I am trying to estimate the variation of a disease incidence by month. The disease incidence should theoretically have a seasonal pattern - Illustrated and exaggerated in the attached dataset.
I am trying to use Poisson regression to evaluate the difference in incidence by month. The underlying population from which the cases derives does not change.
But, I have a problem regarding the exposure, should I include the length of the month as exposure, or is that irrelevant?


Code:
********
** test dataset for 
** monthly variation in incidence
********

* Generate fake dataset
* behaves comparable, but not identical to the real dataset
* consists of:
*    month (1=january, 2 = february etc)
*    number of new cases per month
*    average temperature of the month
*    length of the month measured in days

clear 
input month incidence temperature days 
1    3        0        31
2    5        1        28
3    8        3        31
4    10       6        30
5    15       10       31
6    15       14       30
7    20       20       31
8    30       15       31
9    15       12       30    
10   10       5        31
11    8       3        30
12    5       1        31
end
    
hist month [weight=incidence ], d        ///
    addplot(line  temperature month, yaxis(2))

poisson incidence i.month,  ir
poisson incidence i.month c.temperatur,  ir
poisson incidence i.month, exposure(days) ir
poisson incidence i.month c.temperatur, exposure(days) ir