Negative binomial model - saturated?

I am trying to run a model for cases (n) to see if area, period, and area*period are significant predictors. I am really only interested in the area*period component because the two variables on their own don't make much sense as I need to compare the cases over time between the 3 areas.

So for some of the instances, I run the model and get zeros for some of the goodness of fit parameters and then get p=1.000 in the LR statistics output. I was told that this is correct for my data and that it means the model is saturated. So when one has a saturated model is there anything to do? I can't really break down the data in any way in this case. Or do you just have to ignore this type of result or what? Sorry, I've never had a scenario like this. Thanks!
I have area, period, popsum, and nsum, as well as log of popsum, which is used as the offset term for the regression. So I put:

proc genmod data=want;
class period area;
model nsum= area period area*period / dist=NB link=log offset=logpop type1;

I have 51 observations (17 years for each of the 3 study areas) and for area 1 nsum is around 3,300 for each year, area 2 nsum is around 14,000, and for area 3 nsum is around 10,000. The log says 'fitting saturated model' and I get p=1.000 for the interaction term under LR Statistics for Type 1 Analysis. In the output with the actual estimates, period*area shows p around 0.96-0.99 for each year and areas 1/2 (with area 3 as the reference group).

I tried doing Poisson instead of NB and it still says the model is saturated, but gives a p value for the interaction term instead of p=1.000, so not totally sure which way I should be going (and if I should be including type 1 or type 3 analysis). Thanks!