Integer-valued time series are often modeled with Markov models or hidden Markov models (HMM). However, when the series represents count data it is often subject to excess zeros. In this case, usual distributions such as binomial or Poisson are unable to estimate the zero mass correctly. In order to overcome this issue, we introduce zero-inflated distributions in the hidden Markov model. The empirical results on simulated and real data show good convergence properties, while excess zeros are better estimated than with classical HMM.