  In a prior study, a novel deterministic compartmental model known as the
SEIHRK model was introduced, shedding light on the pivotal role of test kits as
an intervention strategy for mitigating epidemics. Particularly in
heterogeneous networks, it was empirically demonstrated that strategically
distributing a limited number of test kits among nodes with higher degrees
substantially diminishes the outbreak size. The network's dynamics were
explored under varying values of infection rate. In this research, we expand
upon these findings to investigate the influence of migration on infection
dynamics within distinct communities of the network. Notably, we observe that
nodes equipped with test kits and those without tend to segregate into two
separate clusters when coupling strength is low, but beyond a critical
threshold coupling coefficient, they coalesce into a unified cluster. Building
on this clustering phenomenon, we develop a reduced equation model and
rigorously validate its accuracy through comprehensive simulations. We show
that this property is observed in both complete and random graphs.
