  Zero inflation is a common nuisance while monitoring disease progression over
time. This article proposes a new observation driven model for zero inflated
and over-dispersed count time series. The counts given the past history of the
process and available information on covariates is assumed to be distributed as
a mixture of a Poisson distribution and a distribution degenerate at zero, with
a time dependent mixing probability, $\pi_t$. Since, count data usually suffers
from overdispersion, a Gamma distribution is used to model the excess
variation, resulting in a zero inflated Negative Binomial (NB) regression model
with mean parameter $\lambda_t$. Linear predictors with auto regressive and
moving average (ARMA) type terms, covariates, seasonality and trend are fitted
to $\lambda_t$ and $\pi_t$ through canonical link generalized linear models.
Estimation is done using maximum likelihood aided by iterative algorithms, such
as Newton Raphson (NR) and Expectation and Maximization (EM). Theoretical
results on the consistency and asymptotic normality of the estimators are
given. The proposed model is illustrated using in-depth simulation studies and
a dengue data set.
