带有二次传染和常数移入的结核病模型的研究
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摘要
在传统的结核病模型中,一般把人群分成四种类型:易感者,潜伏者,染病者,治愈者.随着结核病的发展和人们对结核病的进一步了解,结核病出现了最新的几个特点:二次感染,未治愈比率和抗药菌珠的出现,潜伏者也能传染等情况.本文在传统的结核病模型的基础上,结合结核病的最新的特点,主要研究了新特点下的结核病的两个传染模型;同时,结合结核病的最新的特点,给出了单、双菌珠下的结核病的传染模型.
在第二章中考虑了二次感染,未治愈比率和部分免疫在结核病的传播中起到的重要作用,建立了具有二次传染的未治愈的结核病模型.通过具有二次传染的未治愈的结核病模型的研究,得到了在基本再生数R0 ,并且在R0 < 1,系统存在多地方病平衡点。同时,通过分析表明,未治愈率可能是现今结核病不绝灭的一个很重要的原因.
在第三章中在结核病传播的数学模型的动力学系统基础上,利用Hurwitz判据和复合矩阵理论,讨论了具有常数移入的结核病传播的数学模型的动力学性质,研究了具有常数移入的结核病模型的地方病平衡点的局部和全局渐近稳定性,得到了具有常数移入的结核病模型的地方病平衡点是局部和全局渐近稳定的充分条件.
在第四章中结合结核病的最新的特点,建立了单、双菌珠下的结核病的传染模型.
In traditional Tuberculosis models,the host population is divided into the following four classes: susceptible, exposed, infectious and treated individuals.With Tuberculosis developing and being known about deeply, Tuberculosis has some following new characteristics: the appearing of reinfection, untreaed rate and drug-resistant strains ,and that exposed individuals can also be infectious.Combining with these new characteristics,the paper mainly proposes two Tuberculosis models on the basis of traditional Tuberculosis models.Meanwhile, combined with these new characteristics, Tuberculosis models of one-strain and two-strains are developed.
In chapter 2,with the imoportant influence of reinfection, untreaed rate,and partle immuned rate ,an untread Tuberculosis model with reinfection is developed.By proposing this model,the basic reproductive number R0 is obtained,and when R0 < 1,the multipie endemic equilibria may exists in the system.Meanwhile,untreaed rate is an important factor for today Tuberculosis prevalence.
In chapter 3,on the basis of the dynamical system on a Tuberculosis spread model, the paper mainly discusses the dynamical behavior of a Tuberculosis spread model with constant immigration.The local and global asymptotically stability of the endemic equilibrium on this model is studied. It is proved by using the hurwitz criterion and the compound matrix theory. Sufficient conditions for the local and global asymptotically stability of the endemic equilibrium on this model are obtained.
In chapter 4, combined with these new characteristics, Tuberculosis models of one-strain and two-strains are developed.
在第二章中考虑了二次感染,未治愈比率和部分免疫在结核病的传播中起到的重要作用,建立了具有二次传染的未治愈的结核病模型.通过具有二次传染的未治愈的结核病模型的研究,得到了在基本再生数R0 ,并且在R0 < 1,系统存在多地方病平衡点。同时,通过分析表明,未治愈率可能是现今结核病不绝灭的一个很重要的原因.
在第三章中在结核病传播的数学模型的动力学系统基础上,利用Hurwitz判据和复合矩阵理论,讨论了具有常数移入的结核病传播的数学模型的动力学性质,研究了具有常数移入的结核病模型的地方病平衡点的局部和全局渐近稳定性,得到了具有常数移入的结核病模型的地方病平衡点是局部和全局渐近稳定的充分条件.
在第四章中结合结核病的最新的特点,建立了单、双菌珠下的结核病的传染模型.
