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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.), Tuberculosis: A Clinical Handbook, Chapman and Hall, London, 1995, 54-101.
doi: 10.1007/978-1-4899-2869-6_4. |
[2] |
R. Anderson and R. May, Population biology of infectious diseases I, Nature, 280 (1979), 361-367. |
[3] |
R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991. |
[4] |
F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations, Funkcial. Ekvac., 31 (1988), 331-347. |
[5] |
E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model, in: T.G. Hallam, L.J. Gross, S.A. Levin (eds.), Mathematical Ecology, World Scientific, Teaneck NJ, (1988), 317-342. |
[6] |
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. |
[7] |
N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications, in: Lecture Notes in Mathematics, Vol. 35, Springer, Berlin, 1967. |
[8] |
J. Chin, Control of Communicable Diseases Manual, American Public Health Association, Washington, 1999. |
[9] |
L. Chow, M. Fan and Z. Feng, Dynamics of a multigroup epidemiological model with group-targeted vaccination strategies, J. Theor. Biol., 291 (2011), 56-64.
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), 365-382.
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), 1762-1769.
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), 292-324.
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), 259-284. |
[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), 2793-2802.
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., vol. 99, Springer, New York, 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), 247-249.
doi: 10.1016/S0035-9203(00)90306-7. |
[17] |
H. W. Hethcote, An immunization model for a heterogeneous population, Theor. Popu. Biol., 14 (1978), 338-349.
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, vol. 1473, Springer-Verlag, berlin, 1991. |
[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), 835-854.
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), 75-83.
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), 221-236.
doi: 10.1016/0025-5564(76)90125-5. |
[22] |
J. P. Lasalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 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), 2434-2448.
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), 1-20.
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), 38-47.
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), 119-127.
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, Washington, 1994. |
[28] |
R. K. Miller, Nonlinear Volterra Integral Equations, W.A. Benjamin Inc., New York, 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), 1581-1592.
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), 280-286.
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, Lecture Notes in Biomathematics, Springer, 57 (1995), 185-189.
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), 205-219.
doi: 10.3934/mbe.2007.4.205. |
[33] |
P. van den Driessche and X. Zou, Modeling relapse in infectious diseases, Math. Biosci., 207 (2007), 89-103.
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), 1051-1053. |
[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), 99-116.
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), 235-258.
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), 500-510.
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.), Tuberculosis: A Clinical Handbook, Chapman and Hall, London, 1995, 54-101.
doi: 10.1007/978-1-4899-2869-6_4. |
[2] |
R. Anderson and R. May, Population biology of infectious diseases I, Nature, 280 (1979), 361-367. |
[3] |
R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991. |
[4] |
F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations, Funkcial. Ekvac., 31 (1988), 331-347. |
[5] |
E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model, in: T.G. Hallam, L.J. Gross, S.A. Levin (eds.), Mathematical Ecology, World Scientific, Teaneck NJ, (1988), 317-342. |
[6] |
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979. |
[7] |
N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications, in: Lecture Notes in Mathematics, Vol. 35, Springer, Berlin, 1967. |
[8] |
J. Chin, Control of Communicable Diseases Manual, American Public Health Association, Washington, 1999. |
[9] |
L. Chow, M. Fan and Z. Feng, Dynamics of a multigroup epidemiological model with group-targeted vaccination strategies, J. Theor. Biol., 291 (2011), 56-64.
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), 365-382.
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), 1762-1769.
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), 292-324.
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), 259-284. |
[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), 2793-2802.
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., vol. 99, Springer, New York, 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), 247-249.
doi: 10.1016/S0035-9203(00)90306-7. |
[17] |
H. W. Hethcote, An immunization model for a heterogeneous population, Theor. Popu. Biol., 14 (1978), 338-349.
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, vol. 1473, Springer-Verlag, berlin, 1991. |
[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), 835-854.
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), 75-83.
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), 221-236.
doi: 10.1016/0025-5564(76)90125-5. |
[22] |
J. P. Lasalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 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), 2434-2448.
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), 1-20.
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), 38-47.
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), 119-127.
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, Washington, 1994. |
[28] |
R. K. Miller, Nonlinear Volterra Integral Equations, W.A. Benjamin Inc., New York, 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), 1581-1592.
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), 280-286.
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, Lecture Notes in Biomathematics, Springer, 57 (1995), 185-189.
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), 205-219.
doi: 10.3934/mbe.2007.4.205. |
[33] |
P. van den Driessche and X. Zou, Modeling relapse in infectious diseases, Math. Biosci., 207 (2007), 89-103.
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), 1051-1053. |
[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), 99-116.
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), 235-258.
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), 500-510.
doi: 10.1016/j.matcom.2008.02.007. |
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