  In this work, we aim to understand the influence of the heterogeneity of
infection rates on the Susceptible-Infected-Susceptible (SIS) epidemic
spreading. Employing the classic SIS model as the benchmark, we study the
influence of the independently identically distributed infection rates on the
average fraction of infected nodes in the metastable state. The log-normal,
gamma and a newly designed distributions are considered for infection rates. We
find that, when the recovery rate is small, i.e.\ the epidemic spreads out in
both homogeneous and heterogeneous cases: 1) the heterogeneity of infection
rates on average retards the virus spreading, and 2) a larger even-order moment
of the infection rates leads to a smaller average fraction of infected nodes,
but the odd-order moments contribute in the opposite way; when the recovery
rate is large, i.e.\ the epidemic may die out or infect a small fraction of the
population, the heterogeneity of infection rates may enhance the probability
that the epidemic spreads out. Finally, we verify our conclusions via
real-world networks with their heterogeneous infection rates. Our results
suggest that, in reality the epidemic spread may not be so severe as the
classic SIS model indicates, but to eliminate the epidemic is probably more
difficult.