In traditional Tuberculosis models,the host population is divided into the following four classes: susceptible, exposed, infectious and treated individuals.With Tuberculosis developing and being known about deeply, Tuberculosis has some following new characteristics: the appearing of reinfection, untreaed rate and drug-resistant strains ,and that exposed individuals can also be infectious.Combining with these new characteristics,the paper mainly proposes two Tuberculosis models on the basis of traditional Tuberculosis models.Meanwhile, combined with these new characteristics, Tuberculosis models of one-strain and two-strains are developed.
In chapter 2,with the imoportant influence of reinfection, untreaed rate,and partle immuned rate ,an untread Tuberculosis model with reinfection is developed.By proposing this model,the basic reproductive number R0 is obtained,and when R0 < 1,the multipie endemic equilibria may exists in the system.Meanwhile,untreaed rate is an important factor for today Tuberculosis prevalence.
In chapter 3,on the basis of the dynamical system on a Tuberculosis spread model, the paper mainly discusses the dynamical behavior of a Tuberculosis spread model with constant immigration.The local and global asymptotically stability of the endemic equilibrium on this model is studied. It is proved by using the hurwitz criterion and the compound matrix theory. Sufficient conditions for the local and global asymptotically stability of the endemic equilibrium on this model are obtained.
In chapter 4, combined with these new characteristics, Tuberculosis models of one-strain and two-strains are developed.
引文
[1] Geneva.Global Tuberculosis Control. WHO report 2001. 2001.
[2] Carlos Castillo-Chavez, Baojun Song. Dynamical models of tuberculosis and their applications.mathematical biosciences and engineering, 2004 (1): 361-404.
[3] Scorpin A,Zhang Y.. Resistance,and mobile elements in mycobacteria. Nature Med, 1994(1):293-298.
[4] Artin C,Ranes M,Giequl B.Plasmids. Antibiotic resistance,and mobile elements in mycobacteria. Moleclar biology of the Mycobacteria.Surrcy University Press. 1990(2):121--138.
[5] Baojun Song ,Carlos Castillo-Chavez,Juan Pablo Aparicio.Tuberculosis models with fast and slow dynamics: the role of close and causal contacts. Mathematical Biosciences. 2002(180):187-205.
[6] ROSE, D. E., ZERBE, G. O., LANTZ, S. O. & BAILEY, W. C. Establishing priority during investigation of tuberculosis contacts, Am. Rev. Respir. Dis.1979(119):603-609.
[7] JUAN P. APARICIO, ANGEL F. CAPURRO, CARLOS. Transmission and Dynamics of Tuberculosis on Generalized Households. J. theor. Biol. 2000(206): 327-341.
[8] C.Connell McCluskey ,P.van den.Driessche, Global Analysis of Two TB models. Journal of Dynamics and Differential Equations. 2004(1).
[9] CASTILLO-CHAVEZ, C. & FENG, Z. Mathematical models for the disease dynamics of tuberculosis. World Scientific.1998.
[10] CASTILLO-CHAVEZ, C.&FENG, Z. To treat or not to treat, the case of tuberculosis. J. Math. Biol. 1997(35):629-659.
[11] Salyers, A. A., Whitt, D. D. Bacterial Pathogenesis. Washington, D.C. 1994:391-396.
[12] CASTILLO-CHAVEZ, C.&FENG, Z.&CAPURRO, A. F, A distributed delay model for tuberculosis. Biometric Unit Technical Report BU-1389-M, Cornell University.1997.
[13] FENG, Z., CASTILLO-CHAVEZ, C. & CAPURRO, A. A model for tuberculosis with exogenuous reinfection, theor. Pop. Biol. 2000(57):235-247.
[14] BLOWER, S. M., SMALL, P. M. & HOPWELL, P. C. Control strategies for tuberculosis epidemics: new models for old problems. Science. 1996(273): 497-500.
[15] Elad Ziv, Charles L. Daley, Sally M. Blower. Early Therapy for Latent Tuberculosis Infection. American Journal of Epidemiology. 2000(153).
[16] S.M.Blower, P.M.Small, P. C. Hopewell, Control Strategies for Tuberculosis Epidemics: New Models For Old Problems. Science. 1996(273).
