**3. Monitoring in-control nonhomogeneous counts**

For most of the real-world cases, TBE distribution may vary due to different circumstances while not experiencing an outbreak. For instance, if we aggregate the TBEs in daily intervals, occurring intervals for the events may vary based on the day of the week, and even working and non-working days may affect the distribution of TBEs. As a result, we often face nonhomogeneous count processes. Hence, we define an adaptive exponentially weighted move average (AEWMA) statistic for nonhomogeneous daily counts as

$$ae\_i = \max\left(0, ac\_i/h\_c(ARL\_0, scale, shape, a) + (1 - a)ae\_{i-1}\right) \tag{2}$$

where *aei* is the AEWMA statistic at time *t*, and *ae*<sup>0</sup> ¼ *E c*ð Þ*<sup>i</sup> =hc*ð Þ *ARL*0, *scale*, *shape*, *α* for days *i* ¼ 1, 2, … . The other notations are as defined earlier. To control the false discovery rate, we set *hc*ð Þ *ARL*<sup>0</sup> ¼ *c*, *scale*, *shape*, *α* so that a desired*ARL*<sup>0</sup> is achieved. An outbreak is flagged approximately whenever *aei* >1.

The *α*ð Þ 0<*α* <1 in Eq. (1) determines the level of memory of past observations in this average *aei*. Smaller *α* values retain more memory of past counts; therefore, small *α* values are efficient at retaining enough memory of past counts to have the power to flag smaller outbreaks. However, larger values of *α* are needed when there is a larger size outbreak, because a shorter range of memory is adequate to build enough power to detect the outbreak. In addition, the shorter memory for the EWMA average with larger values of *α* is less inclined to be influenced by too many past in-control counts.
