastic processes converge to deterministic curves on the space of measure-valued graphons. Measure-valued graphons, introduced by Lovász and Szegedy in \cite{lovasz2010decorated}, are a refinement of the concept of graphons that can distinguish between two infinite exchangeable arrays that give rise to the same graphon limit. We introduce new metrics on this space which provide us with a natural notion of convergence for our limit theorems. This notion is equivalent to the convergence of infinite-exchangeable arrays. Under suitable assumptions and a specified time-scaling, the Metropolis chain admits a diffusion limit as the number of vertices go to infinity. We then demonstrate that, in an appropriately formulated zero-noise limit, the stochastic process of adjacency matrices of this diffusion converges to a deterministic gradient flow curve on the space of graphons introduced in\cite{Oh2023}. A novel feature of this approach is that it provides a precise exponential convergence rate for the Metropolis chain in a certain limiting regime. The connection between a natural Metropolis chain commonly used in exponential random graph models and gradient flows on graphons, to the best of our knowledge, is new in the literature as well.