  Consider a Markovian SIR epidemic model in a homogeneous community. To this
model we add a rate at which individuals are tested, and once an infectious
individual tests positive it is isolated and each of their contacts are traced
and tested independently with some fixed probability. If such a traced
individual tests positive it is isolated, and the contact tracing is iterated.
This model is analysed using large population approximations, both for the
early stage of the epidemic when the "to-be-traced components" of the epidemic
behaves like a branching process, and for the main stage of the epidemic where
the process of to-be-traced components converges to a deterministic process
defined by a system of differential equations. These approximations are used to
quantify the effect of testing and of contact tracing on the effective
reproduction numbers (for the components as well as for the individuals), the
probability of a major outbreak, and the final fraction getting infected. Using
numerical illustrations when rates of infection and natural recovery are fixed,
it is shown that Test-and-Trace strategy is effective in reducing the
reproduction number. Surprisingly, the reproduction number for the branching
process of components is not monotonically decreasing in the tracing
probability, but the individual reproduction number is conjectured to be
monotonic as expected. Further, in the situation where individuals also
self-report for testing, the tracing probability is more influential than the
screening rate (measured by the fraction infected being screened).
