  We propose a novel infection spread model based on a random connection graph
which represents connections between $n$ individuals. Infection spreads via
connections between individuals and this results in a probabilistic cluster
formation structure as well as a non-i.i.d. (correlated) infection status for
individuals. We propose a class of two-step sampled group testing algorithms
where we exploit the known probabilistic infection spread model. We investigate
the metrics associated with two-step sampled group testing algorithms. To
demonstrate our results, for analytically tractable exponentially split cluster
formation trees, we calculate the required number of tests and the expected
number of false classifications in terms of the system parameters, and identify
the trade-off between them. For such exponentially split cluster formation
trees, for zero-error construction, we prove that the required number of tests
is $O(\log_2n)$. Thus, for such cluster formation trees, our algorithm
outperforms any zero-error non-adaptive group test, binary splitting algorithm,
and Hwang's generalized binary splitting algorithm. Our results imply that, by
exploiting probabilistic information on the connections of individuals, group
testing can be used to reduce the number of required tests significantly even
when infection rate is high, contrasting the prevalent belief that group
testing is useful only when infection rate is low.
