一类具有阶段结构和logistic输入的传染病模型的稳定性
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摘要
研究了一类具有阶段结构和logistic输入的SIR传染病模型.将种群分为成年、幼年,并且假定只有成年个体可以染病.通过Hurtwiz判据、Bendixson-Dulac判别法及构造恰当的Lyapunov函数,获得了疾病的无病平衡点和地方病平衡点的局部渐近稳定性和全局渐近稳定性.研究表明:当基本再生数R0<1且满足一定的条件时,疾病将被消除;当基本再生数R0>1时,疾病持续流行并将成为一种地方病.
In this paper,a class of SIR epidemic model with stage structure and logistic input was studied.The population was divided into adult,youth,and assume only adult individuals can be infected.Through Hurtwiz criterion,Bendixson-Dulac criterion and its constructing suitable Lyapunov function,the disease of the disease-free equilibrium and the endemic equilibrium of locally asymptotic stability and global asymptotic stability.The results showed that when the basic rep roductive number R0<1,the disease will be eliminated;When the basic regeneration number R0>1,the disease continues to spread and will become endemic.
In this paper,a class of SIR epidemic model with stage structure and logistic input was studied.The population was divided into adult,youth,and assume only adult individuals can be infected.Through Hurtwiz criterion,Bendixson-Dulac criterion and its constructing suitable Lyapunov function,the disease of the disease-free equilibrium and the endemic equilibrium of locally asymptotic stability and global asymptotic stability.The results showed that when the basic rep roductive number R0<1,the disease will be eliminated;When the basic regeneration number R0>1,the disease continues to spread and will become endemic.
引文
[1]刘芳,胡志兴.一类具有阶段结构的SIS传染病模型的稳定性[J].科学技术与工程,2008,8(2):1671-1819.
[2]金瑜,张勇,王稳地.一类具有阶段结构的传染病模型[J].西南师范大学学报(自然科学版),2003,28(6):863-868.
[3]程晓云,胡志兴.一类具有非线性传染率的阶段结构传染病模型[J].数学的实践与认识,2009,39(7):14-23.
[4]程晓云,胡志兴.一类具有一般发生率的阶段结构传染病模型的稳定性[J].大学数学,2016,32(3):118-123.
[5]吴红良,薛亚奎.一类具有logistic增长的SIR时滞传染病模型的稳定性分析[J].数学的实践与认识,2018,48(7):286-292.
[6]徐为坚.具有种群logistic增长及饱和传染率的SIS模型的稳定性和Hopf分支[J].数学物理学报,2008,28A(3):578-584.
[7]傅金波,陈兰荪,程荣福.具有logistic增长和治疗的SIRS传染病模型的后向分支[J].吉林大学学报(自然科学版),2015,53(6):1166-1170.
[8]程晓云,胡志兴.一类具有一般发生率的阶段结构传染病模型的稳定性[J].大学数学,2016,32(3):118-123.
[9]马知恩,周义仓,李承治.常微分方程定性与稳定性方法[M].北京:科学出版社,2016.
[10]马知恩,周义仓.传染病动力学的数学建模与研究[M].北京:科学出版社,2004.
[11]LIU S Q,CHEN L S.Recent progress on stage-structed population dynamics[J].Mathematical and Computer Modelling,2002,36(3):1319-1360.
[2]金瑜,张勇,王稳地.一类具有阶段结构的传染病模型[J].西南师范大学学报(自然科学版),2003,28(6):863-868.
[3]程晓云,胡志兴.一类具有非线性传染率的阶段结构传染病模型[J].数学的实践与认识,2009,39(7):14-23.
[4]程晓云,胡志兴.一类具有一般发生率的阶段结构传染病模型的稳定性[J].大学数学,2016,32(3):118-123.
[5]吴红良,薛亚奎.一类具有logistic增长的SIR时滞传染病模型的稳定性分析[J].数学的实践与认识,2018,48(7):286-292.
[6]徐为坚.具有种群logistic增长及饱和传染率的SIS模型的稳定性和Hopf分支[J].数学物理学报,2008,28A(3):578-584.
[7]傅金波,陈兰荪,程荣福.具有logistic增长和治疗的SIRS传染病模型的后向分支[J].吉林大学学报(自然科学版),2015,53(6):1166-1170.
[8]程晓云,胡志兴.一类具有一般发生率的阶段结构传染病模型的稳定性[J].大学数学,2016,32(3):118-123.
[9]马知恩,周义仓,李承治.常微分方程定性与稳定性方法[M].北京:科学出版社,2016.
[10]马知恩,周义仓.传染病动力学的数学建模与研究[M].北京:科学出版社,2004.
[11]LIU S Q,CHEN L S.Recent progress on stage-structed population dynamics[J].Mathematical and Computer Modelling,2002,36(3):1319-1360.