  Several concepts that model processes of spreading (of information, disease,
objects, etc.) in graphs or networks have been studied. In many contexts, we
assume that some vertices of a graph $G$ are contaminated initially, before the
process starts. By the $q$-forcing rule, a contaminated vertex having at most
$q$ uncontaminated neighbors enforces all the neighbors to become contaminated,
while by the $p$-percolation rule, an uncontaminated vertex becomes
contaminated if at least $p$ of its neighbors are contaminated. In this paper,
we consider sets $S$ that are at the same time $q$-forcing sets and
$p$-percolating sets, and call them $(p,q)$-spreading sets. Given positive
integers $p$ and $q$, the minimum cardinality of a $(p,q)$-spreading set in $G$
is a $(p,q)$-spreading number, $\sigma_{(p,q)}(G)$, of $G$. While $q$-forcing
sets have been studied in a dozen of papers, the decision version of the
corresponding graph invariant has not been considered earlier, and we fill the
gap by proving its NP-completeness. This, in turn, enables us to prove the
NP-completeness of the decision version of the $(p,q)$-spreading number in
graphs for an arbitrary choice of $p$ and $q$. Again, for every $p\in
\mathbb{N}$ and $q\in\mathbb{N}\cup\{\infty\}$, we find a linear-time algorithm
for determining the $(p,q)$-spreading number of a tree. In addition, we present
a lower and an upper bound on the $(p,q)$-spreading number of a tree and
characterize extremal families of trees. In the case of square grids, we
combine some known results and new results on $(2,1)$-spreading and
$(4,q)$-spreading to obtain $\sigma_{(p,q)}(P_m\Box P_n)$ for all $(p,q)\in
(\mathbb{N}\setminus\{3\})\times (\mathbb{N}\cup\{\infty\})$ and all
$m,n\in\mathbb{N}$.
