  Snake modules introduced by Mukhin and Young form a family of modules of
quantum affine algebras. The aim of this paper is to prove that the
Hernandez-Leclerc conjecture about monoidal categorifications of cluster
algebras is true for prime snake modules of types $A_{n}$ and $B_{n}$. We prove
that prime snake modules are real. We introduce $S$-systems consisting of
equations satisfied by the $q$-characters of prime snake modules of types
$A_{n}$ and $B_{n}$. Moreover, we show that every equation in the $S$-system of
type $A_n$ (respectively, $B_n$) corresponds to a mutation in the cluster
algebra $\mathscr{A}$ (respectively, $\mathscr{A}'$) constructed by Hernandez
and Leclerc and every prime snake module of type $A_n$ (respectively, $B_n$)
corresponds to some cluster variable in $\mathscr{A}$ (respectively,
$\mathscr{A}'$). In particular, this proves that the Hernandez-Leclerc
conjecture is true for all prime snake modules of types $A_{n}$ and $B_{n}$.
