Equivalence between Non-Markovian and Markovian Dynamics in Epidemic Spreading Processes

A general formalism is introduced to allow the steady state of non-Markovian processes on networks to be reduced to equivalent Markovian processes on the same substrates. The example of an epidemic spreading process is considered in detail, where all the non-Markovian aspects are shown to be captured within a single parameter, the effective infection rate. Remarkably, this result is independent of the topology of the underlying network, as demonstrated by numerical simulations on two-dimensional lattices and various types of random networks. Furthermore, an analytic approximation for the effective infection rate is introduced, which enables the calculation of the critical point and of the critical exponents for the non-Markovian dynamics.

Figure 1

Sketch of the infection mechanism from node $j$ to node $i$. Infection events triggered by node $j$ (represented by stars in the figure) are ineffective if node $i$ is already infected.

Figure 2

Steady-state prevalence ${\rho }^{st}$ as a function of the effective infection rate, for different values of the exponent ${\alpha }_I$ controlling the interevent time infection distribution and different network substrates. Symbols represent the effective infection rate ${\lambda }_{\mathrm{eff}}$, extracted by numerical simulations; continuous lines represent the approximate rate ${\lambda }_{\mathrm{app}}$.