  This paper is concerned with stochastic SIR and SEIR epidemic models on
random networks in which individuals may rewire away from infected neighbors at
some rate $\omega$ (and reconnect to non-infectious individuals with
probability $\alpha$ or else simply drop the edge if $\alpha=0$), so-called
preventive rewiring. The models are denoted SIR-$\omega$ and SEIR-$\omega$, and
we focus attention on the early stages of an outbreak, where we derive
expression for the basic reproduction number $R_0$ and the expected degree of
the infectious nodes $E(D_I)$ using two different approximation approaches. The
first approach approximates the early spread of an epidemic by a branching
process, whereas the second one uses pair approximation. The expressions are
compared with the corresponding empirical means obtained from stochastic
simulations of SIR-$\omega$ and SEIR-$\omega$ epidemics on Poisson and
scale-free networks. Without rewiring of exposed nodes, the two approaches
predict the same epidemic threshold and the same $E(D_I)$ for both types of
epidemics, the latter being very close to the mean degree obtained from
simulated epidemics over Poisson networks. Above the epidemic threshold,
pairwise models overestimate the value of $R_0$ computed from simulations,
which turns out to be very close to the one predicted by the branching process
approximation. When exposed individuals also rewire with $\alpha > 0$ (perhaps
unaware of being infected), the two approaches give different epidemic
thresholds, with the branching process approximation being more in agreement
with simulations.
