  Recent analysis of early COVID-19 data from China showed that the number of
confirmed cases followed a subexponential power-law increase, with a growth
exponent of around 2.2 [B.\,F.~Maier, D.~Brockmann, {\it Science} {\bf 368},
742 (2020)]. The power-law behavior was attributed to a combination of
effective containment and mitigation measures employed as well as behavioral
changes by the population. In this work, we report a random walk Monte Carlo
simulation study of proximity-based infection spread. Control interventions
such as lockdown measures and mobility restrictions are incorporated in the
simulations through a single parameter, the size of each step in the random
walk process. The step size $l$ is taken to be a multiple of $\langle r
\rangle$, which is the average separation between individuals. Three temporal
growth regimes (quadratic, intermediate power-law and exponential) are shown to
emerge naturally from our simulations. For $l = \langle r \rangle$, we get
intermediate power-law growth exponents that are in general agreement with
available data from China. On the other hand, we obtain a quadratic growth for
smaller step sizes $l \lesssim \langle r \rangle/2 $, while for large $l$ the
growth is found to be exponential. %Together with available data, these results
suggest that the early containment of the disease within China was close to
optimal. We further performed a comparative case study of early fatality data
(under varying levels of lockdown conditions) from three other countries,
India, Brazil and South Africa. We show that reasonable agreement with these
data can be obtained by incorporating small-world-like connections in our
simulations.
