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Global asymptotic properties of staged models with multiple progression pathways for infectious diseases
1.  Department of Applied Mathematics and Computer Science, Samara Nayanova Academia, Molodogvardeyskaya 196, 443001, Samara, Russian Federation 
2.  MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick 
References:
[1] 
R. M. Anderson, G. F. Medley, R. M. May and A. M. Johnson, A preliminary study of the transmission dynamics of the human immunodeficiency virus (HIV), the causative agent of AIDS, IMA J. Math. Med. Biol., 3 (1986), 229263. Google Scholar 
[2] 
R. M. Anderson and R. M. May, "Infectious Diseases in Humans: Dynamics and Control," Oxford University Press, Oxford, 1991. Google Scholar 
[3] 
E. A. Barbashin, "Introduction to the Theory of Stability," WoltersNoordhoff, Groningen, 1970. Google Scholar 
[4] 
E. Beretta and V. Capasso, On the general structure of epidemic systems. Global asymptotic stability, Computers & Mathematics with Applications, 12A (1986), 677694. Google Scholar 
[5] 
E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model, in "Mathematical Ecology" (eds. T. G. Hallam, L. J. Gross and S. A. Levin), World Scientific Publ., Teaneck, NJ, (1988), 317342. Google Scholar 
[6] 
O. Diekmann, J. A. P. Heesterbek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365382. Google Scholar 
[7] 
Z. Feng and H. R. Thieme, Endemic model with arbitrarily distributed periods of infection I. Fundamental properties of the model, SIAM J. Appl. Math. 61 (2000), 803833. Google Scholar 
[8] 
P. Georgescu and Y.H. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal,, SIAM J. Appl. Math., 67 (): 337. Google Scholar 
[9] 
B.S. Goh, "Management and Analysis of Biological Populations," Elsevier Science, Amsterdam, 1980. Google Scholar 
[10] 
A. B. Gumel, C. C. McCluskey and P. van den Driessche, Mathematical study of a stagedprogression HIV model with imperfect vaccine, Bull. Math. Biol., 68 (2006), 21052128. Google Scholar 
[11] 
H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng., 3 (2006), 513525. Google Scholar 
[12] 
H. Guo and M. Y. Li, Global dynamics of a stagedprogression model with amelioration for infectious diseases, J. Biol. Dynamics, 2 (2008), 154168. Google Scholar 
[13] 
H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599653. Google Scholar 
[14] 
H. W. Hethcote, J. W. VanArk and I. M. Longini Jr., A simulation model of AIDS in San Francisco: I. Model formulation and parameter estimation, Math. Biosci., 106 (1991), 203222. Google Scholar 
[15] 
G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 11921207. Google Scholar 
[16] 
J. M. Hyman, J. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV, Math. Biosci., 155 (1999), 77109. Google Scholar 
[17] 
W. O. Kermack and A. G. McKendrick, A Contribution to the mathematical theory of epidemics, Proc. Roy. Soc. Lond. A, 115 (1927), 700721. Google Scholar 
[18] 
A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Math. Med. Biol., 21 (2004), 7583. Google Scholar 
[19] 
A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879883. Google Scholar 
[20] 
A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with nonlinear transmission, Bull. Math. Biol., 68 (2006), 615626. Google Scholar 
[21] 
A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 18711886. Google Scholar 
[22] 
A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose dependent parasite reproduction and virulence, and nonlinear incidence rate, Math. Med. Biol., 26 (2009), 225239. Google Scholar 
[23] 
A. Korobeinikov, Stability of ecosystem: Global properties of a general preypredator model, Math. Med. Biol., 26 (2009), 309321. Google Scholar 
[24] 
A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull. Math. Biol., 71 (2009), 7583. doi: 10.1007/s115380079196y. Google Scholar 
[25] 
J. P. LaSalle, "The Stability of Dynamical Systems," SIAM, Philadelphia, 1976. Google Scholar 
[26] 
X. Lin, H. W. Hethcote and P. van den Driessche, An epidemiological model for HIV/AIDS with proportional recruitment, Math. Biosci., 118 (1993), 181195. Google Scholar 
