American Institute of Mathematical Sciences

Advanced
2011, 8(4): 1019-1034. doi: 10.3934/mbe.2011.8.1019

Global asymptotic properties of staged models with multiple progression pathways for infectious diseases

Andrey V. Melnik 1, and  Andrei Korobeinikov 2,

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

Received  October 2010 Revised  March 2011 Published  August 2011

We consider global asymptotic properties of compartment staged-progression models for infectious diseases with long infectious period, where there are multiple alternative disease progression pathways and branching. For example, these models reflect cases when there is considerable difference in virulence, or when only a part of the infected individuals undergoes a treatment whereas the rest remains untreated. Using the direct Lyapunov method, we establish sufficient and necessary conditions for the existence and global stability of a unique endemic equilibrium state, and for the stability of an infection-free equilibrium state.
Keywords: global stability, mass action, compartment model, direct Lyapunov method, Lyapunov function, endemic equilibrium state, Infectious disease, HIV..
Mathematics Subject Classification: Primary: 92D30; Secondary: 34D2.
Citation: Andrey V. Melnik, Andrei Korobeinikov. Global asymptotic properties of staged models with multiple progression pathways for infectious diseases. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1019-1034. doi: 10.3934/mbe.2011.8.1019
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), 229-263.  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," Wolters-Noordhoff, Groningen, 1970. Google Scholar

[4]

E. Beretta and V. Capasso, On the general structure of epidemic systems. Global asymptotic stability, Computers & Mathematics with Applications, 12-A (1986), 677-694. 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), 317-342.  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), 365-382.  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), 803-833.  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 staged-progression HIV model with imperfect vaccine, Bull. Math. Biol., 68 (2006), 2105-2128.  Google Scholar

[11]

H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng., 3 (2006), 513-525.  Google Scholar

[12]

H. Guo and M. Y. Li, Global dynamics of a staged-progression model with amelioration for infectious diseases, J. Biol. Dynamics, 2 (2008), 154-168.  Google Scholar

[13]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  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), 203-222. 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), 1192-1207.  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), 77-109. 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), 700-721. Google Scholar

[18]

A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Math. Med. Biol., 21 (2004), 75-83. Google Scholar

[19]

A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883.  Google Scholar

[20]

A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626.  Google Scholar

[21]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886.  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), 225-239. Google Scholar

[23]

A. Korobeinikov, Stability of ecosystem: Global properties of a general prey-predator model, Math. Med. Biol., 26 (2009), 309-321.  Google Scholar

[24]

A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull. Math. Biol., 71 (2009), 75-83. doi: 10.1007/s11538-007-9196-y.  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), 181-195.  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), 1141-1145.  Google Scholar

[29]

C. C. McCluskey, A model of HIV/AIDS with staged progression and amelioration, Math. Biosci., 181 (2003), 1-16.  Google Scholar

[30]

C. C. McCluskey, Lyapunov functions for tuberculosis models with fast and slow progression, Math. Biosci. Eng., 3 (2006), 603-614.  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), 518-535.  Google Scholar

[32]

C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610.  Google Scholar

[33]

C. C. McCluskey, Complete global stability for an SIR epidemic model with delay - distributed or discrete, Nonlinear Anal. - Real, 11 (2010), 55-59.  Google Scholar

[34]

C. C. McCluskey, P. Magal and G. F. Webb, Liapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140.  Google Scholar

[35]

D. Morgan, C. Mahe, B. Mayanja, J. M. Okongo, R. Lubega and J. A. Whitworth, HIV-1 infection in rural Africa: Is there a difference in median time to AIDS and survival compared with that in industrialized countries, AIDS, 16 (2002), 597-632. 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), 99-111.  Google Scholar

[37]

G. Röst and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 5 (2008), 389-402. Google Scholar

[38]

Y. Takeuchi, "Global Dynamical Properties of Lotka-Volterra Systems," World Scientific, Singapore, 1996. Google Scholar

[39]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. Google Scholar

[40]

J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 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), 229-263.  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," Wolters-Noordhoff, Groningen, 1970. Google Scholar

[4]

E. Beretta and V. Capasso, On the general structure of epidemic systems. Global asymptotic stability, Computers & Mathematics with Applications, 12-A (1986), 677-694. 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), 317-342.  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), 365-382.  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), 803-833.  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 staged-progression HIV model with imperfect vaccine, Bull. Math. Biol., 68 (2006), 2105-2128.  Google Scholar

[11]

H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng., 3 (2006), 513-525.  Google Scholar

[12]

H. Guo and M. Y. Li, Global dynamics of a staged-progression model with amelioration for infectious diseases, J. Biol. Dynamics, 2 (2008), 154-168.  Google Scholar

[13]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  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), 203-222. 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), 1192-1207.  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), 77-109. 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), 700-721. Google Scholar

[18]

A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Math. Med. Biol., 21 (2004), 75-83. Google Scholar

[19]

A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883.  Google Scholar

[20]

A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626.  Google Scholar

[21]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886.  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), 225-239. Google Scholar

[23]

A. Korobeinikov, Stability of ecosystem: Global properties of a general prey-predator model, Math. Med. Biol., 26 (2009), 309-321.  Google Scholar

[24]

A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull. Math. Biol., 71 (2009), 75-83. doi: 10.1007/s11538-007-9196-y.  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), 181-195.  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), 1141-1145.  Google Scholar

[29]

C. C. McCluskey, A model of HIV/AIDS with staged progression and amelioration, Math. Biosci., 181 (2003), 1-16.  Google Scholar

[30]

C. C. McCluskey, Lyapunov functions for tuberculosis models with fast and slow progression, Math. Biosci. Eng., 3 (2006), 603-614.  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), 518-535.  Google Scholar

[32]

C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610.  Google Scholar

[33]

C. C. McCluskey, Complete global stability for an SIR epidemic model with delay - distributed or discrete, Nonlinear Anal. - Real, 11 (2010), 55-59.  Google Scholar

[34]

C. C. McCluskey, P. Magal and G. F. Webb, Liapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140.  Google Scholar

[35]

D. Morgan, C. Mahe, B. Mayanja, J. M. Okongo, R. Lubega and J. A. Whitworth, HIV-1 infection in rural Africa: Is there a difference in median time to AIDS and survival compared with that in industrialized countries, AIDS, 16 (2002), 597-632. 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), 99-111.  Google Scholar

[37]

G. Röst and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 5 (2008), 389-402. Google Scholar

[38]

Y. Takeuchi, "Global Dynamical Properties of Lotka-Volterra Systems," World Scientific, Singapore, 1996. Google Scholar

[39]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. Google Scholar

[40]

J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two infinite delays, Math. Med. Biol., 2011. (doi:10.1093/imammb/dqr009) Google Scholar

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