  During the COVID pandemic, periods of exponential growth of the disease have
been mitigated by containment measures that in different occasions have
resulted in a power-law growth of the number of cases. The first observation of
such behaviour has been obtained from 2020 late spring data coming from China
by Ziff and Ziff in Ref. [1]. After this important observation the power-law
scaling (albeit with different exponents) has also been observed in other
countries during periods of containment of the spread. Early interpretations of
these results suggest that this phenomenon might be due to spatial effects of
the spread. Here we show that temporal modulations of infectivity of
individuals due to containment measures can also cause power-law growth of the
number of cases over time. To this end we propose a stochastic well-mixed
Susceptible-Infected-Removed (SIR) model of epidemic spreading in presence of
containment measures resulting in time-dependent infectivity and we explore the
statistical properties of the resulting branching process at criticality. We
show that at criticality it is possible to observe power-law growth of the
number of cases with exponents ranging between one and two. Our asymptotic
analytical results are confirmed by extensive Monte Carlo simulations. Although
these results do not exclude that spatial effects might be important in
modulating the power-law growth of the number of cases at criticality, this
work shows that even well-mixed populations may already feature non-trivial
power-law exponents at criticality.
