具有饱和治疗率的SIR传染病模型的后向分支
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摘要
传染病模型的渐近行为已经被很多人研究。通常情况下,基本再生数是决定疾病流行与否的阈值。如果它小于1,无病平衡点是全局稳定的且疾病灭绝;如果它大于1,正平衡点是全局稳定的且发展为地方病。在这种情况下,从无病平衡点到正平衡点引起的分支是向前的。近年来,由于社群具有不同的感染性、非线性发生率和年龄结构等原因,许多关于传染病模型的论文发现了后向分支。在这种情况下,基本再生数不能完全描述疾病消除的效应,而这种效应能被转向点的关键参数描述,得到控制疾病的阈值对于确认后向分支是重要的。基于此,本文中我们研究了具有饱和治疗函数的传染病模型,通过数学分析和数值模拟主要得到以下结论:
1.当感染者治疗延滞的效应弱时,基本再生数是控制疾病的强阈值。当感染者治疗延滞的效应强时,后向分支将发生,对于消除疾病来说基本再生数小于1是不足的。
2.当后向分支发生时,转向点关键值是控制疾病的新阈值。
3.当基本再生数减少到一定程度时,无病平衡点是全局稳定的。
4.数学结果表明给病人及时的治疗、提高治疗率,和减少传染的协同因素对控制疾病是有效的。
最后,我们结合前人的一些工作,提出了今后努力的方向。
The asymptotic behavior of epidemic models has been studied by many researchers. It is common that a basic reproduction number is a threshold which determines the outcome of the disease. If it is below 1, the disease-free equilibrium is globally stable and the disease dies out. while if it is greater than 1, a unique endemic equilibrium is globally stable. So the bifurcation leading from a disease free equilibrium to an endemic equilibrium is forward. In recent years, papers found backward bifurcations due to social groups with different susceptibilities, nonlinear incidences, and age structures in epidemic models and so on. In this case, the basic reproduction number does not describe the necessary elimination effort; rather the effort is described by the value of the critical parameter at the turning point. Thus, it is important to identify backward bifurcations to obtain thresholds for the control of diseases. In this paper, we studied an epidemic model with saturated treatment function. Through mathematical analysis and numerical simulation, we obtain the following main results:
1 .When the effect of the infected being delayed for treatment is weak, the basic reproduction number being unity is a strict threshold for the control of the disease. However, when this delayed effect is strong, a backward bifurcation will take place. Thus it is not enough for us to drive the basic reproduction number below one to eradicate the disease.
2.When the backward bifurcation takes place, there is a critical value R~c at the turning point which can be taken as a new threshold for the control of the disease.
3.The disease-free equilibrium is globally stable if the basic reproduction number reduces further to some degree.
4. Mathematical results in this paper suggest that giving the patients timely treatment, improving the cure efficiency and decreasing the infective coefficient are all valid methods for the control of disease.
Finally, based on the works of early researchers, we suggest our effort in the future.
1.当感染者治疗延滞的效应弱时,基本再生数是控制疾病的强阈值。当感染者治疗延滞的效应强时,后向分支将发生,对于消除疾病来说基本再生数小于1是不足的。
2.当后向分支发生时,转向点关键值是控制疾病的新阈值。
3.当基本再生数减少到一定程度时,无病平衡点是全局稳定的。
4.数学结果表明给病人及时的治疗、提高治疗率,和减少传染的协同因素对控制疾病是有效的。
最后,我们结合前人的一些工作,提出了今后努力的方向。
The asymptotic behavior of epidemic models has been studied by many researchers. It is common that a basic reproduction number is a threshold which determines the outcome of the disease. If it is below 1, the disease-free equilibrium is globally stable and the disease dies out. while if it is greater than 1, a unique endemic equilibrium is globally stable. So the bifurcation leading from a disease free equilibrium to an endemic equilibrium is forward. In recent years, papers found backward bifurcations due to social groups with different susceptibilities, nonlinear incidences, and age structures in epidemic models and so on. In this case, the basic reproduction number does not describe the necessary elimination effort; rather the effort is described by the value of the critical parameter at the turning point. Thus, it is important to identify backward bifurcations to obtain thresholds for the control of diseases. In this paper, we studied an epidemic model with saturated treatment function. Through mathematical analysis and numerical simulation, we obtain the following main results:
1 .When the effect of the infected being delayed for treatment is weak, the basic reproduction number being unity is a strict threshold for the control of the disease. However, when this delayed effect is strong, a backward bifurcation will take place. Thus it is not enough for us to drive the basic reproduction number below one to eradicate the disease.
