Negative Binomial Regression: Offset Variable and Dispersion Parameter

My case is that previously it's assumed that the counts of events follows a negative binomial distribution, and the annualized exacerbation rate is 1 with a dispersion parameter of 1.5, and all 500 patients will be followed up for 1 year. But due to some operation reasons, the follow-up period have to be shorten from 1 year to around 0.5 year for each patient. My understanding is that the annualized exacerbation rate is still 1, just including 0.5 year as a offset variable when doing the NB regression (correct me if I am wrong). My question is will the dispersion parameter be impacted due to the fact that the followup period become shorter? (I have to do some simulation using R, i need to know how should the dispersion parameter as well as the mean of the NB distribution be setup).

Thank you.

Particularly point (1) can be addressed by simulating data separately for each month to see the imapct. For that you can exploit that the negative binomial is a gamma-Poisson mixture. I.e. you can create a random patient effect $$u_i$$ from a gamma distribution and then simulation each month as $$Y_{ij} \sim \text{Poisson}(u_i \times \mu_j / 12)$$, if the assumed annualized rate in month $$j$$ is $$\mu_j$$.