Spatiotemporal dynamics of periodic waves in SIR model with driving factors

Abstract

The world is facing Covid-19 waves and the overall pattern of confirmed cases shows periodic oscillations. In this paper, we investigate the spatio-temporal spread of Covid-19 in the network-organized SIR model with an extrinsic incubation period of the driving factors. Firstly, Our analysis shows the occurrence of Hopf bifurcation and periodic outbreaks which is consistent with the actual spread of Covid-19. And we investigate periodic waves on spatial scales using Turing instability, and the spread of infected individuals increases the localized hot spots. We study the effect of the incubation period, more incubation periods generate Turing instability resulting in periodic outbreaks. There is an occurrence of bursting states at peaks of periodic waves due to small diffusion of infected and susceptible, which means stable and unstable areas try to convert each other due to high competition among nodes. Also, We note the disappearance of these bursts when infected and susceptible individuals' movements are easier, thus dominance of infected individuals prevails everywhere. Effective policy interventions and seasonality can cause periodic perturbations in the model, and therefore we study the impact of these perturbations on the spread of Covid-19. Periodic perturbations on the driving factors, infected individuals show co-existing spatial patterns. Chaotic outbreak becomes periodic outbreaks through alternating periodic or period-2 outbreaks as we regulate the amplitude and frequency of infected individuals. In short, regulations can erase period-2 and chaotic spread through policy interventions.

Nonequilibrium phase transitions in metapopulation models of infectious diseases on heterogeneous networks

“…However, the diffusion process may affect the stationary behavior of the epidemic model in the small-size network under some specific control parameters. Zheng et al found periodic outbreaks via Turing instabilities caused by the diffusion, delay, and driving factors [14][15][16]. The equations for network-averaged densities are written as…”

“…In the metapopulation models, we consider that individuals randomly move to connected nodes after all processes are completed in every node. Recently, Zheng et al presented the effects of the diffusion, delay, and driving factors on the stationary behavior of epidemic model on the small-size network with 100 nodes by calculating the eigenvalues of the Laplacian matrix of the network [14][15][16]. Infection processes are assumed to be frequency-dependent, which means that infection rate does not depend on the population [4].…”

Hopf bifurcation and patterns in a modified SIR model

“…Zheng et al constructed a network-organized SIR model to show the effects of the network structured entropy and diffusion on the bifurcation and Turing instability. They explained the dynamical mechanism of the periodic outbreak and endemic diseases through wavenumber [15], after that, the influences of directed network [16], driving factors [17] and time-delay network [18] on the pattern formation of epidemic model are given. Pattern formation provides new insight into the spread of infectious diseases, and relationships between pattern formation and the region of infectious diseases were introduced in the SIR reaction-diffusion model, which provided an optimal control method for the epidemic [19].…”

Hopf bifurcation and patterns in a modified SIR model

“…Zheng et al constructed a network-organized SIR model to show the effects of the network structured entropy and diffusion on the bifurcation and Turing instability. They explained the dynamical mechanism of the periodic outbreak and endemic diseases through wavenumber [15], after that, the influences of directed network [16], driving factors [17] and timedelay network [18] on the pattern formation of epidemic model are given. Pattern formation provides new insight into the spread of infectious diseases, and relationships between pattern formation and the region of infectious diseases were introduced in the SIR reactiondiffusion model, which provided an optimal control method for the epidemic [19].…”