  Bern\'ath and Gerbner in 2007 introduced $(p,q)$-chain intersecting families
of subsets of an $n$-element underlying set. Those have the property that for
any $p$-chain $A_1\subsetneq A_2\subsetneq \dots \subsetneq A_p$ and $q$-chain
$B_1\subsetneq B_2\subsetneq \dots \subsetneq B_q$, we have $A_p\cap B_q\neq
\emptyset$. Bern\'ath and Gerbner determined the largest cardinality of such
families. They also introduced strongly $(p,q)$-chain intersecting families,
where $A_p\cap B_1\neq \emptyset$ and totally $(p,q)$-chain intersecting
families, where $A_1\cap B_1\neq \emptyset$. They obtained some partial results
on the maximum cardinality of such families. We extend those results by
determining the largest cardinality of strongly $(p,q)$-chain intersecting
families if $n$ is sufficiently large, and by determining the largest
cardinality of totally $(2,2)$-chain intersecting families.
