Pattern formation and bifurcation analysis of delay induced fractional-order epidemic spreading on networks

TL;DR

This paper investigates how delay and fractional-order dynamics influence pattern formation in network-based epidemic models, revealing conditions for Turing instability and the effects of network parameters on spatiotemporal patterns.

Contribution

It introduces a novel delayed fractional-order SIRS model on networks and analyzes how delay, network degree, and diffusion affect pattern formation and stability.

Findings

Delays induce system instability and periodic fluctuations.

Network degree influences oscillatory spatial patterns.

Fractional-order dynamics can suppress pattern formation.

Abstract

The spontaneous emergence of ordered structures, known as Turing patterns, in complex networks is a phenomenon that holds potential applications across diverse scientific fields, including biology, chemistry, and physics. Here, we present a novel delayed fractional-order susceptible-infected-recovered-susceptible (SIRS) reaction-diffusion model functioning on a network, which is typically used to simulate disease transmission but can also model rumor propagation in social contexts. Our theoretical analysis establishes the Turing instability resulting from delay, and we support our conclusions through numerical experiments. We identify the unique impacts of delay, average network degree, and diffusion rate on pattern formation. The primary outcomes of our study are: (i) Delays cause system instability, mainly evidenced by periodic temporal fluctuations; (ii) The average network degree produces periodic oscillatory states in uneven spatial distributions; (iii) The combined influence of diffusion rate and delay results in irregular oscillations in both time and space. However, we also find that fractional-order can suppress the formation of spatiotemporal patterns. These findings are crucial for comprehending the impact of network structure on the dynamics of fractional-order systems.