[17] Christopher Dye, Geoffrey P Garnett, Karen Sleeman, Brian G Williams. Prospects for worldwide tuberculosis control under the WHO DOTS strategy. Tuberculosis,the lancet. 1998(352).
[18] Christopher Dye, Brian G Williams, Criteria for the control of drug-resistant tuberculosis.PNAS. 2000(97):8180-8185.
[19] Christopher Dye, Zhao Fengzeng, Suzanne Scheele, Brian G Williams, Evaluating the impact of tuberculosis control:number of deaths prevented by short-course chemotherapy in China, International Journal of Epidemiology. 2000(29):558-564.
[20] G. Kolata, First documented cases of TB passed on airliner is reported by the U.S., New York Times.1995.
[21] J. Raffalii, K.A. Sepkowitz, D. Armstrong, Community-based outbreaks of tuberculosis primitive matrices, Arch. Int. Med. 156(1996):1053.
[22] E.Lincoln.Epidemics of tuberculosis Matrices,Adv.Tuberc. 1965 (14).
[23] BLOWER, S. M., MCLEAN, A. R., PORCO, T. C., SMALL,P. M., HOPWELL, P. C., SANCHEZ, M. A. & MOSS, A. R. The intrinsic transmission dynamics of tuberculosis epidemics. Nat. 1995(1):815-821.
[24] BLOOM, B.Tuberculosis Pathogenesis and Control, WA, U.S.A.: ASM. 1994.
[25] P. A. Selwyn, D. Hartel, & V. A. Lewis et al. A prospective study of the risk of tuberculosis among intravenous drug users with human immunodeficiency virus infection. N. Engl. J. 1989(320):545-550.
[26] E. Jung, S. Lenhart, Z. Feng, Optimal control of treatments in a two-strain tuberculosis model.DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES. 2002(2):473-482
[27] Carlos Franco-Paredes , Nadine Rouphael , Carlos del Rio, JoseIgnacio Santos-Preciado. strategies to prevent tuberculosis in the new millennium: from BCG to new vaccine candidates, International Journal of Infectious Diseases. 2006(10):93-102.
[28] Castillo-Chavez,C, Feng ,Z.Optimal vaccination strategies for TB in age-structure populations. Submitted Journal of Mathematical Biology. 1996.
[29] Castillo-Chavez,C, , Hethcote, H. W., Andreason, V. Levin, S. A., Liu, W. Epidemiological models with age structure, proportionate mixing, and cross-immunity. J. Math. Biol. 1989(27): 233-258.
[30] Christopher Dye, Brian G. Williams. Criteria for the control of drug-resistant tuberculosis.PNAS.2000.
[31] Sally M. Blower, Angela R. Mclean, Travis C. porco, Peter M. Small, Philip C.Hopewe,Melissa A. Sanchez Andrew R. Moss.The instrinsic transmission dynamics of tuberculosis epidemics. Nature medicine.1995 (1).
[32] Baojun Song , Carlos Castillo-Chavez, Juan Pablo Aparicio. Global Dynamics of TB Models with Density Dependent Demography. IMA.2002 (126):275-294.
[33] Brian,M.Murphy,Benjamin H,Singer schoana Anderson,Denise Kirschner. Comparing epidemic tuberculosis in demographically distinct heterogeneous populations.Mathematical Bioscience. Mathematical Bioscience. 2002(180):161-185.
[34] H.T.Waaler&M.A.Gese&S.Anderson. The use of mathematical models in the study of the epidemiology of tuberculosis. Am.J.Publ.Health.1962 (52):1002-1013.
[35] S.Brogger. Systems analysis in tuberculosis control: a model. Amer.Resp. 1967(95):419-434.
[36] C.S.ReVele,W.R.Lynn,&F.Feldmann. Mathematical models for the economic allocation of tuberculosis control activities in developing nations. Am.Rev.RRRespir. 1967(96):893-909.
[37] C.S.ReVele. The economics allocation of tuberculosis control activities in developing nations. Ph.D.Thesis. Cornell University. 1967.
