The phase-shift of the composite waveform generated by combining sine and cosine components of the above equation can be approximated as the arctangent of β 2 /β 3 , and can be used to estimate the timing of peak disease incidence [22] . We also included model terms that controlled for longer term trends in incidence of invasive pneumococcal disease, which may have reflected the initiation of surveillance, the introduction of public funding for conjugate pneumococcal vaccination [12] , changes in medical diagnostic practices, or other long-term changes in real or apparent pneumococcal epidemiology. As yearon-year trends in disease occurrence reflected a non-linear increase in disease risk, we used models that incorporated both linear and quadratic yearly terms.