The frog model is a classical branching model for the spread of an infection. In this model, sleeping frogs are placed on the vertices of a graph and initially one vertex is activated. Active frogs move as simple random walks and wake up all sleeping frogs they encounter. Motivated by the addition of an immunological response, we present an extension of this model in which sleeping frogs must be visited a random number of times, i.i.d. as some $I$, before they awaken, and the active frogs that attempt to wake them up are killed.

We examine the propagation speed and demonstrate that the frogs spread ballistically for a $2+\varepsilon$-moment assumption on $I$, but sublinearly for a more heavy-tailed $I$. By constructing a series of renewal times, at which the front becomes independent of the past, we are able to derive a shape theorem under more technical assumptions.