[38] Michael Y.Li,John R.Graef,Liancheng Wang,Janos Karsai, Global dynamics of a SEIR model with varying total population size. Mathematical Biosciences. 1999(160) :191-213.
[39] Sinder,D.E.,Raviglione,M.,and Kochi,M.,Global burden of tuberculosis, in “Tuberculosis: Pathogenesis,Protection and Control”(B.R.Bloom,Ed.),ASM Press,Washington,DC.(1994).
[40] P.van den Ariessche,James Watmough,Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,Math.Biosci.2002(180):29-48.
[41] A.Berman,RJ.Plemmons,Nonnegative Matrices in the Mathematical Sciences,Academic Press,New York.1970:135-137.
[42] O.Diekmann,J.A.P.Heesterbeek,J.A.J.Metz,On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations,J.Math.Biol. 1990(28):365.
[43] R.M.Anderson,R.M.May.Infectious Diseases of Humans.Oxford University. Oxford.1991:17.
[44] R.A.Horn,C.R.Johnson,Topics in Matrix Analysis,Cambridge University, Cambridge.1991:127.
[45] Castillo-Chavez,C.,and Thieme,R.H.,Asymptotically autonomous epidemic models,in “Proceeding Third International Conference on Mathematical Population Dynamics(O.Arino and M.Kimmd,Eds.),1995:33-50.
[46] F. Brauer and P. Van den Driessche. Models for transmission of disease with immigration of infectives[J], Math Biosci. 2001 (171): 143–154.
[47] G. Li and Z. Jin, Global stability of an SEIR epidemic model with infectious force in latent infected and immune period[J], Chaos, Solitons & Fractals .2005(25): 1177–1184.
[48] M.Y. Li and J.S. Muldowney, A geometric approach to the global-stability problems[J], SIAM J Math Anal. 1996 (27): 1070–1083.
[49] M.Y. Li, Dulac criteria for autonomous systems having an invariant affine manifold[J], J Math Anal Appl 1996(199): 374–390.
[50] M.Y. Li and J.S. Muldowney, On R.A. Smith’s autonomous convergence theorem[J], Rocky Mount J Math. 1995(25): 365–379.
[51] W.A. Coppel, Stability and asymptotic behavior of differential equation[M], Heath,Boston. 1965:41
[52] J.S. Muldowney, Compound matrices and ordinary differential equations[J], Rocky Mount J Math. 1990(20): 857–872.
[53] R.H. Martin Jr., Logarithmic norms and projections applied to linear differential systems[J], J Math Anal Appl 1974(45): 432–454.
[2] Carlos Castillo-Chavez, Baojun Song. Dynamical models of tuberculosis and their applications.mathematical biosciences and engineering, 2004 (1): 361-404.
[3] Scorpin A,Zhang Y.. Resistance,and mobile elements in mycobacteria. Nature Med, 1994(1):293-298.
[4] Artin C,Ranes M,Giequl B.Plasmids. Antibiotic resistance,and mobile elements in mycobacteria. Moleclar biology of the Mycobacteria.Surrcy University Press. 1990(2):121--138.
[5] Baojun Song ,Carlos Castillo-Chavez,Juan Pablo Aparicio.Tuberculosis models with fast and slow dynamics: the role of close and causal contacts. Mathematical Biosciences. 2002(180):187-205.
[6] ROSE, D. E., ZERBE, G. O., LANTZ, S. O. & BAILEY, W. C. Establishing priority during investigation of tuberculosis contacts, Am. Rev. Respir. Dis.1979(119):603-609.
[7] JUAN P. APARICIO, ANGEL F. CAPURRO, CARLOS. Transmission and Dynamics of Tuberculosis on Generalized Households. J. theor. Biol. 2000(206): 327-341.
[8] C.Connell McCluskey ,P.van den.Driessche, Global Analysis of Two TB models. Journal of Dynamics and Differential Equations. 2004(1).
