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Limit theorems for contact processes with cooperative mechanisms on homogeneous trees and complete graphs

  • School of Science, Beijing Jiaotong University, Beijing 100044, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2020.06.011
  • Received Date: 01 March 2020
  • Accepted Date: 08 July 2020
  • Rev Recd Date: 08 July 2020
  • Publish Date: 30 June 2020
    Abstract

  • Abstract

    Several limit theorems for contact processes with cooperation mechanisms were given from the case where the base maps are homogeneous trees and complete graphs. First, the solution of the limit equation of the contact process with the cooperation mechanism was obtained under the homogeneous tree at a given point and a given time. Next, by means of the idea of the fixed point of the differential equation, the critical value of the cooperation mechanism parameter β was obtained. Then, changing the base map to the complete map, the contact process with the cooperation mechanism was studied, and the density of diseased points at a given moment was obtained when the dimensionality tended to infinity. Finally, as a special case of the contact process of the mechanism, the contact process under the classic mechanism (that is β=0) was reviewed, and the limit function of the number of diseased points was derived.

    Abstract

    Several limit theorems for contact processes with cooperation mechanisms were given from the case where the base maps are homogeneous trees and complete graphs. First, the solution of the limit equation of the contact process with the cooperation mechanism was obtained under the homogeneous tree at a given point and a given time. Next, by means of the idea of the fixed point of the differential equation, the critical value of the cooperation mechanism parameter β was obtained. Then, changing the base map to the complete map, the contact process with the cooperation mechanism was studied, and the density of diseased points at a given moment was obtained when the dimensionality tended to infinity. Finally, as a special case of the contact process of the mechanism, the contact process under the classic mechanism (that is β=0) was reviewed, and the limit function of the number of diseased points was derived.
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    • References(24)
  • [1]
    HARRIS T E. Contact interactions on a lattice[J]. Ann Probab, 1974, 2(6): 969-988.
    [2]
    LIGGETT T M. Interacting Particle Systems[M]. New York: Springer,1985.
    [3]
    LIGGETT T M. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes[M]. New York: Springer,1999.
    [4]
    SUDBURY A. Rigorous lower bounds for the critical infection rate in the diffusive contact process[J]. J Appl Probab, 2001, 38(4): 1074-1078.
    [5]
    纪瑞瑞. 一维单边接触过程性质的研究[D]. 芜湖：安徽师范大学,2010.
    [6]
    李建全, 娄洁, 娄梅枝. 离散的SI 和SIS 传染病模型的研究[J]. 应用数学和力学,2008, 29(1):104-110.
    LI Jianquan, LOU Jie,LOU Meizhi.Study of some discrete SI and SIS epidemic models[J].Applied Mathematics and Mechanics,2008, 29(1):104-110.
    [7]
    PEMANTLE R. The contact process on trees[J]. Ann Probab, 1992, 20(4): 2089-2116.
    [8]
    ZHANG Y. The complete convergence theorem of the contact process on trees[J]. The Annals of Probability, 1996, 24: 1408-1443.
    [9]
    SALZANO M,SCHONMANN R H. A new proof that for the contact process on homogeneous trees local survival implies complete convergence[J]. The Annals of Probability, 1998, 26: 1251-1258.
    [10]
    曹宇. 传染病动力学模型研究[D]. 沈阳：东北大学,2014.
    [11]
    XUE X. An improved upper bound for the critical value of the contact process on Zd with d≥3[J]. Electron Commun Probab, 2018,23: No.77.
    [12]
    CRANSTON M, MOUNTFORD T,MOURRAT J C, et al. The contact process on finite homogeneous trees revisited[J]. ALEA Lat Am J Probab Math Stat, 2014, 11: 385-408.
    [13]
    张刚强. 关于基本接触过程临界值的新估计[J]. 华中理工大学学报,1999,27(6):94-96.
    ZHANG Gangqiang. Estimation of the critical value in the basic contact process[J]. J Huazhong Univ of Sci & Tech,1999,27(6):94-96.
    [14]
    丁万鼎, 朱作宾. 基本接触过程临界值的新估计[J]. 安徽师范大学学报(自然科学版),1984(1):3-8.
    [15]
    AIZENMAN M, JUNG P. On the critical behavior at the lower phase transition of the contact process[J]. ALEA Lat Am J Probab Math Stat, 2007, 3: 301-320.
