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Title: 一個具擴散性的SIR模型之行進波解
Traveling wave solutions for a diffusive SIR model
Authors: 余陳宗
Yu, Chen Tzung
Contributors: 符聖珍
Fu, Sheng Chen
Yu, Chen Tzung
Keywords: 行進波解
SIR model
traveling wave solution
Date: 2016
Issue Date: 2016-12-07 10:46:54 (UTC+8)
Abstract: 本篇論文討論的是SIR模型的反應擴散方程
         s_t = d_1 s_xx − βsi/(s + i),
         i_t = d_2 i_xx + βsi/(s + i) − γi,
         r_t = d_3 r_xx + γi,
之行進波的存在性,其中模型描述的是在一個封閉區域裡流行疾病爆發的狀態。這裡的 β 是傳播係數,γ 是治癒或移除(即死亡)速率,s 是未被傳染個體數,i 是傳染源個體數,d_1、d_2、d_3分別為其擴散之係數。
  我們將使用Schauder不動點定理(Schauder fixed point theorem)、Arzela-Ascoli定理和最大值原理(maximum principle)來證明:該系統存在最小速度為c=c*:=2√(d2( β - γ ))之行進波解。我們的結果回答了[11]裡所提出的開放式問題。
 In this thesis, we study the existence of traveling waves of a reaction-diffusion equation for a diffusive epidemic SIR model
         s_t = d_1 s_xx − βsi/(s + i),
         i_t = d_2 i_xx + βsi/(s + i) − γi,
         r_t = d_3 r_xx + γi,
which describes an infectious disease outbreak in a closed population. Here β is the transmission coefficient, γ is the recovery or remove rate, and s, i, and r rep-resent numbers of susceptible individuals, infected individuals, and removed individuals, respectively, and d_1, d_2, and d_3 are their diffusion rates. We use the Schauder fixed point theorem, the Arzela-Ascoli theorem, and the maximum principle to show that this system has a traveling wave solution with minimum speed c=c*:=2√(d2( β - γ )). Our result answers an open problem proposed in [11].
Reference: [1]  Shangbing Ai and Wenzhang Huang. Travelling waves for a reaction–diffusion system in population dynamics and epidemiology. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 135(4):663–675, 07 2007.

[2]  Steven R. Dunbar. Travelling wave solutions of diffusive Lotka-Volterra equations. J. Math. Biol., 17(1):11–32, 1983.

[3]  Steven R. Dunbar. Traveling wave solutions of diffusive Lotka-Volterra equations: a het-eroclinic connection in R4. Trans. Amer. Math. Soc., 286(2):557–594, 1984.

[4]  Sheng-Chen Fu. The existence of traveling wave fronts for a reaction-diffusion system modelling the acidic nitrate-ferroin reaction. Quart. Appl. Math., 72(4):649??64, 2014.

[5]  Sheng-Chen Fu. Traveling waves for a diffusive SIR model with delay. J. Math. Anal. Appl., 435(1):20–37, 2016.

[6]  Philip Hartman. Ordinary differential equations, volume 38 of Classics in Applied Math-ematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Corrected reprint of the second (1982) edition [Birkhäuser, Boston, MA; MR0658490 (83e: 34002)], With a foreword by Peter Bates.

[7]  Yuzo Hosono and Bilal Ilyas. Existence of traveling waves with any positive speed for a diffusive epidemic model. Nonlinear World, 1(3):277–290, 1994.

[8]  Yuzo Hosono and Bilal Ilyas. Traveling waves for a simple diffusive epidemic model. Math. Models Methods Appl. Sci., 5(7):935–966, 1995.

[9]  Wenzhang Huang. Traveling waves for a biological reaction-diffusion model. J. Dynam. Differential Equations, 16(3):745–765, 2004.

[10] Anders Källén. Thresholds and travelling waves in an epidemic model for rabies. Nonlin-ear Anal., 8(8):851–856, 1984.

[11] Xiang-Sheng Wang, Haiyan Wang, and Jianhong Wu. Traveling waves of diffusive predator-prey systems: disease outbreak propagation. Discrete Contin. Dyn. Syst., 32(9): 3303–3324, 2012.
Description: 碩士
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Data Type: thesis
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