一类具有阶段结构的传染病模型的全局分析
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摘要
考虑疾病仅在成年个体间传播,并且成年个体的增长受到密度制约,本文建立了一类具有双线性发生率和阶段结构的传染病模型.文中得到了种群增长的基本再生数和疾病传播的基本再生数,通过构造Lyapunov函数证明了平衡点的全局稳定性,通过数值模拟验证了所获得的结果.结果显示,两类基本再生数完全确定了模型的动力学性态,通过降低传染率和增大染病者移除率可以降低疾病基本再生数.
An epidemic model with bilinear incidence and stage structure is established by assuming that the infection can transmit only among adult individuals and that the growth of adult individuals is density-dependent. The basic reproduction numbers of the population growth and the infection transmission are both obtained, the global stability of equilibriums is established by constructing Lyapunov functions, and the theoretical results are verified through numerical simulation. In conclusion, the dynamic behaviors of this model is completely determined by its two basic reproduction numbers, and the basic reproduction numbers of infection transmission can be depressed through decreasing the incidence or increasing the remove rate.
An epidemic model with bilinear incidence and stage structure is established by assuming that the infection can transmit only among adult individuals and that the growth of adult individuals is density-dependent. The basic reproduction numbers of the population growth and the infection transmission are both obtained, the global stability of equilibriums is established by constructing Lyapunov functions, and the theoretical results are verified through numerical simulation. In conclusion, the dynamic behaviors of this model is completely determined by its two basic reproduction numbers, and the basic reproduction numbers of infection transmission can be depressed through decreasing the incidence or increasing the remove rate.
引文
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[13] Saito Y. Effects of cannibalism on a basic stage structure[J]. Applied Mathematics&Computation, 2010,217(5):2133-2141
[14] Kostova T, Li J, Friedman M. Two models for competition between age classes[J]. Mathematical Biosciences,1999, 157(1-2):65-89
[15] Baer S M, Kooi B W. Multiparametric bifurcation analysis of a basic two-stage population model[J]. SIAM Journal on Applied Mathematics, 2006, 66(4):1339-1365
[16] Lyapunov A M. The general problem of the stability of motion[J]. International Journal of Control, 1992,31(3):353-354
[17] LaSalle J P. The Stability of Dynamical Systems[M]. New Jersey:Hamilton Press, 1976
[2]马知恩,周义仓,王稳地,等.传染病动力学的数学建模与研究[M].北京:科学出版社,2004Ma Z E, Zhou Y C, Wang W D, et al. Mathematical Modeling and Dynamics of Infectious Disease[M].Beijing:Science Press, 2004
[3] Xiao Y N, Chen L S. An SIS epidemic model with stage structure and a delay[J]. Acta Mathematicae Applicatae Sinica, 2002, 18(4):607-618
[4]石瑞青,陈兰荪.具有阶段结构和时滞的幼年染病单种群模型研究[J].大连理工大学学报,2010, 50(2):304-308Shi R Q, Chen L S. Delayed stage-structured single-species model with disease in infant[J]. Journal of Dalian University of Technology, 2010, 50(2):304-308
[5] Liu L, Li X, Zhuang K. Bifurcation analysis on a delayed SIS epidemic model with stage structure[J].Electronic Journal of Differential Equations, 2007, 2007(77):249-266
[6] Jia J W, Li Q Y. Qualitative analysis of an SIR epidemic model with stage structure[J]. Applied Mathematics&Computation, 2007, 193(1):106-115
[7] Teng Z D, Nie L F, Xu J B. Dynamical behaviors of a discrete SIS epidemic model with standard incidence and stage structure[J]. Advances in Difference Equations, 2013, 2013(1):1-23
[8]原存德,胡宝安.具有阶段结构的SI传染病模型[J].应用数学学报,2002, 25(2):193-203Yuan C D, Hu B A. A SI epidemic model with two-stage structure[J]. Acta Mathematicae Applicatae Sinica, 2002, 25(2):193-203
[9] Wu C F, Weng P X. Stability analysis of a SIS model with stage structured and distributed maturation delay[J]. Nonlinear Analysis:Theory Methods&Application, 2009, 71(12):e892-e901
[10] Zhang F F, Jin Z, Sun G Q. Bifurcation analysis of a delayed epidemic model[J]. Applied Mathematics&Computation, 2010, 216(3):753-767
[11] Zhang T L, Liu J L, Teng Z D. Stability of Hopf bifurcation of a delayed SIRS epidemic model with stage structure[J]. Nonlinear Analysis:Real World Application, 2010, 11(1):293-306
[12] Zhang T L, Liu J L, Teng Z D. Bifurcation analysis of a delayed SIS epidemic model with stage structure[J].Chaos Solitons&Fractals, 2009, 40(2):563-576
[13] Saito Y. Effects of cannibalism on a basic stage structure[J]. Applied Mathematics&Computation, 2010,217(5):2133-2141
[14] Kostova T, Li J, Friedman M. Two models for competition between age classes[J]. Mathematical Biosciences,1999, 157(1-2):65-89
[15] Baer S M, Kooi B W. Multiparametric bifurcation analysis of a basic two-stage population model[J]. SIAM Journal on Applied Mathematics, 2006, 66(4):1339-1365
[16] Lyapunov A M. The general problem of the stability of motion[J]. International Journal of Control, 1992,31(3):353-354
[17] LaSalle J P. The Stability of Dynamical Systems[M]. New Jersey:Hamilton Press, 1976