  Contact tracing is an important control strategy for containing Ebola
epidemics. From a theoretical perspective, explicitly incorporating contact
tracing with disease dynamics presents challenges, and population level effects
of contact tracing are difficult to determine. In this work, we formulate and
analyze a mechanistic SEIR type outbreak model which considers the key features
of contact tracing, and we characterize the impact of contact tracing on the
effective reproduction number, $\mathcal R_e$, of Ebola. In particular, we
determine how relevant epidemiological properties such as incubation period,
infectious period and case reporting, along with varying monitoring protocols,
affect the efficacy of contact tracing. In the special cases of either perfect
monitoring of traced cases or perfect reporting of all cases, we derive simple
formulae for the critical proportion of contacts that need to be traced in
order to bring the effective reproduction number $\mathcal R_e$ below one.
Also, in either case, we show that $\mathcal R_e$ can be expressed completely
in terms of observable reported case/tracing quantities, namely $\mathcal
R_e=k\dfrac{(1-q)}{q}+k_m$ where $k$ is the number of secondary traced infected
contacts per primary untraced reported case, $k_m$ is the number of secondary
traced infected contacts per primary traced reported case and $(1-q)/q$ is the
odds that a reported case is not a traced contact. These formulae quantify
contact tracing as both an intervention strategy that impacts disease spread
and a probe into the current epidemic status at the population level. Data from
the West Africa Ebola outbreak is utilized to form real-time estimates of
$\mathcal R_e$, and inform our projections of the impact of contact tracing,
and other control measures, on the epidemic trajectory.
