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Global analysis on a class of multi-group SEIR model with latency and relapse
1. | School of Mathematical Science, Heilongjiang University, Harbin 150080, China |
2. | Department of Mathematics, Tongji University, Shanghai 200092, China |
References:
[1] |
E. I. M. Abter, O. Schaening, R. L. Barbour and L. I. Lutwick, Tuberculosis in the adult,, in: L.I. Lutwick (eds.), (1995), 54.
doi: 10.1007/978-1-4899-2869-6_4. |
[2] |
R. Anderson and R. May, Population biology of infectious diseases I,, Nature, 280 (1979), 361. Google Scholar |
[3] |
R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control,, Oxford University Press, (1991). Google Scholar |
[4] |
F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations,, Funkcial. Ekvac., 31 (1988), 331.
|
[5] |
E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model,, in: T.G. Hallam, (1988), 317.
|
[6] |
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, Academic Press, (1979).
|
[7] |
N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications,, in: Lecture Notes in Mathematics, (1967).
|
[8] |
J. Chin, Control of Communicable Diseases Manual,, American Public Health Association, (1999). Google Scholar |
[9] |
L. Chow, M. Fan and Z. Feng, Dynamics of a multigroup epidemiological model with group-targeted vaccination strategies,, J. Theor. Biol., 291 (2011), 56.
doi: 10.1016/j.jtbi.2011.09.020. |
[10] |
O. Diekmann, J. A. P. Heesterbeek 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.
doi: 10.1007/BF00178324. |
[11] |
C. R. Driver, S. S. Munsiff, J. Li, N. Kundamal and S. S. Osahan, Relapse in persons treated for drug-susceptible tuberculosis in a population with high coinfection with human immunodeficiency virus in New York city,, Clin. Inf. Dis., 33 (2001), 1762.
doi: 10.1086/323784. |
[12] |
Z. Feng, W. Huang and C. Castillo-Chavez, Global behavior of a multigroup sis epidemic model with age structure,, J. Differential Equations, 218 (2005), 292.
doi: 10.1016/j.jde.2004.10.009. |
[13] |
H. B. Guo, M. Y. Li and Z. S. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models,, Canad. Appl. Math. Quart., 14 (2006), 259.
|
[14] |
H. B. Guo, M. Y. Li and Z. S. Shuai, A graph-theoretic approach to the method of global Lyapunov functions,, Proc. Amer. Math. Soc., 136 (2008), 2793.
doi: 10.1090/S0002-9939-08-09341-6. |
[15] |
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Appl. Math. Sci., (1993).
doi: 10.1007/978-1-4612-4342-7. |
[16] |
A. D. Harries, N. J. Hargreaves, J. H. Kwanjana and F. M. L. Salaniponi, Relapse and recurrent tuberculosis in the context of a national tuberculosis control programme,, Tran. R. Soc. Trop. Med. Hyg., 94 (2000), 247.
doi: 10.1016/S0035-9203(00)90306-7. |
[17] |
H. W. Hethcote, An immunization model for a heterogeneous population,, Theor. Popu. Biol., 14 (1978), 338.
doi: 10.1016/0040-5809(78)90011-4. |
[18] |
Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay,, in: Lecture Notes in Mathematics, (1473).
|
[19] |
W. Huang, L. Keenth and C. Castillo-Chavez, Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission,, SIAM J. Appl. Math., 52 (1992), 835.
doi: 10.1137/0152047. |
[20] |
A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages,, Bull. Math. Biol., 71 (2009), 75.
doi: 10.1007/s11538-008-9352-z. |
[21] |
A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population,, Math. Biosci., 28 (1976), 221.
doi: 10.1016/0025-5564(76)90125-5. |
[22] |
J. P. Lasalle, The Stability of Dynamical Systems,, Regional Conference Series in Applied Mathematics, (1976).
|
[23] |
M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections,, SIAM J. Appl. Math., 70 (2010), 2434.
doi: 10.1137/090779322. |
[24] |
M. Y. Li and Z. S. Shuai, Global stability problem for coupled systems of differential equations on networks,, J. Differential Equations, 248 (2010), 1.
doi: 10.1016/j.jde.2009.09.003. |
[25] |
M. Y. Li, Z. S. Shuai and C. C. Wang, Global stability of multi-group epidemic models with distributed delays,, J. Math. Anal. Appl., 361 (2010), 38.
doi: 10.1016/j.jmaa.2009.09.017. |
[26] |
S. Liu, S. Wang and L. Wang, Global dynamics of delay epidemic models with nonlinear incidence rate and relapse,, Nonlinear Anal.: RWA, 12 (2011), 119.
