Consequences of non-Markovian healing processes on epidemic models with recurrent infections on networks

Abstract Infectious diseases are marked by recovering time distributions which can be far from the exponential one associated with Markovian/Poisson processes, broadly applied in epidemic models. In the present work, we tackled this problem by investigating a susceptible-infected-recovered-susceptible model on networks with η independent infectious compartments (SI <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi/> <mml:mrow> <mml:mi>η</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> RS), each one with a Markovian dynamics, that leads to a Gamma-distributed recovering time. We analytically develop a theory for the epidemic lifespan on star graphs with a center and K leaves, which mimic hubs on networks, showing that the epidemic lifespan scales with a non-universal power-law. Compared with standard susceptible-infected-recovered-susceptible dynamics, the epidemic lifespan on star graphs is severely reduced as the number of stages increases. In particular, the case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>η</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:math> leads to a finite lifespan. Numerical simulations support the approximated analytical calculations. We investigated the SI <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi/> <mml:mrow> <mml:mi>η</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> RS dynamics on random power-law networks. When the epidemic processes are ruled by a maximum k -core activation, either the epidemic threshold or the epidemic localization pattern are unaltered. When hub mutual activation is at work, the localization is reduced but not sufficiently to alter the threshold scaling with the network size. Therefore, the activation mechanisms remain the same as in the case of Markovian healing.