2.When the backward bifurcation takes place, there is a critical value R~c at the turning point which can be taken as a new threshold for the control of the disease.
3.The disease-free equilibrium is globally stable if the basic reproduction number reduces further to some degree.
4. Mathematical results in this paper suggest that giving the patients timely treatment, improving the cure efficiency and decreasing the infective coefficient are all valid methods for the control of disease.
Finally, based on the works of early researchers, we suggest our effort in the future.
引文
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[3]Anderson, R.M., and May, R.M., Infectious disease of humans, dynamics and control.Oxford University Press, Oxford, 1991.
[4]Anderson, R.M., and May,R.M.(Eds.), Population Biology of Infectious Diseases,Springer-Verlag, Berlin, 1982.
[5]Arino, J., McCluskey, C.C., and Driessche, P.van den., Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J.Appl.Math.2003,64, 260 - 276.
[6]Capasso, V., and Serio, G., A generalization of the Kermack - Mckendrick deterministic epidemic model, Math.Biosci.1978,42,43 - 61.
[7]Castillo-Chavez, C, and Thieme,H.R., Asymptotically autonomous epidemic models, in: O.Arino, et al.(Eds.), Mathematical Population Dynamics: Analysis of Heterogeneity, in: Theory of Epidemics, vol.I, Wuerz, Canada, 1995, pp.33 - 50.
[8]Diekmann,0., and Heesterbeek, J.A.P., Mathematical Epidemiology of Infectious Diseases.Model Building, Analysis and Interpretation, Wiley Ser.Math.Comput.Biol., John Wiley and Sons, Chichester, 2000.
[9]Diekman O, Heesterbeek J A P, Metz J A J, The legacy of Kermack and Mckendrick,in: Mollision(Ed.), Epidemic Models, Their Structure and Relation to Date [M].Cambridge, Cambridge University Press, 1994.
[10]Driessche,P.van den and WatmoughJ., A simple SIS epidemic model with a backward bifurcation, J.Math.Biol.2000,40, 525 - 540.
[11]Dushoff,J., Huang, W., and Castillo-Chavez, C, Backwards bifurcations and catas- trophe in simple models of fatal diseases, J.Math.Biol.1998,36,227 - 248.
[12]Greenhalgh,D.,Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity,Math.ComputModelling.1997,25,85-107.
[13]Hadeler,K.R, and Driessche, P.van den, Backward bifurcation in epidemic control,Math.Biosci.1997,146,15 - 35.
[14]Hethcote ,H.W., The mathematics of infectious disease [J].SIAM Review, 2000,42(2): 599-653.
[15]Hui,J., and Zhu, D.M., Global Stability and Periodicity on SIS Epidemic Models with Backward Bifurcation,Compu.Math.Appl.2005,50,1271-1290.
[16]Kribs-Zaleta,C.M., and Velasco-Hernandez,J.X., A simple vaccination model with multiple endemic states,Math.Bios.2000,164,183-201.
[17]Li,J.Q.,Ma.Z.E., and Zhou,Y.C.,Global analysis of SIS epidemic model with a simple vaccination and multiple endemic equilibria,Acta.Math.Scientia.2006,26,83-93.
[18]Li,X.Z.,Li,W.S., and Ghosh.M.N., Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment,Appl.Math.Compu.2009,210,141-150.
[19]Liu,L., Li,X.G., and Zhuang,K.J., Bifurcation analysis on a delayed SIS epidemic model with stage structure,J.Diff.Equa.2007,77,l-17.
[20]Liu,W.M., Levin, S.A., and Iwasa, Y., Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J.Math.Biol.1986,23 187 - 204.
