  This paper introduces a novel approach for epidemic nowcasting and
forecasting over networks using total variation (TV) denoising, a method
inspired by classical signal processing techniques. Considering a network that
models a population as a set of $n$ nodes characterized by their infection
statuses $Y_i$ and that represents contacts as edges, we prove the consistency
of graph-TV denoising for estimating the underlying infection probabilities
$\{p_i\}_{ i \in \{1,\cdots, n\}}$ in the presence of Bernoulli noise. Our
results provide an important extension of existing bounds derived in the
Gaussian case to the study of binary variables -- an approach hereafter
referred to as one-bit total variation denoising. The methodology is further
extended to handle incomplete observations, thereby expanding its relevance to
various real-world situations where observations over the full graph may not be
accessible. Focusing on the context of epidemics, we establish that one-bit
total variation denoising enhances both nowcasting and forecasting accuracy in
networks, as further evidenced by comprehensive numerical experiments and two
real-world examples. The contributions of this paper lie in its theoretical
developments, particularly in addressing the incomplete data case, thereby
paving the way for more precise epidemic modelling and enhanced surveillance
strategies in practical settings.