[27] 
A. M. Lyapunov, "The General Problem of the Stability of Motion," Taylor & Francis, Ltd., London, 1992. Google Scholar 
[28] 
W. Ma, M. Song and Y. Takeuchi, Global stability for an SIR epidemic model with time delay, Appl. Math. Lett., 17 (2004), 11411145. Google Scholar 
[29] 
C. C. McCluskey, A model of HIV/AIDS with staged progression and amelioration, Math. Biosci., 181 (2003), 116. Google Scholar 
[30] 
C. C. McCluskey, Lyapunov functions for tuberculosis models with fast and slow progression, Math. Biosci. Eng., 3 (2006), 603614. Google Scholar 
[31] 
C. C. McCluskey, Global stability for a class of mass action systems allowing for latency in tuberculosis, J. Math. Anal. Appl., 338 (2008), 518535. Google Scholar 
[32] 
C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603610. Google Scholar 
[33] 
C. C. McCluskey, Complete global stability for an SIR epidemic model with delay  distributed or discrete, Nonlinear Anal.  Real, 11 (2010), 5559. Google Scholar 
[34] 
C. C. McCluskey, P. Magal and G. F. Webb, Liapunov functional and global asymptotic stability for an infectionage model, Appl. Anal., 89 (2010), 11091140. Google Scholar 
[35] 
D. Morgan, C. Mahe, B. Mayanja, J. M. Okongo, R. Lubega and J. A. Whitworth, HIV1 infection in rural Africa: Is there a difference in median time to AIDS and survival compared with that in industrialized countries, AIDS, 16 (2002), 597632. Google Scholar 
[36] 
D. Okuonghae and A. Korobeinikov, Dynamics of tuberculosis: The effect of direct observation therapy strategy (DOTS) in Nigeria, Math. Model. Nat. Phenom., 2 (2006), 99111. Google Scholar 
[37] 
G. Röst and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 5 (2008), 389402. Google Scholar 
[38] 
Y. Takeuchi, "Global Dynamical Properties of LotkaVolterra Systems," World Scientific, Singapore, 1996. Google Scholar 
[39] 
P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 2948. Google Scholar 
[40] 
J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV1 dynamics with two infinite delays, Math. Med. Biol., 2011. (doi:10.1093/imammb/dqr009) Google Scholar 
show all references
References:
[1] 
R. M. Anderson, G. F. Medley, R. M. May and A. M. Johnson, A preliminary study of the transmission dynamics of the human immunodeficiency virus (HIV), the causative agent of AIDS, IMA J. Math. Med. Biol., 3 (1986), 229263. Google Scholar 
[2] 
R. M. Anderson and R. M. May, "Infectious Diseases in Humans: Dynamics and Control," Oxford University Press, Oxford, 1991. Google Scholar 
[3] 
E. A. Barbashin, "Introduction to the Theory of Stability," WoltersNoordhoff, Groningen, 1970. Google Scholar 
[4] 
E. Beretta and V. Capasso, On the general structure of epidemic systems. Global asymptotic stability, Computers & Mathematics with Applications, 12A (1986), 677694. Google Scholar 
[5] 
E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model, in "Mathematical Ecology" (eds. T. G. Hallam, L. J. Gross and S. A. Levin), World Scientific Publ., Teaneck, NJ, (1988), 317342. Google Scholar 
[6] 
O. Diekmann, J. A. P. Heesterbek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365382. Google Scholar 
[7] 
Z. Feng and H. R. Thieme, Endemic model with arbitrarily distributed periods of infection I. Fundamental properties of the model, SIAM J. Appl. Math. 61 (2000), 803833. Google Scholar 
[8] 
P. Georgescu and Y.H. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal,, SIAM J. Appl. Math., 67 (): 337. Google Scholar 
[9] 
B.S. Goh, "Management and Analysis of Biological Populations," Elsevier Science, Amsterdam, 1980. Google Scholar 
[10] 
A. B. Gumel, C. C. McCluskey and P. van den Driessche, Mathematical study of a stagedprogression HIV model with imperfect vaccine, Bull. Math. Biol., 68 (2006), 21052128. Google Scholar 
[11] 
H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng., 3 (2006), 513525. Google Scholar 
[12] 
H. Guo and M. Y. Li, Global dynamics of a stagedprogression model with amelioration for infectious diseases, J. Biol. Dynamics, 2 (2008), 154168. Google Scholar 
