American Institute of Mathematical Sciences

Advanced
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

Jinliang Wang 1, and  Hongying Shu 2,

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.
Keywords: relapse, Lyapunov functional., global stability, latency, Multi-group model.
Mathematics Subject Classification: Primary: 92B05, 34D23; Secondary: 34D2.
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.), (1995), 54.  doi: 10.1007/978-1-4899-2869-6_4.  Google Scholar

[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.   Google Scholar

[5]

E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model,, in: T.G. Hallam, (1988), 317.   Google Scholar

[6]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, Academic Press, (1979).   Google Scholar

[7]

N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications,, in: Lecture Notes in Mathematics, (1967).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay,, in: Lecture Notes in Mathematics, (1473).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

J. P. Lasalle, The Stability of Dynamical Systems,, Regional Conference Series in Applied Mathematics, (1976).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.   Google Scholar

[5]

E. Beretta and V. Capasso, Global stability results for a multigroup SIR epidemic model,, in: T.G. Hallam, (1988), 317.   Google Scholar

[6]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, Academic Press, (1979).   Google Scholar

[7]

N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications,, in: Lecture Notes in Mathematics, (1967).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay,, in: Lecture Notes in Mathematics, (1473).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

J. P. Lasalle, The Stability of Dynamical Systems,, Regional Conference Series in Applied Mathematics, (1976).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Xiaomei Feng, Zhidong Teng, Fengqin Zhang. Global dynamics of a general class of multi-group epidemic models with latency and relapse. Mathematical Biosciences & Engineering, 2015, 12 (1) : 99-115. doi: 10.3934/mbe.2015.12.99
[2]

Yoshiaki Muroya, Toshikazu Kuniya, Yoichi Enatsu. Global stability of a delayed multi-group SIRS epidemic model with nonlinear incidence rates and relapse of infection. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3057-3091. doi: 10.3934/dcdsb.2015.20.3057
[3]

Toshikazu Kuniya, Yoshiaki Muroya. Global stability of a multi-group SIS epidemic model for population migration. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1105-1118. doi: 10.3934/dcdsb.2014.19.1105
[4]

Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with generalized nonlinear incidence and vaccination age. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 977-996. doi: 10.3934/dcdsb.2016.21.977
[5]

Jinhu Xu, Yicang Zhou. Global stability of a multi-group model with vaccination age, distributed delay and random perturbation. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1083-1106. doi: 10.3934/mbe.2015.12.1083
[6]

Shanjing Ren. Global stability in a tuberculosis model of imperfect treatment with age-dependent latency and relapse. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1337-1360. doi: 10.3934/mbe.2017069
[7]

Gunduz Caginalp, Mark DeSantis. Multi-group asset flow equations and stability. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 109-150. doi: 10.3934/dcdsb.2011.16.109
[8]

Jinliang Wang, Xianning Liu, Toshikazu Kuniya, Jingmei Pang. Global stability for multi-group SIR and SEIR epidemic models with age-dependent susceptibility. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2795-2812. doi: 10.3934/dcdsb.2017151
[9]

Chunmei Zhang, Wenxue Li, Ke Wang. Graph-theoretic approach to stability of multi-group models with dispersal. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 259-280. doi: 10.3934/dcdsb.2015.20.259
[10]

Toshikazu Kuniya, Jinliang Wang, Hisashi Inaba. A multi-group SIR epidemic model with age structure. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3515-3550. doi: 10.3934/dcdsb.2016109
[11]

Gang Huang, Edoardo Beretta, Yasuhiro Takeuchi. Global stability for epidemic model with constant latency and infectious periods. Mathematical Biosciences & Engineering, 2012, 9 (2) : 297-312. doi: 10.3934/mbe.2012.9.297
[12]

P. van den Driessche, Lin Wang, Xingfu Zou. Modeling diseases with latency and relapse. Mathematical Biosciences & Engineering, 2007, 4 (2) : 205-219. doi: 10.3934/mbe.2007.4.205
[13]

Yoshiaki Muroya. A Lotka-Volterra system with patch structure (related to a multi-group SI epidemic model). Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 999-1008. doi: 10.3934/dcdss.2015.8.999
[14]

Lili Liu, Xianning Liu, Jinliang Wang. Threshold dynamics of a delayed multi-group heroin epidemic model in heterogeneous populations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2615-2630. doi: 10.3934/dcdsb.2016064
[15]

Rui Wang, Xiaoyue Li, Denis S. Mukama. On stochastic multi-group Lotka-Volterra ecosystems with regime switching. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3499-3528. doi: 10.3934/dcdsb.2017177
[16]

Yoshiaki Muroya, Yoichi Enatsu, Huaixing Li. A note on the global stability of an SEIR epidemic model with constant latency time and infectious period. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 173-183. doi: 10.3934/dcdsb.2013.18.173
[17]

Mohammed Nor Frioui, Tarik Mohammed Touaoula, Bedreddine Ainseba. Global dynamics of an age-structured model with relapse. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2245-2270. doi: 10.3934/dcdsb.2019226
[18]

Deqiong Ding, Wendi Qin, Xiaohua Ding. Lyapunov functions and global stability for a discretized multigroup SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1971-1981. doi: 10.3934/dcdsb.2015.20.1971
[19]

Yinshu Wu, Wenzhang Huang. Global stability of the predator-prey model with a sigmoid functional response. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1159-1167. doi: 10.3934/dcdsb.2019214
[20]

Junyuan Yang, Yuming Chen, Jiming Liu. Stability analysis of a two-strain epidemic model on complex networks with latency. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2851-2866. doi: 10.3934/dcdsb.2016076

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (27)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences