Recovery rate affects the effective epidemic threshold with synchronous updating.

Panpan Shu,Yi-Cheng Zhang,Ming Tang,Wei Wang,Pengcheng Zhao

doi:10.1063/1.4953661

Abstract

Accurate identification of effective epidemic threshold is essential for understanding epidemic dynamics on complex networks. In this paper, we systematically study how the recovery rate affects the susceptible-infected-removed spreading dynamics on complex networks, where synchronous and asynchronous updating processes are taken into account. We derive the theoretical effective epidemic threshold and final outbreak size based on the edge-based compartmental theory. To validate the proposed theoretical predictions, extensive numerical experiments are implemented by using asynchronous and synchronous updating methods. When asynchronous updating method is used in simulations, recovery rate does not affect the final state of spreading dynamics. But with synchronous updating, we find that the effective epidemic threshold decreases with recovery rate, and final outbreak size increases with recovery rate. A good agreement between the theoretical predictions and the numerical results are observed on both synthetic and real-world networks. Our results extend the existing theoretical studies and help us to understand the phase transition with arbitrary recovery rate.

Highlights

Susceptible-infected-recovered (SIR) model on complex networks has been used to model a wide variety of real epidemic spreading.1–3 Examples include the spreads of mumps, varicella, rabies, and Acquired Immune Deficiency Syndrome (AIDS).4 In the SIR model, an infected node can transmit a disease to each of its susceptible neighbors with infection rate b
The developed theory predicts that recovery rate does not affect the spreading dynamics with asynchronous updating, but with synchronous updating, the effective epidemic threshold decreases with the recovery rate, and the final outbreak sizes increase with the recovery rate for a given effective transmission rate
Our theoretical predictions are in a good agreement with the numerical effective epidemic thresholds identified by the variability measure,35,36 which has been confirmed to be effective for identifying the SIR effective epidemic threshold

Summary

INTRODUCTION

Susceptible-infected-recovered (SIR) model on complex networks has been used to model a wide variety of real epidemic spreading. Examples include the spreads of mumps, varicella, rabies, and Acquired Immune Deficiency Syndrome (AIDS). In the SIR model, an infected node can transmit a disease to each of its susceptible neighbors with infection rate b. The above theoretical predictions have been made by considering arbitrary recovery rate l, and pointed out that the value of l does not affect the effective epidemic threshold for continuous-time spreading dynamics. In cooperative games, the cooperators and defectors appear in turn in synchronous simulations, while the matrix always evolves rapidly into a state of overall defection in asynchronous simulations.31 In these two updating methods, how the recovery rate influences the spreading dynamics such as the effective epidemic threshold is long neglected. We develop an edge-based compartmental theory to derive the effective epidemic thresholds for the SIR model with arbitrary recovery rate, in both asynchronous and synchronous updating spreading processes. The proposed theory could be considered as supplementary to the existing theories, and it predicts that the effective epidemic threshold is independent of (decreases with) the recovery rate in asynchronous (synchronous) updating spreading processes. The validity of this numerical identification method for the SIR model has been confirmed in Ref. 37

THEORY

Random regular networks

Scale-free networks

Real-world networks

CONCLUSION AND DISCUSSION