2015, 12(1): 99-115. doi: 10.3934/mbe.2015.12.99

Global dynamics of a general class of multi-group epidemic models with latency and relapse

1. 

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046

2. 

Department of Applied Mathematics, Yuncheng University, Yuncheng 044000, Shanxi

Received  April 2014 Revised  November 2014 Published  December 2014

A multi-group model is proposed to describe a general relapse phenomenon of infectious diseases in heterogeneous populations. In each group, the population is divided into susceptible, exposed, infectious, and recovered subclasses. A general nonlinear incidence rate is used in the model. The results show that the global dynamics are completely determined by the basic reproduction number $R_0.$ In particular, a matrix-theoretic method is used to prove the global stability of the disease-free equilibrium when $R_0\leq1,$ while a new combinatorial identity (Theorem 3.3 in Shuai and van den Driessche [29]) in graph theory is applied to prove the global stability of the endemic equilibrium when $R_0>1.$ We would like to mention that by applying the new combinatorial identity, a graph of 3n (or 2n+m) vertices can be converted into a graph of n vertices in order to deal with the global stability of the endemic equilibrium in this paper.
Citation: 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
References:
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A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, Academic Press, (1979).   Google Scholar

[2]

N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications,, Lecture Notes in Math. 35, (1967).   Google Scholar

[3]

C. Castillo-Charez, W. Huang and J. Li, Competitive exclusion in genorrhea models and other sexually transmitted diseases,, SIAM J. Appl. Math., 56 (1996), 494.  doi: 10.1137/S003613999325419X.  Google Scholar

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M. C. Eisenberg, Z. Shuai, J. H. Tien and P. van den Driessche, A cholera model in a patchy environment with water and human movement,, Math. Biosci., 246 (2013), 105.  doi: 10.1016/j.mbs.2013.08.003.  Google Scholar

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H. I. Freedman, S. Ruan and M. Tang, Uniform persistence and flows near a closed positively invariant set,, J. Dynam. Diff. Equat., 6 (1994), 583.  doi: 10.1007/BF02218848.  Google Scholar

[8]

F. Harary, Graph Theory,, Addison-Wesley, (1969).   Google Scholar

[9]

Q. Hou, Z. Jin and S. Ruan, Dynamics of rabies epidemics and the impact of control efforts in Guangdong Province, China,, J. Theor. Biol., 300 (2012), 39.  doi: 10.1016/j.jtbi.2012.01.006.  Google Scholar

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D. Gao and S. Ruan, A multipatch mararia model with logistic growth population,, SIAM J. Appl. Math., 72 (2012), 819.  doi: 10.1137/110850761.  Google Scholar

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L. J. Gonzalez-Montaner, S. Natal, P. Yongchaiyud and P. Olliaro, et al., Rifabutin for the treatment of newly-diagnosed pulmonary tuberculosis: a multinational, randomized, comparative study versus Rifampicin,, Tuber Lung Dis., 75 (1994), 341.  doi: 10.1016/0962-8479(94)90079-5.  Google Scholar

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H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models,, Can. Appl. Math. Q., 14 (2006), 259.   Google Scholar

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J. Jiang and Z. Qiu, The complete classification for dynamics in a nine-dimensional West Nile Virus model,, SIAM J. Appl. Math., 69 (2009), 1205.  doi: 10.1137/070709438.  Google Scholar

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A. Y. Kim, J. Schulze zur Wiesch, T. Kuntzen , J. Timm and D. E Kaufmann, et al., Impaired Hepatitis C virus-specific T cell responses and recurrent Hepatitis C virus in HIV coinfection,, PLoS Med., 3 (2006).  doi: 10.1371/journal.pmed.0030492.  Google Scholar

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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

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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

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M. L. Lamberta, E. Haskera, A. Van Deuna, D. Roberfroida, M. Boelaerta and P. Van der Stuyft, Recurrence in tuberculosis: Relapse or reinfection?,, Lancet Infect. Dis., 3 (2003), 282.  doi: 10.1016/S1473-3099(03)00607-8.  Google Scholar

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M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size,, Math. Biosci., 160 (1999), 191.  doi: 10.1016/S0025-5564(99)00030-9.  Google Scholar

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M. Y. Li, Z. Shuai and 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

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M.Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks,, J. Diff. Equat., 248 (2010), 1.  doi: 10.1016/j.jde.2009.09.003.  Google Scholar

[22]

S. Liu and L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy,, Math. Biosci. Eng., 7 (2010), 675.  doi: 10.3934/mbe.2010.7.675.  Google Scholar

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S. Liu, S. Wang and L. Wang, Global dynamics of delay epidemic models with nonlinear incidence rate and relapse,, Nonlinear Anal. Real World Appl., 12 (2011), 119.  doi: 10.1016/j.nonrwa.2010.06.001.  Google Scholar

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A. Marzano, S. Gaia, V. Ghisetti, S. Carenzi and A. Premoli, et al., Viral load at the time of liver transplantation and risk of hepatitis B virus recurrence,, Liver Transpl., 11 (2005), 402.  doi: 10.1002/lt.20402.  Google Scholar

