We study a system of random walks, known as the frog model, starting from a profile of independent Poisson($\lambda$) particles per site, with one additional active particle planted at some vertex $o$ of a finite connected simple graph $G=(V,E)$. Initially, only the particles occupying $o$ are active. Active particles perform $t\in N\cup \{\infty \}$ steps of the walk they picked before vanishing and activate all inactive particles they hit. This system is often taken as a model for the spread of an epidemic over a population. Let $Rt$ be the set of vertices which are visited by the process, when active particles vanish after $t$ steps. We study the susceptibility of the process on the underlying graph, defined as the random quantity $S(G):=inf\{t:Rt=V\}$ (essentially, the shortest particles' lifetime required for the entire population to get infected). We consider the cases that the underlying graph is either a regular expander or a $d$-dimensional torus of side length $n$ (for all $d\ge 1$) $Td(n)$ and determine the asymptotic behavior of $\mathcal{S} $ up to a constant factor. In fact, throughout we allow the particle density $\lambda$ to depend on $n$ and for $d\ge 2$ we determine the asymptotic behavior of $S(Td(n))$ up to smaller order terms for a wide range of $\lambda =\lambda n$.