  The frog model is a stochastic model for the spreading of an epidemic on a
graph, in which a dormant particle starts to perform a simple random walk on
the graph and to awake other particles, once it becomes active. We study two
versions of the frog model on the complete graph with $N + 1$ vertices. In the
first version we consider, active particles have geometrically distributed
lifetimes. In the second version, the displacement of each awakened particle
lasts until it hits a vertex already visited by the process. For each model, we
prove that as $N \to \infty$, the trajectory of the process is well
approximated by a three-dimensional discrete-time dynamical system. We also
study the long-term behavior of the corresponding deterministic systems.

Probability