具有接种项且考虑医院病床数的SVIS模型的性态分析
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摘要
建立具有接种项且考虑医院病床数的SVIS模型,并对其动力学性态进行了分析。发现:基本再生数R_0是疫苗接种率φ的函数,并且当传染率较大或者病床数目较小时,系统会出现后向分支,即当R_0小于1时,系统会出现两个正平衡点或者无正平衡点;当系统存在两个正平衡点时,其中染病者数量较小的是鞍点,染病者数量较大的为非鞍点。当R_0小于1时,通过增加病床数和减少疾病的传染率,可以消除疾病。
An SVIS model with vaccination and the impact of the number of hospital beds is established,its dynamic behavior is studied. For the model,it is found that the basic reproduction number R_0 is a function of the vaccination φ,and the system undergoes backward bifurcation if the incidence rate is big or the number of beds is small,i. e.,when R_0< 1 there exist two positive equilibria or none in the model. If there exist two positive equilibria,then the lower infected one is a hyperbolic saddle,and the higher one is an anti-saddle. The epidemic can be eliminated by increasing the number of beds or decreasing the value of incidence rate when R_0< 1.
An SVIS model with vaccination and the impact of the number of hospital beds is established,its dynamic behavior is studied. For the model,it is found that the basic reproduction number R_0 is a function of the vaccination φ,and the system undergoes backward bifurcation if the incidence rate is big or the number of beds is small,i. e.,when R_0< 1 there exist two positive equilibria or none in the model. If there exist two positive equilibria,then the lower infected one is a hyperbolic saddle,and the higher one is an anti-saddle. The epidemic can be eliminated by increasing the number of beds or decreasing the value of incidence rate when R_0< 1.
引文
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[10]SHAN C,ZHU H.Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds[J].Journal of Differential Equations,2014,257(5):1662-1688.
[2]HETHCOTE H W.The mathematics of infectious diseases[J].SIAM Review,2000,42(4):599-653.
[3]KRIBS-ZALETA C M,VELASCO-HERNNDEZ J X.A simple vaccination model with multiple endemic states[J].Mathematical Biosciences,2000,164(2):183-201.
[4]ARINO J,MCCLUSKEY C,VAN DEN DRIESSCHE P.Global results for an epidemic model with vaccination that exhibits backward bifurcation[J].SIAM Journal on Applied Mathematics,2003,64(1):260-276.
[5]ALEXANDER M E,MOGHADAS S M.Periodicity in an epidemic model with a generalized non-linear incidence[J].Mathematical Biosciences,2004,189(1):75-96.
[6]LIU X N,TAKEUCHI Y,IWAMI S.SVIR epidemic models with vaccination strategies[J].Journal of Theoretical Biology,2008,253(1):1-11.
[7]XIAO Y N,TANG S Y.Dynamics of infection with nonlinear incidence in a simple vaccination model[J].Nonlinear Analysis:Real World Applications,2010,11(5):4154-4163.
[8]MAGPANTAY F M G,RIOLO M A,DE CELLES M D,et al,Epidemiological consequences of imperfect vaccines for immunizing infections[J].SIAM Journal on Applied Mathematics,2014,74(6):1810-1830.
[9]VAN DEN DRIESSCHE P,WATMOUGH J.Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission[J].Mathematical Biosciences,2002,180(1-2):29-48.
[10]SHAN C,ZHU H.Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds[J].Journal of Differential Equations,2014,257(5):1662-1688.