[9] CASTILLO-CHAVEZ, C. & FENG, Z. Mathematical models for the disease dynamics of tuberculosis. World Scientific.1998.
[10] CASTILLO-CHAVEZ, C.&FENG, Z. To treat or not to treat, the case of tuberculosis. J. Math. Biol. 1997(35):629-659.
[11] Salyers, A. A., Whitt, D. D. Bacterial Pathogenesis. Washington, D.C. 1994:391-396.
[12] CASTILLO-CHAVEZ, C.&FENG, Z.&CAPURRO, A. F, A distributed delay model for tuberculosis. Biometric Unit Technical Report BU-1389-M, Cornell University.1997.
[13] FENG, Z., CASTILLO-CHAVEZ, C. & CAPURRO, A. A model for tuberculosis with exogenuous reinfection, theor. Pop. Biol. 2000(57):235-247.
[14] BLOWER, S. M., SMALL, P. M. & HOPWELL, P. C. Control strategies for tuberculosis epidemics: new models for old problems. Science. 1996(273): 497-500.
[15] Elad Ziv, Charles L. Daley, Sally M. Blower. Early Therapy for Latent Tuberculosis Infection. American Journal of Epidemiology. 2000(153).
[16] S.M.Blower, P.M.Small, P. C. Hopewell, Control Strategies for Tuberculosis Epidemics: New Models For Old Problems. Science. 1996(273).
[17] Christopher Dye, Geoffrey P Garnett, Karen Sleeman, Brian G Williams. Prospects for worldwide tuberculosis control under the WHO DOTS strategy. Tuberculosis,the lancet. 1998(352).
[18] Christopher Dye, Brian G Williams, Criteria for the control of drug-resistant tuberculosis.PNAS. 2000(97):8180-8185.
[19] Christopher Dye, Zhao Fengzeng, Suzanne Scheele, Brian G Williams, Evaluating the impact of tuberculosis control:number of deaths prevented by short-course chemotherapy in China, International Journal of Epidemiology. 2000(29):558-564.
[20] G. Kolata, First documented cases of TB passed on airliner is reported by the U.S., New York Times.1995.
[21] J. Raffalii, K.A. Sepkowitz, D. Armstrong, Community-based outbreaks of tuberculosis primitive matrices, Arch. Int. Med. 156(1996):1053.
[22] E.Lincoln.Epidemics of tuberculosis Matrices,Adv.Tuberc. 1965 (14).
[23] BLOWER, S. M., MCLEAN, A. R., PORCO, T. C., SMALL,P. M., HOPWELL, P. C., SANCHEZ, M. A. & MOSS, A. R. The intrinsic transmission dynamics of tuberculosis epidemics. Nat. 1995(1):815-821.
[24] BLOOM, B.Tuberculosis Pathogenesis and Control, WA, U.S.A.: ASM. 1994.
[25] P. A. Selwyn, D. Hartel, & V. A. Lewis et al. A prospective study of the risk of tuberculosis among intravenous drug users with human immunodeficiency virus infection. N. Engl. J. 1989(320):545-550.
[26] E. Jung, S. Lenhart, Z. Feng, Optimal control of treatments in a two-strain tuberculosis model.DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES. 2002(2):473-482
[27] Carlos Franco-Paredes , Nadine Rouphael , Carlos del Rio, JoseIgnacio Santos-Preciado. strategies to prevent tuberculosis in the new millennium: from BCG to new vaccine candidates, International Journal of Infectious Diseases. 2006(10):93-102.
[28] Castillo-Chavez,C, Feng ,Z.Optimal vaccination strategies for TB in age-structure populations. Submitted Journal of Mathematical Biology. 1996.
[29] Castillo-Chavez,C, , Hethcote, H. W., Andreason, V. Levin, S. A., Liu, W. Epidemiological models with age structure, proportionate mixing, and cross-immunity. J. Math. Biol. 1989(27): 233-258.
[30] Christopher Dye, Brian G. Williams. Criteria for the control of drug-resistant tuberculosis.PNAS.2000.
