Here y t denotes the number of deaths at week t. ns(t), ns(temp t ), and ns(humd t ) denote the natural cubic spline smoothing functions of time, weekly average temperature and relative humidity. The natural spline smoothing function was used to remove the small variation while maintaining the major trend of each variable, i.e. to make them smoother. The aim of smoothing was to increase the efficiency in estimating the model coefficients [20] . Given the high correlation between temperature and vapor pressure, the smoothing function of vapor pressure was not added in order to avoid collinearity between the variables. The adequacy of this core model was evaluated by the absence of any obvious pattern in partial autocorrelation functions of its residuals (Additional file 1: Figure S1 ). Second, the weekly proportions of specimens positive for influenza A or B were then added into the core model as a variable for influenza virus activity to obtain a main effect model.