We find the steady state of the system by solving the set of equations $$ \begin{equation*} w(t=\infty) = Ww(t=\infty), \end{equation*} $$ which is an eigenvalue problem with eigenvalue equal to one! This set of equations reads $$ \begin{align} W_{11}w_1(t=\infty) +W_{12}w_2(t=\infty) +W_{13}w_3(t=\infty)+ W_{14}w_4(t=\infty)=&w_1(t=\infty) \nonumber \\ W_{21}w_1(t=\infty) + W_{22}w_2(t=\infty) + W_{23}w_3(t=\infty)+ W_{24}w_4(t=\infty)=&w_2(t=\infty) \nonumber \\ W_{31}w_1(t=\infty) + W_{32}w_2(t=\infty) + W_{33}w_3(t=\infty)+ W_{34}w_4(t=\infty)=&w_3(t=\infty) \nonumber \\ W_{41}w_1(t=\infty) + W_{42}w_2(t=\infty) + W_{43}w_3(t=\infty)+ W_{44}w_4(t=\infty)=&w_4(t=\infty) \nonumber \\ \tag{2} \end{align} $$ with the constraint that $$ \begin{equation*} \sum_i w_i(t=\infty) = 1, \end{equation*} $$ yielding as solution $$ \begin{equation*} \hat{w}(t=\infty)= \left(\begin{array}{c}0.244318 \\ 0.319602 \\ 0.056818 \\ 0.379261 \end{array} \right). \end{equation*} $$