# American Institute of Mathematical Sciences

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:
 [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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##### 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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