doi: 10.1016/j.nonrwa.2010.06.001. |
[27] |
S. W. Martin, Livestock Disease Eradication: Evaluation of the Cooperative State Federal Bovine Tuberculosis Eradication Program,, National Academy Press, (1994). Google Scholar |
[28] |
R. K. Miller, Nonlinear Volterra Integral Equations,, W.A. Benjamin Inc., (1971).
|
[29] |
H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission,, Nonlinear Anal.: RWA, 13 (2012), 1581.
doi: 10.1016/j.nonrwa.2011.11.016. |
[30] |
R. Sun and J. Shi, Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates,, Appl. Math. Comput., 218 (2011), 280.
doi: 10.1016/j.amc.2011.05.056. |
[31] |
H. R. Thieme, Local stability in epidemic models for heterogeneous populations,, in: Mathematics in Biology and Medicine, 57 (1995), 185.
doi: 10.1007/978-3-642-93287-8_26. |
[32] |
P. van den Driessche, L. Wang and X. Zou, Modeling diseases with latency and relapse,, Math. Biosci. Eng., 4 (2007), 205.
doi: 10.3934/mbe.2007.4.205. |
[33] |
P. van den Driessche and X. Zou, Modeling relapse in infectious diseases,, Math. Biosci., 207 (2007), 89.
doi: 10.1016/j.mbs.2006.09.017. |
[34] |
K. E. VanLandingham, H. B. Marsteller, G. W. Ross and F. G. Hayden, Relapse of herpes simplex encephalitis after conventional acyclovir therapy,, J. Amer. Med. Assoc., 259 (1988), 1051. Google Scholar |
[35] |
J. Wang, J. Pang and X. Liu, Modelling diseases with relapse and nonlinear incidence of infection: a multi-group epidemic model,, J. Biol. Dyn., 8 (2014), 99.
doi: 10.1080/17513758.2014.912682. |
[36] |
J. Wang, J. Zu, X. Liu, G. Huang and J. Zhang, Global dynamics of a multi-group epidemic model with general relapse distribution and nonlinear incidence rate,, J. Biol. Syst., 20 (2012), 235.
doi: 10.1142/S021833901250009X. |
[37] |
Z. Zhao, L. Chen and X. Song, Impulsive vaccination of SEIR epidemic model with time delay and nonlinear incidence rate,, Math. Comput. Simul., 79 (2008), 500.
doi: 10.1016/j.matcom.2008.02.007. |
show all references
References:
[1] |
E. I. M. Abter, O. Schaening, R. L. Barbour and L. I. Lutwick, Tuberculosis in the adult,, in: L.I. Lutwick (eds.), (1995), 54.
doi: 10.1007/978-1-4899-2869-6_4. |
[2] |
R. Anderson and R. May, Population biology of infectious diseases I,, Nature, 280 (1979), 361. Google Scholar |
[3] |
R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control,, Oxford University Press, (1991). Google Scholar |
[4] |
F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations,, Funkcial. Ekvac., 31 (1988), 331.
|
[5] |
E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model,, in: T.G. Hallam, (1988), 317.
|
[6] |
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, Academic Press, (1979).
|
[7] |
N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications,, in: Lecture Notes in Mathematics, (1967).
|
[8] |
J. Chin, Control of Communicable Diseases Manual,, American Public Health Association, (1999). Google Scholar |
[9] |
L. Chow, M. Fan and Z. Feng, Dynamics of a multigroup epidemiological model with group-targeted vaccination strategies,, J. Theor. Biol., 291 (2011), 56.
doi: 10.1016/j.jtbi.2011.09.020. |
[10] |
O. Diekmann, J. A. P. Heesterbeek 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.
doi: 10.1007/BF00178324. |
[11] |
C. R. Driver, S. S. Munsiff, J. Li, N. Kundamal and S. S. Osahan, Relapse in persons treated for drug-susceptible tuberculosis in a population with high coinfection with human immunodeficiency virus in New York city,, Clin. Inf. Dis., 33 (2001), 1762.
doi: 10.1086/323784. |
[12] |
Z. Feng, W. Huang and C. Castillo-Chavez, Global behavior of a multigroup sis epidemic model with age structure,, J. Differential Equations, 218 (2005), 292.
doi: 10.1016/j.jde.2004.10.009. |
[13] |
H. B. Guo, M. Y. Li and Z. S. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models,, Canad. Appl. Math. Quart., 14 (2006), 259.
|
[14] |
H. B. Guo, M. Y. Li and Z. S. Shuai, A graph-theoretic approach to the method of global Lyapunov functions,, Proc. Amer. Math. Soc., 136 (2008), 2793.
