  As of November 2020, the number of COVID-19 cases is increasing rapidly in
many countries. In Europe, the virus spread slowed considerably in the late
spring due to strict lockdown, but a second wave of the pandemic grew
throughout the fall. In this study, we first reconstruct the time evolution of
the effective reproduction numbers ${\cal R}(t)$ for each country by
integrating the equations of the classic SIR model. We cluster countries based
on the estimated ${\cal R}(t)$ through a suitable time series dissimilarity.
The result suggests that simple dynamical mechanisms determine how countries
respond to changes in COVID-19 case counts. Inspired by these results, we
extend the SIR model to include a social response to explain the number $X(t)$
of new confirmed daily cases. As a first-order model, we assume that the social
response is on the form $d_t{\cal R}=-\nu (X-X^*)$, where $X^*$ is a threshold
for response. The response rate $\nu$ depends on whether $X^*$ is below or
above this threshold, on three parameters $\nu_1,\;\nu_2,\,\nu_3,$, and on $t$.
When $X<X^*$, $\nu= \nu_1$, describes the effect of relaxed intervention when
the incidence rate is low. When $X>X^*$, $\nu=\nu_2\exp{(-\nu_3t)}$, models the
impact of interventions when incidence rate is high. The parameter $\nu_3$
represents the fatigue, i.e., the reduced effect of intervention as time
passes. The proposed model reproduces typical evolving patterns of COVID-19
epidemic waves observed in many countries. Estimating the parameters
$\nu_1,\,\nu_2,\,\nu_3$ and initial conditions, such as ${\cal R}_0$, for
different countries helps to identify important dynamics in their social
responses. One conclusion is that the leading cause of the strong second wave
in Europe in the fall of 2020 was not the relaxation of interventions during
the summer, but rather the general fatigue to interventions developing in the
fall.