    [16]
    PETERSON J. The contact process on the complete graph with random vertex-dependent infection rates[J]. Stochastic Processes and Their Applications, 2011, 121: 609-629.
    [17]
    ARMBRUSTER B, BECK E. Elementary proof of convergence to the mean-field model for the SIR process[J]. J Math Biol，2017，75：327-339.
    [18]
    王高雄, 周之铭，朱思铭,等. 常微分方程[M]. 第3 版. 北京: 高等教育出版社, 2006: 219.
    [19]
    PASTOR-SATORRAS R, VESPIGNANI A. Epidemic spreading in scale-free networks[J]. Physical Review Letters, 2001, 86: 3200-3203.
    [20]
    PASTOR-SATORRAS R, VESPIGNANI A. Epidemic dynamics and endemic states in complex networks[J]. Physical Review E, 2001, 63: 066117.
    [21]
    李彦. 复杂网络上的相变问题研究[D]. 上海：上海交通大学,2014.
    [22]
    XUE X. Priority of the result in “Mean field limit for survival probability of the high-dimensional contact process”[J]. Statist Probab Lett, 2019, 148: 133.
    [23]
    KURTZ T G. Strong approximation theorems for density dependent Markov chains[J]. Stochastic Processes and Their Applications, 1978, 6: 223-240.
    [24]
    张恭庆, 郭懋正. 泛函分析讲义(下册)[M]. 北京: 北京大学出版社, 1990: 45-130.)
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    [1]
    HARRIS T E. Contact interactions on a lattice[J]. Ann Probab, 1974, 2(6): 969-988.
    [2]
    LIGGETT T M. Interacting Particle Systems[M]. New York: Springer,1985.
    [3]
    LIGGETT T M. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes[M]. New York: Springer,1999.
    [4]
    SUDBURY A. Rigorous lower bounds for the critical infection rate in the diffusive contact process[J]. J Appl Probab, 2001, 38(4): 1074-1078.
    [5]
    纪瑞瑞. 一维单边接触过程性质的研究[D]. 芜湖：安徽师范大学,2010.
    [6]
    李建全, 娄洁, 娄梅枝. 离散的SI 和SIS 传染病模型的研究[J]. 应用数学和力学,2008, 29(1):104-110.
    LI Jianquan, LOU Jie,LOU Meizhi.Study of some discrete SI and SIS epidemic models[J].Applied Mathematics and Mechanics,2008, 29(1):104-110.
    [7]
    PEMANTLE R. The contact process on trees[J]. Ann Probab, 1992, 20(4): 2089-2116.
    [8]
    ZHANG Y. The complete convergence theorem of the contact process on trees[J]. The Annals of Probability, 1996, 24: 1408-1443.
    [9]
    SALZANO M,SCHONMANN R H. A new proof that for the contact process on homogeneous trees local survival implies complete convergence[J]. The Annals of Probability, 1998, 26: 1251-1258.
    [10]
    曹宇. 传染病动力学模型研究[D]. 沈阳：东北大学,2014.
    [11]
    XUE X. An improved upper bound for the critical value of the contact process on Zd with d≥3[J]. Electron Commun Probab, 2018,23: No.77.
    [12]
    CRANSTON M, MOUNTFORD T,MOURRAT J C, et al. The contact process on finite homogeneous trees revisited[J]. ALEA Lat Am J Probab Math Stat, 2014, 11: 385-408.
    [13]
    张刚强. 关于基本接触过程临界值的新估计[J]. 华中理工大学学报,1999,27(6):94-96.
    ZHANG Gangqiang. Estimation of the critical value in the basic contact process[J]. J Huazhong Univ of Sci & Tech,1999,27(6):94-96.
    [14]
    丁万鼎, 朱作宾. 基本接触过程临界值的新估计[J]. 安徽师范大学学报(自然科学版),1984(1):3-8.
    [15]
    AIZENMAN M, JUNG P. On the critical behavior at the lower phase transition of the contact process[J]. ALEA Lat Am J Probab Math Stat, 2007, 3: 301-320.
    [16]
    PETERSON J. The contact process on the complete graph with random vertex-dependent infection rates[J]. Stochastic Processes and Their Applications, 2011, 121: 609-629.
    [17]
    ARMBRUSTER B, BECK E. Elementary proof of convergence to the mean-field model for the SIR process[J]. J Math Biol，2017，75：327-339.
    [18]
    王高雄, 周之铭，朱思铭,等. 常微分方程[M]. 第3 版. 北京: 高等教育出版社, 2006: 219.
    [19]
    PASTOR-SATORRAS R, VESPIGNANI A. Epidemic spreading in scale-free networks[J]. Physical Review Letters, 2001, 86: 3200-3203.
    [20]
    PASTOR-SATORRAS R, VESPIGNANI A. Epidemic dynamics and endemic states in complex networks[J]. Physical Review E, 2001, 63: 066117.
    [21]
    李彦. 复杂网络上的相变问题研究[D]. 上海：上海交通大学,2014.
    [22]
    XUE X. Priority of the result in “Mean field limit for survival probability of the high-dimensional contact process”[J]. Statist Probab Lett, 2019, 148: 133.
    [23]
    KURTZ T G. Strong approximation theorems for density dependent Markov chains[J]. Stochastic Processes and Their Applications, 1978, 6: 223-240.
    [24]
    张恭庆, 郭懋正. 泛函分析讲义(下册)[M]. 北京: 北京大学出版社, 1990: 45-130.)
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