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Y. Muroya, Y. Enatsu and T. Kuniya, Global stability for a multi-group SIRS epidemic model with varying population sizes,, Nonlinear Anal. Real World Appl., 14 (2013), 1693.  doi: 10.1016/j.nonrwa.2012.11.005.  Google Scholar

[26]

H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission,, Nonlinear Anal. Real World Appl., 13 (2012), 1581.  doi: 10.1016/j.nonrwa.2011.11.016.  Google Scholar

[27]

Z. Shuai and P. van den Driessche, Global dynamics of cholera models with differential infectivity,, Math. Biosci., 234 (2011), 118.  doi: 10.1016/j.mbs.2011.09.003.  Google Scholar

[28]

Z. Shuai and P. van den Driessche, Impact of heterogeneity on the dynamics of an SEIR epidemic model,, Math. Biosci. Eng., 9 (2012), 393.  doi: 10.3934/mbe.2012.9.393.  Google Scholar

[29]

Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functious,, SIAM J. Appl. Math., 73 (2013), 1513.  doi: 10.1137/120876642.  Google Scholar

[30]

H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511530043.  Google Scholar

[31]

P. Sonnenberg, J. Murray, J. R Glynn, S. Shearer and B. Kambashi, et al., HIV-1 and recurrence, relapse, and reinfection of tuberculosis after cure: a cohort study in South African mineworkers,, Lancet, 358 (2001), 1687.  doi: 10.1016/S0140-6736(01)06712-5.  Google Scholar

[32]

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

[33]

A. R. Tuite, J. H. Tien, M. Eisenberg, D. J. D. Earn and J. Ma, et al., Cholera epidemic in Haiti, 2010: Using a transmission model to explain spatial spread of disease and identify optimal control interventions,, Ann. Internal Med., 154 (2011), 593.  doi: 10.7326/0003-4819-154-9-201105030-00334.  Google Scholar

[34]

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.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[35]

P. van den Driessche, L. Wang and X. Zou, Modeling disease with latencecy and relapse,, Math. Biosci. Eng., 4 (2007), 205.  doi: 10.3934/mbe.2007.4.205.  Google Scholar

[36]

P. van den Driessche and X. Zou, Modeling relapse in infectious disease,, Math. Biosci., 207 (2007), 89.  doi: 10.1016/j.mbs.2006.09.017.  Google Scholar

[37]

Y. Yuan and J. Bélair, Threshold dynamics in an SEIRS model with latency and temporary immunity,, J. Math. Biol., 69 (2014), 875.  doi: 10.1007/s00285-013-0720-4.  Google Scholar

[38]

J. Zhang, Z. Jin, G. Sun, X. Sun and S. Ruan, Modeling seasonal Rabies epidemics in china,, Bull. Math. Biol., 74 (2012), 1226.  doi: 10.1007/s11538-012-9720-6.  Google Scholar

show all references

References:
[1]

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

[2]

N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications,, Lecture Notes in Math. 35, (1967).   Google Scholar

[3]

C. Castillo-Charez, W. Huang and J. Li, Competitive exclusion in genorrhea models and other sexually transmitted diseases,, SIAM J. Appl. Math., 56 (1996), 494.  doi: 10.1137/S003613999325419X.  Google Scholar

[4]

C. Castillo-Charez, W. Huang and J. Li, Competitive exclusion and coexistence of multiple strains in an SIS STD model,, SIAM J. Appl. Math., 59 (1999), 1790.  doi: 10.1137/S0036139997325862.  Google Scholar

[5]

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

[6]

M. C. Eisenberg, Z. Shuai, J. H. Tien and P. van den Driessche, A cholera model in a patchy environment with water and human movement,, Math. Biosci., 246 (2013), 105.  doi: 10.1016/j.mbs.2013.08.003.  Google Scholar

[7]

H. I. Freedman, S. Ruan and M. Tang, Uniform persistence and flows near a closed positively invariant set,, J. Dynam. Diff. Equat., 6 (1994), 583.  doi: 10.1007/BF02218848.  Google Scholar

[8]

F. Harary, Graph Theory,, Addison-Wesley, (1969).   Google Scholar

[9]

Q. Hou, Z. Jin and S. Ruan, Dynamics of rabies epidemics and the impact of control efforts in Guangdong Province, China,, J. Theor. Biol., 300 (2012), 39.  doi: 10.1016/j.jtbi.2012.01.006.  Google Scholar

[10]

D. Gao and S. Ruan, A multipatch mararia model with logistic growth population,, SIAM J. Appl. Math., 72 (2012), 819.  doi: 10.1137/110850761.  Google Scholar

[11]

L. J. Gonzalez-Montaner, S. Natal, P. Yongchaiyud and P. Olliaro, et al., Rifabutin for the treatment of newly-diagnosed pulmonary tuberculosis: a multinational, randomized, comparative study versus Rifampicin,, Tuber Lung Dis., 75 (1994), 341.  doi: 10.1016/0962-8479(94)90079-5.  Google Scholar

[12]

H. Guo, M. Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models,, Can. Appl. Math. Q., 14 (2006), 259.   Google Scholar

