Stationary Patterns of a Cross-Diffusion Epidemic Model

Cai, Yongli; Dong-xuan, Chi; Liu, Wenbin; Wang, Weiming

doi:10.1155/2013/852698

Cited by 6 publications

(10 citation statements)

References 52 publications

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“…Cross-diffusion of the susceptible is also applied to the spatial epidemic model. This term represents the tendency of the susceptible to stay away from the infected since they are able to recognize the infected [4,5,12,24,37,41,43]. The results of this study indicate that susceptible cross-diffusion has an influence on the spread of an infectious disease and the pattern dynamics.…”

Section: Introductionmentioning

confidence: 93%

“…To prevent this to happen again, there must be a study on the behavior patterns of people moving who are still infected, from one place to another especially to areas with high density of the susceptible such as for work or study. As a result, this study addresses a spatial epidemic model, not only with cross-diffusion of the susceptible as in previous studies [4,5,12,24,37,43], but also with cross-diffusion of the infected to illustrate the abovementioned situation. The diffusion coefficient can be positive, zero or negative [16,37,40].…”

Section: Introductionmentioning

confidence: 95%

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Outbreak spatial pattern formation based on an SI model with the infected cross-diffusion term

Triska¹,

Gunawan²,

Nur’aini³

2022

J. Math. Computer Sci.

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This study is to discuss the pattern formations of a spatial epidemic model with cross-diffusion of the susceptible and infected groups simultaneously. The infected cross-diffusion term described the situation that the infected was allowed to move to areas with high density of the susceptible such as for work or study, especially after the pandemic. Turing analysis was applied to the model and yielded the conditions for Turing instability corresponding to the model. The amplitude equations were also given by the support of multiple-scale analysis, which then provided information about the stability of the patterns near the Turing bifurcation point. Numerical simulations revealed that there were five types of patterns, such as the spots, spots-stripes, stripes, stripes-holes, and holes. The holes indicated a disease outbreak in a region, while the spots showed nonoutbreak. Furthermore, numerical simulations were carried out by varying the cross-diffusion coefficients of the susceptible and infected. The simulation results showed once the cross-diffusion coefficient of the infected was bigger than the susceptible, then an outbreak in a region was triggered. The results of this study showed that the movement of infected had a significant role in the spread of an infectious disease that could lead to another wave of pandemic. By using Turing analysis as a tool, as well as predator-prey model as the basis of movement theory, this paper tries to fill in the gaps in the discussion about the movement of infected people to areas with high density of the susceptible.

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Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 95%

Outbreak spatial pattern formation based on an SI model with the infected cross-diffusion term

Triska¹,

Gunawan²,

Nur’aini³

2022

J. Math. Computer Sci.

View full text Add to dashboard Cite

show abstract

“…Many studies indicate that spatial epidemiology with selfdiffusion has become a principal scientific discipline aiming at understanding the causes and consequences of spatial heterogeneity in disease transmission [3]. In these studies, reaction-diffusion equations have been intensively used to describe spatiotemporal dynamics.…”

Section: Introductionmentioning

confidence: 99%

“…Turing's revolutionary idea was that the passive diffusion could interact with chemical reaction in such a way that even if the reaction by itself has no symmetrybreaking capabilities, diffusion can destabilize the symmetric solutions with the result that the system with diffusion has them [12]. In these studies [3,[13][14][15][16][17][18][19][20], via standard linear analysis, the authors obtained the conditions of Turing instability, and, via numerical simulation, they showed the pattern formation induced by self-diffusion or cross-diffusion and found that model dynamics exhibits a diffusion controlled formation growth to stripes, spots, and coexistence or chaos pattern replication.…”

Section: Introductionmentioning

confidence: 99%

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The Existence of Positive Nonconstant Steady States in a Reaction: Diffusion Epidemic Model

Yuan

Wang

2013

Abstract and Applied Analysis

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Fully-Discrete Lyapunov Consistent Discretizations for Parabolic Reaction-Diffusion Equations with r Species

Jahdali,

Fernández,

Dalcin

et al. 2024

Commun. Appl. Math. Comput.

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Reaction-diffusion equations model various biological, physical, sociological, and environmental phenomena. Often, numerical simulations are used to understand and discover the dynamics of such systems. Following the extension of the nonlinear Lyapunov theory applied to some class of reaction-diffusion partial differential equations (PDEs), we develop the first fully discrete Lyapunov discretizations that are consistent with the stability properties of the continuous parabolic reaction-diffusion models. The proposed framework provides a systematic procedure to develop fully discrete schemes of arbitrary order in space and time for solving a broad class of equations equipped with a Lyapunov functional. The new schemes are applied to solve systems of PDEs, which arise in epidemiology and oncolytic M1 virotherapy. The new computational framework provides physically consistent and accurate results without exhibiting scheme-dependent instabilities and converging to unphysical solutions. The proposed approach represents a capstone for developing efficient, robust, and predictive technologies for simulating complex phenomena.

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Stationary Patterns of a Cross-Diffusion Epidemic Model

Cited by 6 publications

References 52 publications

Outbreak spatial pattern formation based on an SI model with the infected cross-diffusion term

Outbreak spatial pattern formation based on an SI model with the infected cross-diffusion term

The Existence of Positive Nonconstant Steady States in a Reaction: Diffusion Epidemic Model

Fully-Discrete Lyapunov Consistent Discretizations for Parabolic Reaction-Diffusion Equations with r Species

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