  In this paper we conduct a simulation study of the spread of an epidemic like
COVID-19 with temporary immunity on finite spatial and non-spatial network
models. In particular, we assume that an epidemic spreads stochastically on a
scale-free network and that each infected individual in the network gains a
temporary immunity after its infectious period is over. After the temporary
immunity period is over, the individual becomes susceptible to the virus again.
When the underlying contact network is embedded in Euclidean geometry, we model
three different intervention strategies that aim to control the spread of the
epidemic: social distancing, restrictions on travel, and restrictions on
maximal number of social contacts per node. Our first finding is that on a
finite network, a long enough average immunity period leads to extinction of
the pandemic after the first peak, analogous to the concept of "herd immunity".
For each model, there is a critical average immunity length $L_c$ above which
this happens. Our second finding is that all three interventions manage to
flatten the first peak (the travel restrictions most efficiently), as well as
decrease the critical immunity length $L_c$, but elongate the epidemic.
However, when the average immunity length $L$ is shorter than $L_c$, the price
for the flattened first peak is often a high second peak: for limiting the
maximal number of contacts, the second peak can be as high as 1/3 of the first
peak, and twice as high as it would be without intervention. Thirdly,
interventions introduce oscillations into the system and the time to reach
equilibrium is, for almost all scenarios, much longer. We conclude that
network-based epidemic models can show a variety of behaviors that are not
captured by the continuous compartmental models.
