  In this study, we address three important challenges related to disease
transmissions such as the COVID-19 pandemic, namely, (a) providing an early
warning to likely exposed individuals, (b) identifying individuals who are
asymptomatic, and (c) prescription of optimal testing when testing capacity is
limited. First, we present a dynamic-graph based SEIR epidemiological model in
order to describe the dynamics of the disease propagation. Our model considers
a dynamic network that accounts for the interactions between individuals over
time, such as the ones obtained by manual or automated contact tracing, and
uses a diffusion-reaction mechanism to describe the state dynamics. This
dynamic graph model helps identify likely exposed/infected individuals to whom
we can provide early warnings, even before they display any symptoms and/or are
asymptomatic. Moreover, when the testing capacity is limited compared to the
population size, reliable estimation of individual's health state and disease
transmissibility using epidemiological models is extremely challenging. Thus,
estimation of state uncertainty is paramount for both eminent risk assessment,
as well as for closing the tracing-testing loop by optimal testing
prescription. Therefore, we propose the use of arbitrary Polynomial Chaos
Expansion, a popular technique used for uncertainty quantification, to
represent the states, and quantify the uncertainties in the dynamic model. This
design enables us to assign uncertainty of the state of each individual, and
consequently optimize the testing as to reduce the overall uncertainty given a
constrained testing budget. These tools can also be used to optimize vaccine
distribution to curb the disease spread when limited vaccines are available. We
present a few simulation results that illustrate the performance of the
proposed framework, and estimate the impact of incomplete contact tracing data.
