We model monthly disease counts on an age-time grid using the two-dimensional varying-coefficient Poisson regression. Since the marginal profile of counts shows a very strong and varying annual cycle, sine and cosine regressors model periodicity, but their coefficients are allowed to vary smoothly over the age and time plane. The coefficient surfaces are estimated using a relatively large tensor product B-spline basis. Smoothness is tuned using difference penalties on the rows and columns of the tensor product coefficients. Heavy over-dispersion occurs, making it impossible to use Akaike's information criterion or Bayesian information criterion based on a Poisson likelihood. It is handled by selective weighting of part of the data and by the use of extended quasi-likelihood. Very efficient computation is achieved with fast array algorithms. The model is applied to monthly deaths due to respiratory diseases, for U.S. females during 1959-1998 and for ages 51-100.