Spatiotemporal Dynamics in a Reaction–Diffusion Epidemic Model with a Time-Delay in Transmission

2015 ◽  
Vol 25 (08) ◽  
pp. 1550099 ◽  
Author(s):  
Yongli Cai ◽  
Shuling Yan ◽  
Hailing Wang ◽  
Xinze Lian ◽  
Weiming Wang
Keyword(s):  
Time Delay ◽  
Epidemic Model ◽  
Spiral Wave ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Spatiotemporal Dynamics ◽  
Strong Impact ◽  
Delayed Reaction ◽  
Wave Patterns ◽  
And Diffusion

In this paper, we investigate the effects of time-delay and diffusion on the disease dynamics in an epidemic model analytically and numerically. We give the conditions of Hopf and Turing bifurcations in a spatial domain. From the results of mathematical analysis and numerical simulations, we find that for unequal diffusive coefficients, time-delay and diffusion may induce that Turing instability results in stationary Turing patterns, Hopf instability results in spiral wave patterns, and Hopf–Turing instability results in chaotic wave patterns. Our results well extend the findings of spatiotemporal dynamics in the delayed reaction–diffusion epidemic model, and show that time-delay has a strong impact on the pattern formation of the reaction–diffusion epidemic model.

Hopf Bifurcation and Delay-Induced Turing Instability in a Diffusive lac Operon Model

2016 ◽  
Vol 26 (10) ◽  
pp. 1650167 ◽  
Author(s):  
Xin Cao ◽  
Yongli Song ◽  
Tonghua Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Spatial Patterns ◽  
Turing Instability ◽  
Delayed Feedback ◽  
Spatiotemporal Dynamics ◽  
Positive Equilibrium ◽  
Lac Operon ◽  
Diffusion Effect ◽  
And Diffusion

In this paper, we investigate the dynamics of a [Formula: see text] operon model with delayed feedback and diffusion effect. If the system is without delay or the delay is small, the positive equilibrium is stable so that there are no spatial patterns formed; while the time delay is large enough the equilibrium becomes unstable so that rich spatiotemporal dynamics may occur. We have found that time delay can not only incur temporal oscillations but also induce imbalance in space. With different initial values, the system may have different spatial patterns, for instance, spirals with one head, four heads, nine heads, and even microspirals.

A reaction–diffusion epidemic model with incubation period in almost periodic environments

2020 ◽  
pp. 1-24
Author(s):  
LIZHONG QIANG ◽  
BIN-GUO WANG ◽  
ZHI-CHENG WANG
Keyword(s):  
Time Delay ◽  
Spatial Heterogeneity ◽  
Incubation Period ◽  
Epidemic Model ◽  
Reaction Diffusion ◽  
Fixed Time ◽  
Reaction Diffusion Equations ◽  
Threshold Dynamics ◽  
Almost Periodic ◽  
Periodic Reaction

In this paper, we propose and study an almost periodic reaction–diffusion epidemic model in which disease latency, spatial heterogeneity and general seasonal fluctuations are incorporated. The model is given by a spatially nonlocal reaction–diffusion system with a fixed time delay. We first characterise the upper Lyapunov exponent $${\lambda ^*}$$ for a class of almost periodic reaction–diffusion equations with a fixed time delay and provide a numerical method to compute it. On this basis, the global threshold dynamics of this model is established in terms of $${\lambda ^*}$$ . It is shown that the disease-free almost periodic solution is globally attractive if $${\lambda ^*} < 0$$ , while the disease is persistent if $${\lambda ^*} < 0$$ . By virtue of numerical simulations, we investigate the effects of diffusion rate, incubation period and spatial heterogeneity on disease transmission.

scholarly journals Design and analysis of a discrete method for a time‐delayed reaction–diffusion epidemic model

2020 ◽  
Author(s):  
Jorge E. Macías‐Díaz ◽  
Nauman Ahmed ◽  
Muhammad Jawaz ◽  
Muhammad Rafiq ◽  
Muhammad Aziz ur Rehman
Keyword(s):  
Epidemic Model ◽  
Reaction Diffusion ◽  
Delayed Reaction ◽  
Discrete Method

scholarly journals Pattern Dynamics of Nonlocal Delay SI Epidemic Model with the Growth of the Susceptible following Logistic Mode

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Zun-Guang Guo ◽  
Jing Li ◽  
Can Li ◽  
Juan Liang ◽  
Yiwei Yan
Keyword(s):  
Time Delay ◽  
Epidemic Model ◽  
Turing Instability ◽  
Average Density ◽  
Linear Stability Theory ◽  
Susceptible Population ◽  
Nonlocal Delay ◽  
Pattern Dynamics ◽  
And Control ◽  
Theoretical Results

