  We consider the problem of controlling the spread of harmful items in
networks, such as the contagion proliferation of diseases or the diffusion of
fake news. We assume the linear threshold model of diffusion where each node
has a threshold that measures the node resistance to the contagion. We study
the parameterized complexity of the problem: Given a network, a set of
initially contaminated nodes, and two integers $k$ and $\ell$, is it possible
to limit the diffusion to at most $k$ other nodes of the network by immunizing
at most $\ell$ nodes? We consider several parameters associated to the input,
including: the bounds $k$ and $\ell$, the maximum node degree $\Delta$, the
treewidth, and the neighborhood diversity of the network. We first give $W[1]$
or $W[2]$-hardness results for each of the considered parameters. Then we give
fixed-parameter algorithms for some parameter combinations.
