Figure 2

The time derivative of influenza incidence. Fig. 2(a)–(c) show the total number of new patients within a week for three influenza seasons (2013–14, 2014–15, 2015–16). In Fig. 2(d)–(f), \(d_{t}\), \(s_{t}\), \(\mu _{t}\), and \(\sigma _{t}\) represent the first derivative, second derivative, seasonal cumulative mean and standard deviation, respectively. We confirm that \(d_{t}\) abruptly fluctuates around zero before the onset of influenza outbreak. Then, the difference between \(d_{t}\) and \(\mu _{t}\) increases at the onset of influenza outbreak. In Fig. 2(a)–(c), the onset week t where the condition (2) is satisfied is marked by the blue circle