[13]

J. Jiang and Z. Qiu, The complete classification for dynamics in a nine-dimensional West Nile Virus model,, SIAM J. Appl. Math., 69 (2009), 1205.  doi: 10.1137/070709438.  Google Scholar

[14]

A. Y. Kim, J. Schulze zur Wiesch, T. Kuntzen , J. Timm and D. E Kaufmann, et al., Impaired Hepatitis C virus-specific T cell responses and recurrent Hepatitis C virus in HIV coinfection,, PLoS Med., 3 (2006).  doi: 10.1371/journal.pmed.0030492.  Google Scholar

[15]

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

[16]

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

[17]

M. L. Lamberta, E. Haskera, A. Van Deuna, D. Roberfroida, M. Boelaerta and P. Van der Stuyft, Recurrence in tuberculosis: Relapse or reinfection?,, Lancet Infect. Dis., 3 (2003), 282.  doi: 10.1016/S1473-3099(03)00607-8.  Google Scholar

[18]

J. P. Lasalle, The stability of dynamicals systems,, Reginal Conf. Ser. Appl., (1976).   Google Scholar

[19]

M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size,, Math. Biosci., 160 (1999), 191.  doi: 10.1016/S0025-5564(99)00030-9.  Google Scholar

[20]

M. Y. Li, Z. Shuai and 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

[21]

M.Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks,, J. Diff. Equat., 248 (2010), 1.  doi: 10.1016/j.jde.2009.09.003.  Google Scholar

[22]

S. Liu and L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy,, Math. Biosci. Eng., 7 (2010), 675.  doi: 10.3934/mbe.2010.7.675.  Google Scholar

[23]

S. Liu, S. Wang and L. Wang, Global dynamics of delay epidemic models with nonlinear incidence rate and relapse,, Nonlinear Anal. Real World Appl., 12 (2011), 119.  doi: 10.1016/j.nonrwa.2010.06.001.  Google Scholar

[24]

A. Marzano, S. Gaia, V. Ghisetti, S. Carenzi and A. Premoli, et al., Viral load at the time of liver transplantation and risk of hepatitis B virus recurrence,, Liver Transpl., 11 (2005), 402.  doi: 10.1002/lt.20402.  Google Scholar

[25]

Y. Muroya, Y. Enatsu and T. Kuniya, Global stability for a multi-group SIRS epidemic model with varying population sizes,, Nonlinear Anal. Real World Appl., 14 (2013), 1693.  doi: 10.1016/j.nonrwa.2012.11.005.  Google Scholar

[26]

H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission,, Nonlinear Anal. Real World Appl., 13 (2012), 1581.  doi: 10.1016/j.nonrwa.2011.11.016.  Google Scholar

[27]

Z. Shuai and P. van den Driessche, Global dynamics of cholera models with differential infectivity,, Math. Biosci., 234 (2011), 118.  doi: 10.1016/j.mbs.2011.09.003.  Google Scholar

[28]

Z. Shuai and P. van den Driessche, Impact of heterogeneity on the dynamics of an SEIR epidemic model,, Math. Biosci. Eng., 9 (2012), 393.  doi: 10.3934/mbe.2012.9.393.  Google Scholar

[29]

Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functious,, SIAM J. Appl. Math., 73 (2013), 1513.  doi: 10.1137/120876642.  Google Scholar

[30]

H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511530043.  Google Scholar

[31]

P. Sonnenberg, J. Murray, J. R Glynn, S. Shearer and B. Kambashi, et al., HIV-1 and recurrence, relapse, and reinfection of tuberculosis after cure: a cohort study in South African mineworkers,, Lancet, 358 (2001), 1687.  doi: 10.1016/S0140-6736(01)06712-5.  Google Scholar

[32]

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

[33]

A. R. Tuite, J. H. Tien, M. Eisenberg, D. J. D. Earn and J. Ma, et al., Cholera epidemic in Haiti, 2010: Using a transmission model to explain spatial spread of disease and identify optimal control interventions,, Ann. Internal Med., 154 (2011), 593.  doi: 10.7326/0003-4819-154-9-201105030-00334.  Google Scholar

[34]

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.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[35]

P. van den Driessche, L. Wang and X. Zou, Modeling disease with latencecy and relapse,, Math. Biosci. Eng., 4 (2007), 205.  doi: 10.3934/mbe.2007.4.205.  Google Scholar

[36]

P. van den Driessche and X. Zou, Modeling relapse in infectious disease,, Math. Biosci., 207 (2007), 89.  doi: 10.1016/j.mbs.2006.09.017.  Google Scholar

[37]

Y. Yuan and J. Bélair, Threshold dynamics in an SEIRS model with latency and temporary immunity,, J. Math. Biol., 69 (2014), 875.  doi: 10.1007/s00285-013-0720-4.  Google Scholar

[38]

J. Zhang, Z. Jin, G. Sun, X. Sun and S. Ruan, Modeling seasonal Rabies epidemics in china,, Bull. Math. Biol., 74 (2012), 1226.  doi: 10.1007/s11538-012-9720-6.  Google Scholar

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