具时滞扩散效应的病原体-免疫模型的稳定性及分支

摘要/Abstract

摘要：

为了了解病原体与免疫细胞相互作用过程中存在的扩散因素与时滞因素对其动力学行为的影响，建立了带有齐次Neumann边界条件的具时滞病原体-免疫反应扩散模型.以病原体与免疫细胞的扩散比率和免疫时滞为参数，通过分析该模型在正稳态解处线性化系统特征根的分布，并利用泛函微分方程分支理论，得到正稳态解经历Turing失稳的充要条件以及经历Hopf分支的条件.利用Matlab数值模拟直观地展示了病原体与宿主免疫在临界点附近经历Turing失稳和Hopf分支的动力学行为，并解释了动力学行为所对应的生物医学意义，为控制病原体生长提供了一定的理论支持.

关键词: 病原体免疫, 反应扩散, 时滞, Turing分支, Hopf分支

Abstract:

In order to understand the effects of diffusion and time-delay factors on the dynamics between pathogens and immune cells, a delayed pathogen-immune reaction diffusion model with homogeneous Neumann boundary condition is established. By using the diffusion ratio of pathogen-immune cells and immune delay as two parameters, the characteristic root distribution of the linearized system at the positive steady state is analyzed and the necessary and sufficient conditions for the positive steady state to undergo Turing instability and Hopf bifurcation are obtained by using the bifurcation theory of functional differential equations. In addition, the dynamic behavior close to the critical value of Turing instability and Hopf bifurcation is intuitively shown by Matlab numerical simulation. The biological and medicinal significance of corresponding dynamic behaviors are discussed. Furthermore, the obtained results provide certain theoretical support for controlling the growth of pathogen.

Key words: Pathogen immunity, Reaction diffusion, Delay, Turing bifurcation, Hopf bifurcation

中图分类号:

O175.29

王晶囡,杨德中. 具时滞扩散效应的病原体-免疫模型的稳定性及分支[J]. 数学物理学报, 2021, 41(4): 1204-1217.

Jingnan Wang,Dezhong Yang. Stability and Bifurcation of a Pathogen-Immune Model with Delay and Diffusion Effects[J]. Acta mathematica scientia,Series A, 2021, 41(4): 1204-1217.