[21]Lizana, M., and Rivero, J.Multiparametric bifurcations for a model in epidemiology, J.Math.Biol.1996,35, 21 - 36.
[22]Martcheva,M., and Thieme, H.R., Progression age enhanced backward bifurcation in an epidemic model with superinfection, J.Math.Biol.2003,46,385 - 424.
[23]Ruan,S.,and Wang,W.,Dynamical behavior of an epidemic model with a nonlinear incidence rate,J.Differential Equations 2003,188,135 - 163.
[24]Takeuchi,Y.,Liu,X.,and Cui,J.,Global dynamics of SIS models with transportrelated infection,J.Math.Anal.Appl.2007,329,1460 - 1471.
[25]Takeuchi,Y.,Ma,W.,and Beretta,E.,Global asymptotic properties of a delay SIR epidemic model with finite incubation times,Nonlinear Anal.2000,42,931 - 947.
[26]Wang,W.,and Ma,Z.,Global dynamics of an epidemic model with time delay,Nonlinear Anal.Real World Appl.2002,3,365 - 373.
[27]Xiao,D.,and Ruan,S.,Global analysis of an epidemic model with nonmonotone incidence rate,Math.Biosci.2007,208,419 - 429.
[28]Timothy,C.R.,Jan.M,and Alan,S.P.,Backward bifurcations and multiple equilibria in epidemic models with structured immunity,J.Thero.Biol.2008,252,155-165.
[29]Wang,W.,Backward bifurcation of an epidemic model with treatment,Math.Biosci.2006,201,58 - 71.
[30]Wang,W.,and Ruan,S.,Bifurcation in an epidemic model with constant removal rate of the infectives,J.Math.Anal.Appl.2004,291,775 - 793.
[31]Wu,L.,and Feng,Z.,Homoclinic bifurcation in an SIQR model for childhood diseases,J.Differential Equations 2000,168,150 - 167.
[32]Xu,Z.X.,liu,S,and Wang,H.,Backward bifurcation of an epidemic model with standard incidence rate and treatment rate,Nonlinear Anal.RWA.2008,9,2302 - 2312.
[33]Zhang,X.,and Liu,X.,Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment,Nonlinear Anal.RWA.2009,10,565-575.
[34]Zhang,X.,and Liu,X.N.,Backward bifurcation of an epidemic model with saturated treatment function,J.Math.Anal.Appl.2008,348,433 - 443.
[2]Anderson, R.M., and May, R.M., The invasion, persistence, and spread of infectious disease within animal and plant communities.Phi.Trans.R.Soc.London B.1986,314,533-570
[3]Anderson, R.M., and May, R.M., Infectious disease of humans, dynamics and control.Oxford University Press, Oxford, 1991.
[4]Anderson, R.M., and May,R.M.(Eds.), Population Biology of Infectious Diseases,Springer-Verlag, Berlin, 1982.
[5]Arino, J., McCluskey, C.C., and Driessche, P.van den., Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J.Appl.Math.2003,64, 260 - 276.
[6]Capasso, V., and Serio, G., A generalization of the Kermack - Mckendrick deterministic epidemic model, Math.Biosci.1978,42,43 - 61.
[7]Castillo-Chavez, C, and Thieme,H.R., Asymptotically autonomous epidemic models, in: O.Arino, et al.(Eds.), Mathematical Population Dynamics: Analysis of Heterogeneity, in: Theory of Epidemics, vol.I, Wuerz, Canada, 1995, pp.33 - 50.
[8]Diekmann,0., and Heesterbeek, J.A.P., Mathematical Epidemiology of Infectious Diseases.Model Building, Analysis and Interpretation, Wiley Ser.Math.Comput.Biol., John Wiley and Sons, Chichester, 2000.
[9]Diekman O, Heesterbeek J A P, Metz J A J, The legacy of Kermack and Mckendrick,in: Mollision(Ed.), Epidemic Models, Their Structure and Relation to Date [M].Cambridge, Cambridge University Press, 1994.
[10]Driessche,P.van den and WatmoughJ., A simple SIS epidemic model with a backward bifurcation, J.Math.Biol.2000,40, 525 - 540.
