  We consider the frog model with Bernoulli initial configuration, which is an
interacting particle system on the multidimensional lattice consisting of two
states of particles: active and sleeping. Active particles perform independent
simple random walks. On the other hand, although sleeping particles do not move
at first, they become active and can move around when touched by active
particles. Initially, only the origin has one active particle, and the other
sites have sleeping particles according to a Bernoulli distribution. Then,
starting from the original active particle, active ones are gradually generated
and propagate across the lattice, with time. It is of interest to know how the
propagation of active particles behaves as the parameter of the Bernoulli
distribution varies. In this paper, we treat the so-called time constant
describing the speed of propagation, and prove that the absolute difference
between the time constants for parameters $p,q \in (0,1]$ is bounded from above
and below by multiples of $|p-q|$.