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参考文献 28

1	Naheed A , Singh M , Lucy D . Numerical study of SARS epidemic model with the inclusion of diffusion in the system. Appl Math Comput, 2014, 229, 480- 498
2	Beauchemin C A A , Handel A . A review of mathematical models of influenza a infections within a host or cell culture: lessons learned and challenges ahead. BMC Public Health, 2011, 11 (Suppl 1): S7 doi: 10.1186/1471-2458-11-S1-S7
3	韩祥临, 汪维刚, 莫嘉琪. 流行性病毒传播生态动力学系统. 数学物理学报, 2019, 39A (1): 200- 208 doi: 10.3969/j.issn.1003-3998.2019.01.019
	Han X L , Wang W G , Mo J Q . Bionomics dynamic system for epidemic virus transmission. Acta Math Sci, 2019, 39A (1): 200- 208 doi: 10.3969/j.issn.1003-3998.2019.01.019
4	唐三一, 唐彪, BragazziN L, 等. 新型冠状病毒肺炎疫情数据挖掘与离散随机传播动力学模型分析. 中国科学: 数学, 2020, 50 (8): 1071- 1086
	Tang S Y , Tang B , Bragazzi N L , et al. Analysis of COVID-19 epidemic traced data and stochastic discrete transmission dynamic model. Sci Sin Math, 2020, 50 (8): 1071- 1086
5	程欣欣, 饶亚情, 黄刚. 封闭空间中新型冠状病毒肺炎传播模型: 以日本"钻石公主号"邮轮为例. 数学物理学报, 2020, 40A (2): 540- 544 doi: 10.3969/j.issn.1003-3998.2020.02.024
	Cheng X X , Rao Y Q , Huang G . COVID-19 transmission model in an enclosed space: a case study of Japan Diamond Princess Cruises. Acta Math Sci, 2020, 40A (2): 540- 544 doi: 10.3969/j.issn.1003-3998.2020.02.024
6	Gilchrist M A , Sasaki A . Modeling host-parasite coevolution: a nested approach based on mechanistic models. J Theor Biol, 2002, 218 (3): 289- 308 doi: 10.1006/jtbi.2002.3076
7	Mohtashemi M , Levins R . Transient dynamics and early diagnostics in infectious disease. J Math Biol, 2001, 43 (5): 446- 470 doi: 10.1007/s002850100103
8	Pugliese A , Gandolfi A . A simple model of pathogen-immune dynamics including specific and non-specific immunity. Math Biosci, 2008, 214 (1): 73- 80
9	Wang W , Ma W . Hepatitis C virus infection is blocked by HMGB1: a new nonlocal and time-delayed reaction-diffusion model. Appl Math Comput, 2018, 320, 633- 653
10	Stancevic O , Angstmann C N , Murray J M , et al. Turing patterns from dynamics of early HIV infection. B Math Biol, 2013, 75 (5): 774- 795 doi: 10.1007/s11538-013-9834-5
11	Lee M R , Huang Y T , Lee P I , et al. Healthcare-associated bacteraemia caused by Leuconostoc species at a university hospital in Taiwan between 1995 and 2008. J Hosp Infect, 2011, 78 (1): 45- 49 doi: 10.1016/j.jhin.2010.11.014
12	崔青曼, 袁春营, 李春岭, 等. 主要海水养殖鱼类白点病和盾纤毛虫病防治技术. 水利渔业, 2007, 27 (6): 85- 87 doi: 10.3969/j.issn.1003-1278.2007.06.037
	Cui Q M , Yuan C Y , Li C L , et al. Control of white - spot disease and scuticociliatida disease of some main cultured sea fishes. Reservoir Fisheries, 2007, 27 (6): 85- 87 doi: 10.3969/j.issn.1003-1278.2007.06.037
13	孙汶生. 医学免疫学. 北京: 高等教出版社, 2010
	Sun W S . Medical Immunology. Bei Jing: Higher Education Press, 2010
14	Han X L , Jin Z . A dynamic model of hepatitis B virus with delayed immune response. J North University of China, 2011, 32 (1): 197- 208
15	Bai Z , Peng R , Zhao X Q . A reaction-diffusion malaria model with seasonality and incubation period. J Math Biol, 2018, 77 (1): 201- 228 doi: 10.1007/s00285-017-1193-7
16	Zhu D D , Ren J L , Zhu H P . Spatial-temporal basic reproduction number and dynamics for a dengue disease diffusion model for a dengue disease diffusion model. Math Meth Appl Sci, 2018, 41, 5388- 5403 doi: 10.1002/mma.5085
17	Yamazaki K . Threshold dynamics of reaction-diffusion partial differential equations model of Ebola virus disease. Int J Biomath, 2018, 11 (8): 1850108 doi: 10.1142/S1793524518501085
18	Diggles B K , Lester R J G . Influence of temperature and host species on the development of cryptocaryon irritans. J Parasitol, 1996, 82 (1): 45- 51 doi: 10.2307/3284114
19	Wang K , Wang W , Pang H , et al. Complex dynamic behavior in a viral model with delayed immune response. Physica D, 2007, 226 (2): 197- 208 doi: 10.1016/j.physd.2006.12.001
20	Xie Q , Huang D , Zhang S , et al. Analysis of a viral infection model with delayed immune response. Appl Math Model, 2010, 34 (9): 2388- 2395 doi: 10.1016/j.apm.2009.11.005
21	Niu B , Guo Y X , Du Y F . Hopf bifurcation induced by delay effect in a diffusive tumor-immune system. Int J Bifurcat Chaos, 2018, 28 (11): 1850136 doi: 10.1142/S0218127418501365
22	Canabarro A A , Gléria I M , Lyra M L . Periodic solutions and chaos in a nonlinear model for the delayed cellular immune response. Physica A, 2004, 342 (1/2): 234- 241
23	Wu J H . Theory and Applications of Partial Functional Differential Equations. New York: Springer-Verlag, 1996
24	Jiang W , Wang H , Cao X . Turing instability and Turing-Hopf bifurcation in diffusive Schnakenberg systems with gene expression time delay. J Dynam Differential Equations, 2019, 31 (4): 2223- 2247 doi: 10.1007/s10884-018-9702-y
25	Wang W , Liu Q X , Jin Z . Spatiotemporal complexity of a ratio-dependent predator-prey system. Phys Rev E, 2007, 75 (5): 051913 doi: 10.1103/PhysRevE.75.051913
26	Baurmann M , Gross T , Feudel U . Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations. J Theor Biol, 2007, 245 (2): 220- 229 doi: 10.1016/j.jtbi.2006.09.036
27	Song Y , Jiang H , Liu Q X , et al. Spatiotemporal dynamics of the diffusive Mussel-Algae model near Turing-Hopf bifurcation. SIAM J on Appl Dyn Syst, 2017, 16 (4): 2030- 2062 doi: 10.1137/16M1097560
28	Garvie M R . Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB. B Math Biol, 2007, 69 (3): 931- 956 doi: 10.1007/s11538-006-9062-3

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