‎In this paper‎, ‎we introduce a new integer-valued autoregressive model of first order based on the negative binomial thinning operator‎, ‎where the noises are serially dependent‎. ‎Some statistical properties of the model are discussed‎. ‎The model parameters are estimated by maximum likelihood and Yule-Walker methods‎. ‎By a simulation study‎, ‎the performances of the two estimation methods are studied‎. ‎This survey was carried out to study the efficiency of the new model by applying it on real data‎.

Many of spatial-temporal data, particularly in medicine and disease mapping, are counts. Typically, these types of count data have extra variability that distrusts the classical Poisson model's performance. Therefore, incorporating this variability into the modeling process, plays an essential role in improving the efficiency of spatial-temporal data analysis. For this purpose, in this paper, a new Bayesian spatial-temporal model, called gamma count, with enough flexibility in modeling dispersion is introduced. For implementing statistical inference in the proposed model, the integrated nested Laplace approximation method is applied. A simulation study was performed to evaluate the performance of the proposed model compared to the traditional models. In addition, the application of the model has been demonstrated in analyzing leukemia data in Khorasan Razavi province, Iran.