非生成同质扩散 SIR 狂犬病模型中的动力学和分岔现象

IF 1.9 4区 数学 Q1 MATHEMATICS, APPLIED
Gaoyang She, Fengqi Yi
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引用次数: 0

摘要

SIAM 应用数学杂志》第 84 卷第 2 期第 632-660 页,2024 年 4 月。 摘要在本文中,我们对非退化反应-扩散狂犬病 SIR 模型的时空模式形成和分岔感兴趣,该模型被用来解释在欧洲观察到的流行病模式。首先,通过使用迭代法,我们能够证明抛物线系统的全局存在性和实时解的有界性。其次,对于 ODE,我们分析证明了安德森等人观察到的现象[《自然》,289 (1981),第 765-771 页]:如果承载能力[math]小于某个正数[math],那么狂犬病最终会消亡;如果[math]大于[math],那么狂犬病会流行。此外,如果[math]为某个正数[math],则地方病平衡解(局部渐近)稳定,而如果[math]为某个正数[math],则地方病平衡解不稳定。特别是,在 [math] 时,内生平衡稳定性的丧失会导致霍普夫分岔。最后,对于 PDEs,我们推导出了扩散率的充分条件,因此在这些条件下,内生平衡解和霍普夫分岔空间均质周期解都可能出现图灵不稳定性。一旦解(平衡解或周期解)出现图灵不稳定性,我们就能从数值上观察到系统可能出现新的时空模式。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Dynamics and Bifurcations in a Nondegenerate Homogeneous Diffusive SIR Rabies Model
SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 632-660, April 2024.
Abstract. In this paper, we are interested in the spatiotemporal pattern formations and bifurcations for a nondegenerate reaction-diffusion rabies SIR model which was used to explain the epidemiological patterns observed in Europe. First, by using the iteration methods, we are able to show the global existence and boundedness of in-time solutions of the parabolic system. Second, for the ODEs, we analytically prove the phenomena observed by Anderson et al. [Nature, 289 (1981), pp. 765–771]: if the carrying capacity [math] is smaller than some positive [math], then rabies eventually dies out; if [math] is larger than [math], then the rabies prevails. Moreover, if [math] for some positive [math], then the endemic equilibrium solution is (locally asymptotically) stable, while it is unstable if [math]. In particular, at [math], the loss of the stability of the endemic equilibrium leads to a Hopf bifurcation. Finally, for the PDEs, we derive sufficient conditions on the diffusion rates so that under these conditions, Turing instability of both the endemic equilibrium solution and the Hopf bifurcating spatially homogeneous periodic solutions can occur. Once Turing instability of the solution (equilibrium or periodic solution) occurs, it is observed numerically that the system might have new spatiotemporal patterns.
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来源期刊
CiteScore
3.60
自引率
0.00%
发文量
79
审稿时长
12 months
期刊介绍: SIAM Journal on Applied Mathematics (SIAP) is an interdisciplinary journal containing research articles that treat scientific problems using methods that are of mathematical interest. Appropriate subject areas include the physical, engineering, financial, and life sciences. Examples are problems in fluid mechanics, including reaction-diffusion problems, sedimentation, combustion, and transport theory; solid mechanics; elasticity; electromagnetic theory and optics; materials science; mathematical biology, including population dynamics, biomechanics, and physiology; linear and nonlinear wave propagation, including scattering theory and wave propagation in random media; inverse problems; nonlinear dynamics; and stochastic processes, including queueing theory. Mathematical techniques of interest include asymptotic methods, bifurcation theory, dynamical systems theory, complex network theory, computational methods, and probabilistic and statistical methods.
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