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, New York, 1979.

[2]

N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications, Lecture Notes in Math. 35, Springer, Berlin, 1967.

[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), 494-508. doi: 10.1137/S003613999325419X.

[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), 1790-1811. doi: 10.1137/S0036139997325862.

[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), 365-382. doi: 10.1007/BF00178324.

[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), 105-112. doi: 10.1016/j.mbs.2013.08.003.

[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), 583-600. doi: 10.1007/BF02218848.

[8]

F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1969.

[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), 39-47. doi: 10.1016/j.jtbi.2012.01.006.

[10]

D. Gao and S. Ruan, A multipatch mararia model with logistic growth population, SIAM J. Appl. Math., 72 (2012), 819-841. doi: 10.1137/110850761.

[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), 341-347. doi: 10.1016/0962-8479(94)90079-5.

[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), 259-284.

[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), 1205-1227. doi: 10.1137/070709438.

[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), e492. doi: 10.1371/journal.pmed.0030492.

[15]

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.

[16]

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.

[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), 282-287. doi: 10.1016/S1473-3099(03)00607-8.

[18]

J. P. Lasalle, The stability of dynamicals systems, Reginal Conf. Ser. Appl., SIAM, Philadelphia, 1976.

[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), 191-213. doi: 10.1016/S0025-5564(99)00030-9.

[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), 38-47. doi: 10.1016/j.jmaa.2009.09.017.

[21]

M.Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Diff. Equat., 248 (2010), 1-20. doi: 10.1016/j.jde.2009.09.003.

[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), 675-685. doi: 10.3934/mbe.2010.7.675.

[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), 119-127. doi: 10.1016/j.nonrwa.2010.06.001.

[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), 402-409. doi: 10.1002/lt.20402.

[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), 1693-1704. doi: 10.1016/j.nonrwa.2012.11.005.

[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), 1581-1592. doi: 10.1016/j.nonrwa.2011.11.016.

[27]

Z. Shuai and P. van den Driessche, Global dynamics of cholera models with differential infectivity, Math. Biosci., 234 (2011), 118-126. doi: 10.1016/j.mbs.2011.09.003.

[28]

Z. Shuai and P. van den Driessche, Impact of heterogeneity on the dynamics of an SEIR epidemic model, Math. Biosci. Eng., 9 (2012), 393-411. doi: 10.3934/mbe.2012.9.393.

[29]

Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functious, SIAM J. Appl. Math., 73 (2013), 1513-1532. doi: 10.1137/120876642.

[30]

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

[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), 1687-1693. doi: 10.1016/S0140-6736(01)06712-5.

[32]

R. Sun and J. Shi, Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates, Appl. Math. Comput., 218 (2011), 280-286. doi: 10.1016/j.amc.2011.05.056.

[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), 593-601. doi: 10.7326/0003-4819-154-9-201105030-00334.

[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), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[35]

P. van den Driessche, L. Wang and X. Zou, Modeling disease with latencecy and relapse, Math. Biosci. Eng., 4 (2007), 205-219. doi: 10.3934/mbe.2007.4.205.

[36]

P. van den Driessche and X. Zou, Modeling relapse in infectious disease, Math. Biosci., 207 (2007), 89-103. doi: 10.1016/j.mbs.2006.09.017.

[37]

Y. Yuan and J. Bélair, Threshold dynamics in an SEIRS model with latency and temporary immunity, J. Math. Biol., 69 (2014), 875-904. doi: 10.1007/s00285-013-0720-4.

[38]

J. Zhang, Z. Jin, G. Sun, X. Sun and S. Ruan, Modeling seasonal Rabies epidemics in china, Bull. Math. Biol., 74 (2012), 1226-1251. doi: 10.1007/s11538-012-9720-6.

show all references

References:
[1]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.

[2]

N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications, Lecture Notes in Math. 35, Springer, Berlin, 1967.

[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), 494-508. doi: 10.1137/S003613999325419X.

[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), 1790-1811. doi: 10.1137/S0036139997325862.

[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), 365-382. doi: 10.1007/BF00178324.

[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), 105-112. doi: 10.1016/j.mbs.2013.08.003.

[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), 583-600. doi: 10.1007/BF02218848.

[8]

F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1969.

[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), 39-47. doi: 10.1016/j.jtbi.2012.01.006.

[10]

D. Gao and S. Ruan, A multipatch mararia model with logistic growth population, SIAM J. Appl. Math., 72 (2012), 819-841. doi: 10.1137/110850761.

[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), 341-347. doi: 10.1016/0962-8479(94)90079-5.

[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), 259-284.

[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), 1205-1227. doi: 10.1137/070709438.

[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), e492. doi: 10.1371/journal.pmed.0030492.

[15]

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.

[16]

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.

[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), 282-287. doi: 10.1016/S1473-3099(03)00607-8.

[18]

J. P. Lasalle, The stability of dynamicals systems, Reginal Conf. Ser. Appl., SIAM, Philadelphia, 1976.

[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), 191-213. doi: 10.1016/S0025-5564(99)00030-9.

[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), 38-47. doi: 10.1016/j.jmaa.2009.09.017.

[21]

M.Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Diff. Equat., 248 (2010), 1-20. doi: 10.1016/j.jde.2009.09.003.

[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), 675-685. doi: 10.3934/mbe.2010.7.675.

[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), 119-127. doi: 10.1016/j.nonrwa.2010.06.001.

[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), 402-409. doi: 10.1002/lt.20402.

[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), 1693-1704. doi: 10.1016/j.nonrwa.2012.11.005.

[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), 1581-1592. doi: 10.1016/j.nonrwa.2011.11.016.

[27]

Z. Shuai and P. van den Driessche, Global dynamics of cholera models with differential infectivity, Math. Biosci., 234 (2011), 118-126. doi: 10.1016/j.mbs.2011.09.003.

[28]

Z. Shuai and P. van den Driessche, Impact of heterogeneity on the dynamics of an SEIR epidemic model, Math. Biosci. Eng., 9 (2012), 393-411. doi: 10.3934/mbe.2012.9.393.

[29]

Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functious, SIAM J. Appl. Math., 73 (2013), 1513-1532. doi: 10.1137/120876642.

[30]

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

[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), 1687-1693. doi: 10.1016/S0140-6736(01)06712-5.

[32]

R. Sun and J. Shi, Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates, Appl. Math. Comput., 218 (2011), 280-286. doi: 10.1016/j.amc.2011.05.056.

[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), 593-601. doi: 10.7326/0003-4819-154-9-201105030-00334.

[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), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[35]

P. van den Driessche, L. Wang and X. Zou, Modeling disease with latencecy and relapse, Math. Biosci. Eng., 4 (2007), 205-219. doi: 10.3934/mbe.2007.4.205.

[36]

P. van den Driessche and X. Zou, Modeling relapse in infectious disease, Math. Biosci., 207 (2007), 89-103. doi: 10.1016/j.mbs.2006.09.017.

[37]

Y. Yuan and J. Bélair, Threshold dynamics in an SEIRS model with latency and temporary immunity, J. Math. Biol., 69 (2014), 875-904. doi: 10.1007/s00285-013-0720-4.

[38]

J. Zhang, Z. Jin, G. Sun, X. Sun and S. Ruan, Modeling seasonal Rabies epidemics in china, Bull. Math. Biol., 74 (2012), 1226-1251. doi: 10.1007/s11538-012-9720-6.

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