In this paper, we investigate pattern dynamics of a nonlocal delay SI epidemic model with the growth of susceptible population following logistic mode. Applying the linear stability theory, the condition that the model generates Turing instability at the endemic steady state is analyzed; then, the exact Turing domain is found in the parameter space. Additionally, numerical results show that the time delay has key effect on the spatial distribution of the infected, that is, time delay induces the system to generate stripe patterns with different spatial structures and affects the average density of the infected. The numerical simulation is consistent with the theoretical results, which provides a reference for disease prevention and control.

scholarly journals Pattern Formation in a Predator-Prey Model with Both Cross Diffusion and Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Boli Xie ◽  
Zhijun Wang ◽  
Yakui Xue
Keyword(s):  
Time Delay ◽  
Spatial Dynamics ◽  
Turing Instability ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Real Model ◽  
Diffusion Controlled ◽  
And Diffusion

A predator-prey model with both cross diffusion and time delay is considered. We give the conditions for emerging Turing instability in detail. Furthermore, we illustrate the spatial patterns via numerical simulations, which show that the model dynamics exhibits a delay and diffusion controlled formation growth not only of spots and stripe-like patterns, but also of the two coexist. The obtained results show that this system has rich dynamics; these patterns show that it is useful for the diffusive predation model with a delay effect to reveal the spatial dynamics in the real model.

SPIRALS IN SCALAR REACTION–DIFFUSION EQUATIONS

1995 ◽  
Vol 05 (06) ◽  
pp. 1487-1501 ◽  
Author(s):  
MICHAEL DELLNITZ ◽  
MARTIN GOLUBITSKY ◽  
ANDREAS HOHMANN ◽  
IAN STEWART
Keyword(s):  
Boundary Conditions ◽  
Spiral Wave ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Direct Simulation ◽  
Spiral Pattern ◽  
Wave Patterns ◽  
Single Reaction

Spiral patterns have been observed experimentally, numerically, and theoretically in a variety of systems. It is often believed that these spiral wave patterns can occur only in systems of reaction–diffusion equations. We show, both theoretically (using Hopf bifurcation techniques) and numerically (using both direct simulation and continuation of rotating waves) that spiral wave patterns can appear in a single reaction–diffusion equation [ in u(x, t)] on a disk, if one assumes "spiral" boundary conditions (ur = muθ). Spiral boundary conditions are motivated by assuming that a solution is infinitesimally an Archimedian spiral near the boundary. It follows from a bifurcation analysis that for this form of spirals there are no singularities in the spiral pattern (technically there is no spiral tip) and that at bifurcation there is a steep gradient between the "red" and "blue" arms of the spiral.

scholarly journals Turing Instability of Brusselator in the Reaction-Diffusion Network

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Yansu Ji ◽  
Jianwei Shen
Keyword(s):  
Diffusion Coefficient ◽  
Random Network ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Spontaneous Generation ◽  
Brusselator Model ◽  
Network Connection ◽  
Pattern Dynamics ◽  
Connection Probability ◽  
And Diffusion

Turing instability constitutes a universal paradigm for the spontaneous generation of spatially organized patterns, especially in a chemical reaction. In this paper, we investigated the pattern dynamics of Brusselator from the view of complex networks and considered the interaction between diffusion and reaction in the random network. After a detailed theoretical analysis, we obtained the approximate instability region about the diffusion coefficient and the connection probability of the random network. In the meantime, we also obtained the critical condition of Turing instability in the network-organized system and found that how the network connection probability and diffusion coefficient affect the reaction-diffusion system of the Brusselator model. In the end, the reason for arising of Turing instability in the Brusselator with the random network was explained. Numerical simulation verified the theoretical results.

scholarly journals Pattern formation in a reaction-diffusion rumor propagation system with Allee effect and time delay

2021 ◽  
Author(s):  
Linhe Zhu ◽  
Le He
Keyword(s):  
Diffusion Coefficient ◽  
Time Delay ◽  
Allee Effect ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Small Time ◽  
Diffusion Behavior ◽  
Rumor Propagation ◽  
Reaction Diffusion Model ◽  
Stability And Instability

Abstract This paper analyzes the diffusion behavior of the suspicious and the infected cabins in cyberspace by establishing a rumor propagation reaction diffusion model with Allee effect and time delay. The Turing instability conditions of the system under various conditions are emphatically studied. After considering the delay effect of rumor propagation systems, we have studied the correlation between the stability of the system under the influence of small time delay and the homogeneous system near the equilibrium point, and the critical condition of the delay-induced spatial instability is given. Further considering the possibility of diffusion coefficient changing with time, the critical parameter curves of stability and instability of approximate systems are given by means of Floquet theory, and the necessary conditions of Turing-instability of periodic coefficient are studied. In the numerical simulations, we find that the variation of diffusion coefficient will change the pattern type, and the periodical diffusion behavior will affect the arrangement of the crowd gathering area in the pattern.

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