[13] 
H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599653. Google Scholar 
[14] 
H. W. Hethcote, J. W. VanArk and I. M. Longini Jr., A simulation model of AIDS in San Francisco: I. Model formulation and parameter estimation, Math. Biosci., 106 (1991), 203222. Google Scholar 
[15] 
G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 11921207. Google Scholar 
[16] 
J. M. Hyman, J. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV, Math. Biosci., 155 (1999), 77109. Google Scholar 
[17] 
W. O. Kermack and A. G. McKendrick, A Contribution to the mathematical theory of epidemics, Proc. Roy. Soc. Lond. A, 115 (1927), 700721. Google Scholar 
[18] 
A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Math. Med. Biol., 21 (2004), 7583. Google Scholar 
[19] 
A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879883. Google Scholar 
[20] 
A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with nonlinear transmission, Bull. Math. Biol., 68 (2006), 615626. Google Scholar 
[21] 
A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 18711886. Google Scholar 
[22] 
A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose dependent parasite reproduction and virulence, and nonlinear incidence rate, Math. Med. Biol., 26 (2009), 225239. Google Scholar 
[23] 
A. Korobeinikov, Stability of ecosystem: Global properties of a general preypredator model, Math. Med. Biol., 26 (2009), 309321. Google Scholar 
[24] 
A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull. Math. Biol., 71 (2009), 7583. doi: 10.1007/s115380079196y. Google Scholar 
[25] 
J. P. LaSalle, "The Stability of Dynamical Systems," SIAM, Philadelphia, 1976. Google Scholar 
[26] 
X. Lin, H. W. Hethcote and P. van den Driessche, An epidemiological model for HIV/AIDS with proportional recruitment, Math. Biosci., 118 (1993), 181195. Google Scholar 
[27] 
A. M. Lyapunov, "The General Problem of the Stability of Motion," Taylor & Francis, Ltd., London, 1992. Google Scholar 
[28] 
W. Ma, M. Song and Y. Takeuchi, Global stability for an SIR epidemic model with time delay, Appl. Math. Lett., 17 (2004), 11411145. Google Scholar 
[29] 
C. C. McCluskey, A model of HIV/AIDS with staged progression and amelioration, Math. Biosci., 181 (2003), 116. Google Scholar 
[30] 
C. C. McCluskey, Lyapunov functions for tuberculosis models with fast and slow progression, Math. Biosci. Eng., 3 (2006), 603614. Google Scholar 
[31] 
C. C. McCluskey, Global stability for a class of mass action systems allowing for latency in tuberculosis, J. Math. Anal. Appl., 338 (2008), 518535. Google Scholar 
[32] 
C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603610. Google Scholar 
[33] 
C. C. McCluskey, Complete global stability for an SIR epidemic model with delay  distributed or discrete, Nonlinear Anal.  Real, 11 (2010), 5559. Google Scholar 
[34] 
C. C. McCluskey, P. Magal and G. F. Webb, Liapunov functional and global asymptotic stability for an infectionage model, Appl. Anal., 89 (2010), 11091140. Google Scholar 
[35] 
D. Morgan, C. Mahe, B. Mayanja, J. M. Okongo, R. Lubega and J. A. Whitworth, HIV1 infection in rural Africa: Is there a difference in median time to AIDS and survival compared with that in industrialized countries, AIDS, 16 (2002), 597632. Google Scholar 
[36] 
D. Okuonghae and A. Korobeinikov, Dynamics of tuberculosis: The effect of direct observation therapy strategy (DOTS) in Nigeria, Math. Model. Nat. Phenom., 2 (2006), 99111. Google Scholar 
[37] 
G. Röst and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 5 (2008), 389402. Google Scholar 
[38] 
Y. Takeuchi, "Global Dynamical Properties of LotkaVolterra Systems," World Scientific, Singapore, 1996. Google Scholar 
[39] 
P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 2948. Google Scholar 
[40] 
J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV1 dynamics with two infinite delays, Math. Med. Biol., 2011. (doi:10.1093/imammb/dqr009) Google Scholar 
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