一类新的空间扩散传染病模型的动力学行为
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摘要
建立一类新的由传染病引起严重疾病或并发症的传染病动力学模型.研究该模型平衡点的存在性和稳定性,运用Schauder不动点定理证明连接2个平衡点的行波解的存在性.该结果揭示了由无病情况发展为地方性疾病的一种变化轨迹,解释了当新的传染病爆发,导致疾病不能治愈的情况下,若不对患者做有效隔离,疾病将会蔓延这一现象.最后,通过数值模拟验证了此行波解的存在性.
In this paper,a new epidemic model is established that the infected disease may lose infectiousness and then evolves to a chronic non-infectious disease or more serious disease when it is not cured within a certain time. The existence and stability of equilibria of the model are studied. The existence of traveling wave solutions connecting two equilibria are also proved by using Schauder fixed point theorem. It shows that the infectious disease can be turned to endemic disease. The main results explain the phenomenon that the infectious disease will spread eventually to the whole region when it is not controlled effectively. Finally,numerical simulations illustrate the existence of traveling wave solutions.
In this paper,a new epidemic model is established that the infected disease may lose infectiousness and then evolves to a chronic non-infectious disease or more serious disease when it is not cured within a certain time. The existence and stability of equilibria of the model are studied. The existence of traveling wave solutions connecting two equilibria are also proved by using Schauder fixed point theorem. It shows that the infectious disease can be turned to endemic disease. The main results explain the phenomenon that the infectious disease will spread eventually to the whole region when it is not controlled effectively. Finally,numerical simulations illustrate the existence of traveling wave solutions.
引文
[1]WANG Z Z,GUO Z M.Dynamical behavior of a new epidemiological model[J].J Appl Math,2014,2014:1-9.
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[4]FARIA T,HUANG W Z,WU J.Travelling waves for delayed reaction-diffusion equations with global response[J].Proc RSoc Lond Ser A,2006,462:229-261.
[5]MA S,ZOU X.Existence,uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay[J].J Diff Eqns,2005,217:54-87.
[6]KUANG Y,GOURLEY S A.Wavefronts and global stability in a time-delayed population model with stage structure[J].Proc R Soc Lond Ser A,2003,459:1563-1579.
[7]HUANG W Z.Traveling waves for a biological reaction-diffusion model[J].J Dynam Diff Eqns,2004,16(3):745-765.
[8]HUANG Y H,WENG P X.Traveling waves of a diffusive predator-prey system with general functional response[J].Nonlin Anal Real World Appl,2013,14:940-959.
[9]WU J,ZOU X.Traveling wave fronts of reaction-diffusion systems with delay[J].J Dynam Diff Eqns,2001,13(3):651-687.
[10]HIRSCH M W,SMALE S,DEVANEY R L.Differential equations,dynamical systems and an introduction to chaos[M].Amsterdam:Elsevier Acdemic Press,2004.
[11]HONG K,WENG P X.Stability and traveling waves of a stage-structured predator-prey model with Holling type-II functional response and harvesting[J].Nonlin Anal Real World Appl,2013,14:83-103.
[2]XIAO D,RUAN S.Global analysis of an epidemic model with nonmonotone incidence rate[J].Math Biosci,2007,208(2):419-429.
[3]DUNBAR S.Traveling waves in diffusive predator-prey equations:periodic orbits and point-to periodic heteroclinic orbits[J].SIAM J Appl Math,1986,46:1057-1078.
[4]FARIA T,HUANG W Z,WU J.Travelling waves for delayed reaction-diffusion equations with global response[J].Proc RSoc Lond Ser A,2006,462:229-261.
[5]MA S,ZOU X.Existence,uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay[J].J Diff Eqns,2005,217:54-87.
[6]KUANG Y,GOURLEY S A.Wavefronts and global stability in a time-delayed population model with stage structure[J].Proc R Soc Lond Ser A,2003,459:1563-1579.
[7]HUANG W Z.Traveling waves for a biological reaction-diffusion model[J].J Dynam Diff Eqns,2004,16(3):745-765.
[8]HUANG Y H,WENG P X.Traveling waves of a diffusive predator-prey system with general functional response[J].Nonlin Anal Real World Appl,2013,14:940-959.
[9]WU J,ZOU X.Traveling wave fronts of reaction-diffusion systems with delay[J].J Dynam Diff Eqns,2001,13(3):651-687.
[10]HIRSCH M W,SMALE S,DEVANEY R L.Differential equations,dynamical systems and an introduction to chaos[M].Amsterdam:Elsevier Acdemic Press,2004.
[11]HONG K,WENG P X.Stability and traveling waves of a stage-structured predator-prey model with Holling type-II functional response and harvesting[J].Nonlin Anal Real World Appl,2013,14:83-103.