  In this study we analyze the evolution of the effective reproduction number,
$R$, through a SIR spreading process in heterogeneous networks; Characterizing
its decay process allows to analytically study the effects of countermeasures
on the progress of the virus under heterogeneity, and to optimize their
policies.
  A striking result of recent studies has shown that heterogeneity across
nodes/individuals (or, super-spreading) may have a drastic effect on the
spreading process progression, which may cause a non-linear decrease of $R$ in
the number of infected individuals. We account for heterogeneity and analyze
the stochastic progression of the spreading process. We show that the decrease
of $R$ is, in fact, convex in the number of infected individuals, where this
convexity stems from heterogeneity. The analysis is based on establishing
stochastic monotonic relations between the susceptible populations in varying
times of the spread.
  We demonstrate that the convex behavior of the effective reproduction number
affects the performance of countermeasures used to fight a spread of a virus.
The results are applicable to the control of virus and malware spreading in
computer networks as well. We examine numerically the sensitivity of the Herd
Immunity Threshold (HIT) to the heterogeneity level and to the chosen policy.