doi: 10.1090/S0002-9939-08-09341-6. |
[15] |
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Appl. Math. Sci., (1993).
doi: 10.1007/978-1-4612-4342-7. |
[16] |
A. D. Harries, N. J. Hargreaves, J. H. Kwanjana and F. M. L. Salaniponi, Relapse and recurrent tuberculosis in the context of a national tuberculosis control programme,, Tran. R. Soc. Trop. Med. Hyg., 94 (2000), 247.
doi: 10.1016/S0035-9203(00)90306-7. |
[17] |
H. W. Hethcote, An immunization model for a heterogeneous population,, Theor. Popu. Biol., 14 (1978), 338.
doi: 10.1016/0040-5809(78)90011-4. |
[18] |
Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay,, in: Lecture Notes in Mathematics, (1473).
|
[19] |
W. Huang, L. Keenth and C. Castillo-Chavez, Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission,, SIAM J. Appl. Math., 52 (1992), 835.
doi: 10.1137/0152047. |
[20] |
A. Korobeinikov, Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages,, Bull. Math. Biol., 71 (2009), 75.
doi: 10.1007/s11538-008-9352-z. |
[21] |
A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population,, Math. Biosci., 28 (1976), 221.
doi: 10.1016/0025-5564(76)90125-5. |
[22] |
J. P. Lasalle, The Stability of Dynamical Systems,, Regional Conference Series in Applied Mathematics, (1976).
|
[23] |
M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections,, SIAM J. Appl. Math., 70 (2010), 2434.
doi: 10.1137/090779322. |
[24] |
M. Y. Li and Z. S. Shuai, Global stability problem for coupled systems of differential equations on networks,, J. Differential Equations, 248 (2010), 1.
doi: 10.1016/j.jde.2009.09.003. |
[25] |
M. Y. Li, Z. S. Shuai and C. C. Wang, Global stability of multi-group epidemic models with distributed delays,, J. Math. Anal. Appl., 361 (2010), 38.
doi: 10.1016/j.jmaa.2009.09.017. |
[26] |
S. Liu, S. Wang and L. Wang, Global dynamics of delay epidemic models with nonlinear incidence rate and relapse,, Nonlinear Anal.: RWA, 12 (2011), 119.
doi: 10.1016/j.nonrwa.2010.06.001. |
[27] |
S. W. Martin, Livestock Disease Eradication: Evaluation of the Cooperative State Federal Bovine Tuberculosis Eradication Program,, National Academy Press, (1994). Google Scholar |
[28] |
R. K. Miller, Nonlinear Volterra Integral Equations,, W.A. Benjamin Inc., (1971).
|
[29] |
H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission,, Nonlinear Anal.: RWA, 13 (2012), 1581.
doi: 10.1016/j.nonrwa.2011.11.016. |
[30] |
R. Sun and J. Shi, Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates,, Appl. Math. Comput., 218 (2011), 280.
doi: 10.1016/j.amc.2011.05.056. |
[31] |
H. R. Thieme, Local stability in epidemic models for heterogeneous populations,, in: Mathematics in Biology and Medicine, 57 (1995), 185.
doi: 10.1007/978-3-642-93287-8_26. |
[32] |
P. van den Driessche, L. Wang and X. Zou, Modeling diseases with latency and relapse,, Math. Biosci. Eng., 4 (2007), 205.
doi: 10.3934/mbe.2007.4.205. |
[33] |
P. van den Driessche and X. Zou, Modeling relapse in infectious diseases,, Math. Biosci., 207 (2007), 89.
doi: 10.1016/j.mbs.2006.09.017. |
[34] |
K. E. VanLandingham, H. B. Marsteller, G. W. Ross and F. G. Hayden, Relapse of herpes simplex encephalitis after conventional acyclovir therapy,, J. Amer. Med. Assoc., 259 (1988), 1051. Google Scholar |
[35] |
J. Wang, J. Pang and X. Liu, Modelling diseases with relapse and nonlinear incidence of infection: a multi-group epidemic model,, J. Biol. Dyn., 8 (2014), 99.
doi: 10.1080/17513758.2014.912682. |
[36] |
J. Wang, J. Zu, X. Liu, G. Huang and J. Zhang, Global dynamics of a multi-group epidemic model with general relapse distribution and nonlinear incidence rate,, J. Biol. Syst., 20 (2012), 235.
doi: 10.1142/S021833901250009X. |
[37] |
Z. Zhao, L. Chen and X. Song, Impulsive vaccination of SEIR epidemic model with time delay and nonlinear incidence rate,, Math. Comput. Simul., 79 (2008), 500.
doi: 10.1016/j.matcom.2008.02.007. |
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