Effective intrinsic ergodicity for countable state Markov shifts

For strongly positively recurrent countable state Markov shifts, we bound the distance between an invariant measure and of maximal entropy in terms difference their entropies. This extends earlier result for subshifts finite type, due to Kadyrov. We provide a similar equilibrium measures potentials, pressure difference. with nearly entropy, have new, sharp, bounds. The strong positive recurrence condition is necessary.