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

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 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.

Spatial pattern of a class of SI models driven by cross diffusion

“…以流行病学领域为例, 在 现实生活中易感者有能力识别感染者并且选择远 离或接近他们 [25,26] , 随后研究人员对流行病学模型 的空间格局进行了一系列的研究. Aly等 [27] 研究 了一种具有自扩散和交叉扩散项SIS模型的稳定性 问题, 并通过数值模拟研究了Turing模式. Triska 等 [28] 疫的疾病, 如普通感冒、疟疾等 [29] .…”

The Effects of the Susceptible and Infected Cross-Diffusion Terms on Pattern Formations in an SI Model

“…Nevertheless, allowing infected people to move freely to high-density areas of the susceptible in a pandemic situation is harmful. Therefore, recently, Triska et al in [27] introduced the use of cross-diffusion of the infected in a spatial epidemic model so that the model has complete usage of the diffusion terms. The presence of cross-diffusion of the infected term is certainly not without a biological meaning.…”

“…The idea of the model in [27] was inspired by the diffusion term widely applied by researchers to predator-prey models, with self-diffusion only [28][29][30][31][32][33] and with crossdiffusion as well [34][35][36][37][38][39][40][41]. In describing the behavior of a predator and prey in nature, the predator-prey models in [35,36,39,41] involve cross-diffusion terms completely, namely of the predator and prey.…”

“…In [27], we proposed a spatial epidemic model to capture the phenomenon of the movement of the infected especially to areas of high density of the susceptible for various reasons, such as for work, to access public facilities, or to study. As a consequence, in contrast to the perspective of the predator-prey model in [36,39,41], a negative value to the cross-diffusion coefficient of the infected was applied to imply that the infected move towards a higher density of the susceptible.…”

“…The discussions of a spatial epidemic model in [27] were focused on the model construction to capture the phenomenon of the movement of the susceptible and infected caused by one another simultaneously. Furthermore, patterns dynamics due to variations in the cross-diffusion coefficient of the infected were also discussed, which were verified analytically and numerically.…”