Scaling Behavior of Threshold Epidemics
E. Ben-Naim and P. L. Krapivsky
We study the classic Susceptible-Infected-Recovered (SIR) model for
the spread of an infectious disease. In this stochastic process,
there are two competing mechanism: infection and recovery. Susceptible
individuals may contract the disease from infected individuals, while
infected ones recover from the disease at a constant rate and are
never infected again. Our focus is the behavior at the epidemic
threshold where the rates of the infection and recovery processes
balance. In the infinite population limit, we establish analytically
scaling rules for the time-dependent distribution functions that
characterize the sizes of the infected and the recovered
sub-populations. Using heuristic arguments, we also obtain scaling
laws for the size and duration of the epidemic outbreaks as a function
of the total population. We perform numerical simulations to verify
the scaling predictions and discuss the consequences of these scaling
laws for near-threshold epidemic outbreaks.
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