2016, 13(1): 209-225. doi: 10.3934/mbe.2016.13.209

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

Received  March 2015 Revised  May 2015 Published  October 2015

In this paper, we investigate the global dynamics of a multi-group SEIR epidemic model, allowing heterogeneity of the host population, delay in latency and delay due to relapse distribution for the human population. Our results indicate that when certain restrictions on nonlinear growth rate and incidence are fulfilled, the basic reproduction number $\mathfrak{R}_0$ plays the key role of a global threshold parameter in the sense that the long-time behaviors of the model depend only on $\mathfrak{R}_0$. The proofs of the main results utilize the persistence theory in dynamical systems, Lyapunov functionals guided by graph-theoretical approach.
Citation: Jinliang Wang, Hongying Shu. Global analysis on a class of multi-group SEIR model with latency and relapse. Mathematical Biosciences & Engineering, 2016, 13 (1) : 209-225. doi: 10.3934/mbe.2016.13.209
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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