  The susceptible-infected-recovered (SIR) model and its variants form the
foundation of our understanding of the spread of diseases. Here, each agent can
be in one of three states (susceptible, infected, or recovered), and
transitions between these states follow a stochastic process. The probability
of an agent becoming infected depends on the number of its infected neighbors,
hence all agents are correlated. A common mean-field theory of the same
stochastic process however, assumes that the agents are statistically
independent. This leads to a self-feedback effect in the approximation: when an
agent infects its neighbors, this infection may subsequently travel back to the
original agent at a later time, leading to a self-infection of the agent which
is not present in the underlying stochastic process. We here compute the first
order correction to the mean-field assumption, which takes fluctuations up to
second order in the interaction strength into account. We find that it cancels
the self-feedback effect, leading to smaller infection rates. In the SIR model
and in the SIRS model, the correction significantly improves predictions. In
particular, it captures how sparsity dampens the spread of the disease: this
indicates that reducing the number of contacts is more effective than predicted
by mean-field models.