[11]Dushoff,J., Huang, W., and Castillo-Chavez, C, Backwards bifurcations and catas- trophe in simple models of fatal diseases, J.Math.Biol.1998,36,227 - 248.
[12]Greenhalgh,D.,Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity,Math.ComputModelling.1997,25,85-107.
[13]Hadeler,K.R, and Driessche, P.van den, Backward bifurcation in epidemic control,Math.Biosci.1997,146,15 - 35.
[14]Hethcote ,H.W., The mathematics of infectious disease [J].SIAM Review, 2000,42(2): 599-653.
[15]Hui,J., and Zhu, D.M., Global Stability and Periodicity on SIS Epidemic Models with Backward Bifurcation,Compu.Math.Appl.2005,50,1271-1290.
[16]Kribs-Zaleta,C.M., and Velasco-Hernandez,J.X., A simple vaccination model with multiple endemic states,Math.Bios.2000,164,183-201.
[17]Li,J.Q.,Ma.Z.E., and Zhou,Y.C.,Global analysis of SIS epidemic model with a simple vaccination and multiple endemic equilibria,Acta.Math.Scientia.2006,26,83-93.
[18]Li,X.Z.,Li,W.S., and Ghosh.M.N., Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment,Appl.Math.Compu.2009,210,141-150.
[19]Liu,L., Li,X.G., and Zhuang,K.J., Bifurcation analysis on a delayed SIS epidemic model with stage structure,J.Diff.Equa.2007,77,l-17.
[20]Liu,W.M., Levin, S.A., and Iwasa, Y., Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J.Math.Biol.1986,23 187 - 204.
[21]Lizana, M., and Rivero, J.Multiparametric bifurcations for a model in epidemiology, J.Math.Biol.1996,35, 21 - 36.
[22]Martcheva,M., and Thieme, H.R., Progression age enhanced backward bifurcation in an epidemic model with superinfection, J.Math.Biol.2003,46,385 - 424.
[23]Ruan,S.,and Wang,W.,Dynamical behavior of an epidemic model with a nonlinear incidence rate,J.Differential Equations 2003,188,135 - 163.
[24]Takeuchi,Y.,Liu,X.,and Cui,J.,Global dynamics of SIS models with transportrelated infection,J.Math.Anal.Appl.2007,329,1460 - 1471.
[25]Takeuchi,Y.,Ma,W.,and Beretta,E.,Global asymptotic properties of a delay SIR epidemic model with finite incubation times,Nonlinear Anal.2000,42,931 - 947.
[26]Wang,W.,and Ma,Z.,Global dynamics of an epidemic model with time delay,Nonlinear Anal.Real World Appl.2002,3,365 - 373.
[27]Xiao,D.,and Ruan,S.,Global analysis of an epidemic model with nonmonotone incidence rate,Math.Biosci.2007,208,419 - 429.
[28]Timothy,C.R.,Jan.M,and Alan,S.P.,Backward bifurcations and multiple equilibria in epidemic models with structured immunity,J.Thero.Biol.2008,252,155-165.
[29]Wang,W.,Backward bifurcation of an epidemic model with treatment,Math.Biosci.2006,201,58 - 71.
[30]Wang,W.,and Ruan,S.,Bifurcation in an epidemic model with constant removal rate of the infectives,J.Math.Anal.Appl.2004,291,775 - 793.
[31]Wu,L.,and Feng,Z.,Homoclinic bifurcation in an SIQR model for childhood diseases,J.Differential Equations 2000,168,150 - 167.
[32]Xu,Z.X.,liu,S,and Wang,H.,Backward bifurcation of an epidemic model with standard incidence rate and treatment rate,Nonlinear Anal.RWA.2008,9,2302 - 2312.
[33]Zhang,X.,and Liu,X.,Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment,Nonlinear Anal.RWA.2009,10,565-575.
[34]Zhang,X.,and Liu,X.N.,Backward bifurcation of an epidemic model with saturated treatment function,J.Math.Anal.Appl.2008,348,433 - 443.