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

Volume 27, Issue 1, pp 1--17 http://dx.doi.org/10.22436/jmcs.027.01.01

Publication Date: February 10, 2022 Submission Date: August 14, 2021 Revision Date: October 25, 2021 Accteptance Date: December 01, 2021

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Authors

A. Triska - Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jl. Ganesha No. 10, Bandung, West Java, 40116, Indonesia. A. Y. Gunawan - Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jl. Ganesha No. 10, Bandung, West Java, 40116, Indonesia. N. Nuraini - Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jl. Ganesha No. 10, Bandung, West Java, 40116, Indonesia.

Abstract

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 non-outbreak. 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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Keywords

• Cross-diffusion of the infected
• spatial epidemic model
• Turing pattern
• Turing bifurcation
• amplitude equations

MSC

•  35B36
•  92B05
•  37D99
•  92D25
•  35K57
•  92D30

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