# Is there a single discrete distribution that handles over and under dispersion? [duplicate]

I have some count data I am trying to model. The variance is very close to the mean, so the Poisson distribution for the entire data set seems like a good starting point. I have done and it seems to work reasonably well. But I would like to go further and model the data set over time with time-dependent parameters, and allow the model to go through periods of over and under dispersion. The negative binomial and binomial distributions seem like reasonable choices for each case, respectively. But modeling the data in this way is awkward - I am forced to have switch-points in the model where to goes from binomial to negative binomial as the variance changes between over and under dispersion.

Is there a single model that handles over AND under dispersion? Something that would likely reduce in some limit to a binomial in one limit, and a negative binomial in another limit?

One though I had is that since a negative binomial distribution is just a Gamma-Poisson mixture, simply create a XXX-Poisson mixture, where I need to find a suitable XXX distribution. This will likely result in an integral that cannot be represented in closed form, so I was trying to avoid this if there is already an off-the-shelf distribution to use for such cases.