  Let $G$ be a group and $\varphi$ be an automorphism of $G$. Two elements $x,
y$ of $G$ are said to be $\varphi$-twisted conjugate if $y=gx\varphi(g)^{-1}$
for some $g\in G$. A group $G$ has the $R_{\infty}$-property if the number of
$\varphi$-twisted conjugacy classes is infinite for every automorphism
$\varphi$ of $G$. In this paper we prove that the big mapping class group
$MCG(S)$ possesses the $R_{\infty}$-property under some suitable conditions on
the infinite-type surface $S$. As an application we also prove that the big
mapping class group possesses the $R_\infty$-property if and only if it
satisfies the $S_{\infty}$-property.