[31] Sally M. Blower, Angela R. Mclean, Travis C. porco, Peter M. Small, Philip C.Hopewe,Melissa A. Sanchez Andrew R. Moss.The instrinsic transmission dynamics of tuberculosis epidemics. Nature medicine.1995 (1).
[32] Baojun Song , Carlos Castillo-Chavez, Juan Pablo Aparicio. Global Dynamics of TB Models with Density Dependent Demography. IMA.2002 (126):275-294.
[33] Brian,M.Murphy,Benjamin H,Singer schoana Anderson,Denise Kirschner. Comparing epidemic tuberculosis in demographically distinct heterogeneous populations.Mathematical Bioscience. Mathematical Bioscience. 2002(180):161-185.
[34] H.T.Waaler&M.A.Gese&S.Anderson. The use of mathematical models in the study of the epidemiology of tuberculosis. Am.J.Publ.Health.1962 (52):1002-1013.
[35] S.Brogger. Systems analysis in tuberculosis control: a model. Amer.Resp. 1967(95):419-434.
[36] C.S.ReVele,W.R.Lynn,&F.Feldmann. Mathematical models for the economic allocation of tuberculosis control activities in developing nations. Am.Rev.RRRespir. 1967(96):893-909.
[37] C.S.ReVele. The economics allocation of tuberculosis control activities in developing nations. Ph.D.Thesis. Cornell University. 1967.
[38] Michael Y.Li,John R.Graef,Liancheng Wang,Janos Karsai, Global dynamics of a SEIR model with varying total population size. Mathematical Biosciences. 1999(160) :191-213.
[39] Sinder,D.E.,Raviglione,M.,and Kochi,M.,Global burden of tuberculosis, in “Tuberculosis: Pathogenesis,Protection and Control”(B.R.Bloom,Ed.),ASM Press,Washington,DC.(1994).
[40] P.van den Ariessche,James Watmough,Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,Math.Biosci.2002(180):29-48.
[41] A.Berman,RJ.Plemmons,Nonnegative Matrices in the Mathematical Sciences,Academic Press,New York.1970:135-137.
[42] O.Diekmann,J.A.P.Heesterbeek,J.A.J.Metz,On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations,J.Math.Biol. 1990(28):365.
[43] R.M.Anderson,R.M.May.Infectious Diseases of Humans.Oxford University. Oxford.1991:17.
[44] R.A.Horn,C.R.Johnson,Topics in Matrix Analysis,Cambridge University, Cambridge.1991:127.
[45] Castillo-Chavez,C.,and Thieme,R.H.,Asymptotically autonomous epidemic models,in “Proceeding Third International Conference on Mathematical Population Dynamics(O.Arino and M.Kimmd,Eds.),1995:33-50.
[46] F. Brauer and P. Van den Driessche. Models for transmission of disease with immigration of infectives[J], Math Biosci. 2001 (171): 143–154.
[47] G. Li and Z. Jin, Global stability of an SEIR epidemic model with infectious force in latent infected and immune period[J], Chaos, Solitons & Fractals .2005(25): 1177–1184.
[48] M.Y. Li and J.S. Muldowney, A geometric approach to the global-stability problems[J], SIAM J Math Anal. 1996 (27): 1070–1083.
[49] M.Y. Li, Dulac criteria for autonomous systems having an invariant affine manifold[J], J Math Anal Appl 1996(199): 374–390.
[50] M.Y. Li and J.S. Muldowney, On R.A. Smith’s autonomous convergence theorem[J], Rocky Mount J Math. 1995(25): 365–379.
[51] W.A. Coppel, Stability and asymptotic behavior of differential equation[M], Heath,Boston. 1965:41
[52] J.S. Muldowney, Compound matrices and ordinary differential equations[J], Rocky Mount J Math. 1990(20): 857–872.
[53] R.H. Martin Jr., Logarithmic norms and projections applied to linear differential systems[J], J Math Anal Appl 1974(45): 432–454.