  We consider the continuous-time frog model on $\mathbb{Z}$. At time $t = 0$,
there are $\eta (x)$ particles at $x\in \mathbb{Z}$, each of which is
represented by a random variable. In particular, $(\eta(x))_{x \in \mathbb{Z}
}$ is a collection of independent random variables with a common distribution
$\mu$, $\mu(\mathbb{Z}_+) = 1$. The particles at the origin are active, all
other ones being assumed as dormant, or sleeping. Active particles perform a
simple symmetric continuous-time random walk in $\mathbb{Z} $ (that is, a
random walk with $\exp(1)$-distributed jump times and jumps $-1$ and $1$, each
with probability $1/2$), independently of all other particles. Sleeping
particles stay still until the first arrival of an active particle to their
location; upon arrival they become active and start their own simple random
walks. Different sets of conditions are given ensuring explosion, respectively
non-explosion, of the continuous-time frog model. Our results show in
particular that if $\mu$ is the distribution of $e^{Y \ln Y}$ with a
non-negative random variable $Y$ satisfying $\mathbb{E} Y < \infty$, then a.s.
no explosion occurs. On the other hand, if $a \in (0,1)$ and $\mu$ is the
distribution of $e^X$, where $\mathbb{P} \{X \geq t \} = t^{-a}$, $t \geq 1$,
then explosion occurs a.s. The proof relies on a certain type of comparison to
a percolation model which we call totally asymmetric discrete inhomogeneous
Boolean percolation.
