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2012, 9(2): 393-411. doi: 10.3934/mbe.2012.9.393

## Impact of heterogeneity on the dynamics of an SEIR epidemic model

 1 Department of Mathematics and Statistics, University of Victoria, Victoria, B.C., V8W 3R4, Canada, Canada

Received  April 2011 Revised  August 2011 Published  March 2012

An SEIR epidemic model with an arbitrarily distributed exposed stage is revisited to study the impact of heterogeneity on the spread of infectious diseases. The heterogeneity may come from age or behavior and disease stages, resulting in multi-group and multi-stage models, respectively. For each model, Lyapunov functionals are used to show that the basic reproduction number $\mathcal{R}_0$ gives a sharp threshold. If $\mathcal{R}_0\leq 1$, then the disease-free equilibrium is globally asymptotically stable and the disease dies out from all groups or stages. If $\mathcal{R}_0>1$, then the disease persists in all groups or stages, and the endemic equilibrium is globally asymptotically stable.
Citation: Zhisheng Shuai, P. van den Driessche. Impact of heterogeneity on the dynamics of an SEIR epidemic model. Mathematical Biosciences & Engineering, 2012, 9 (2) : 393-411. doi: 10.3934/mbe.2012.9.393
##### References:
 [1] H. Andersson and T. Britton, Heterogeneity in epidemic models and its effect on the spread of infection,, J. App. Prob., 35 (1998), 651. Google Scholar [2] R. M. Anderson and R. M. May, Spatial, temporal, and genetic heterogeneity in host populations and the design of immunization programmes,, IMA J. Math. Appl. Med. Biol., 1 (1984), 233. doi: 10.1093/imammb/1.3.233. Google Scholar [3] R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control,", Oxford University Press, (1992). 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] A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Sciences,", Computer Science and Applied Mathematics, (1979). Google Scholar [6] K. L. Cooke and P. van den Driessche, Analysis of an SEIRS epidemic model with two delays,, J. Math. Biol., 35 (1996), 240. doi: 10.1007/s002850050051. Google Scholar [7] 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 [8] Z. Feng, D. Xu and H. Zhao, Epidemiological models with non-exponentially distributed disease stages and applications to disease control,, Bull. Math. Biol., 69 (2007), 1511. doi: 10.1007/s11538-006-9174-9. Google Scholar [9] H. Guo and M. Y. Li, Global dynamics of a staged-progression model with amelioration for infectious diseases,, J. Biol. Dyn., 2 (2008), 154. doi: 10.1080/17513750802120877. Google Scholar [10] 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 [11] H. Guo, M. Y. Li and Z. 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 [12] T. J. Hagenaars, C. A. Donnelly and N. M. Ferguson, Spatial heterogeneity and the persistence of infectious diseases,, J. Theor. Biol., 229 (2004), 349. doi: 10.1016/j.jtbi.2004.04.002. Google Scholar [13] J. K. Hale and S. M. V. Lunel, "Introduction to Functional-Differential Equations,", Appl. Math. Sci., (1993). Google Scholar [14] H. W. Hethcote, H. W. Stech, and P. van den Driessche, Stability analysis for models of diseases without immunity,, J. Math. Biol., 13 (1981), 185. doi: 10.1007/BF00275213. Google Scholar [15] H. W. Hethcote and D. W. Tudor, Integral equation models for endemic infectious diseases,, J. Math. Biol., 9 (1980), 37. doi: 10.1007/BF00276034. Google Scholar [16] Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay,", Lectures Notes in Math., (1473). Google Scholar [17] G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence,, J. Math. Biol., 63 (2011), 125. doi: 10.1007/s00285-010-0368-2. Google Scholar [18] G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections,, SIAM J. App. Math., 70 (2010), 2693. doi: 10.1137/090780821. Google Scholar [19] J. M. Hyman, J. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV,, Math. Biosci., 155 (1999), 77. doi: 10.1016/S0025-5564(98)10057-3. Google Scholar [20] A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models,, Math. Med. Biol., 21 (2004), 75. doi: 10.1093/imammb/21.2.75. Google Scholar [21] A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence,, Math. Biosci. Eng., 1 (2004), 57. doi: 10.3934/mbe.2004.1.57. Google Scholar [22] 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 [23] J. P. LaSalle, "The Stability of Dynamical Systems,", With an appendix: \emph{Limiting equations and stability of nonautonomous ordinary differential equations}, (1976). Google Scholar [24] M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology,, Math. Biosci., 125 (1995), 155. doi: 10.1016/0025-5564(95)92756-5. Google Scholar [25] M. Y. Li and J. S. Muldowney, A geometric approach to global stability,, SIAM J. Math. Anal., 27 (1996), 1070. Google Scholar [26] M. Y. Li, J. S. Muldowney and P. van den Driessche, Global stability of SEIRS models in epidemiology,, Can. Appl. Math. Q., 7 (1999), 409. Google Scholar [27] 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 [28] M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, Bull. Math. Bio., 72 (2010), 1429. doi: 10.1007/s11538-010-9503-x. Google Scholar [29] M. Y. Li and Z. 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 [30] 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 [31] 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 [32] W. M. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates,, J. Math. Biol., 25 (1987), 359. doi: 10.1007/BF00277162. Google Scholar [33] A. L. Lloyd and R. M. May, Spatial heterogeneity in epidemic models,, J. Theor. Biol., 179 (1996), 1. doi: 10.1006/jtbi.1996.0042. Google Scholar [34] C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 6 (2009), 603. doi: 10.3934/mbe.2009.6.603. Google Scholar [35] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay--distributed or discrete,, Nonlinear Anal. Real World Appl., 11 (2010), 55. doi: 10.1016/j.nonrwa.2008.10.014. Google Scholar [36] R. K. Miller, "Nonlinear Volterra Integral Equations,", Mathematics Lecture Note Series, (1971). Google Scholar [37] M. E. J. Newman, The spread of epidemic disease on networks,, Phys. Rev. E, 66 (2002). doi: 10.1103/PhysRevE.66.016128. Google Scholar [38] C. P. Simon and J. A. Jacquez, Reproduction numbers and the stability of equilibria of SI models for heterogeneous populations,, SIAM J. Appl. Math., 52 (1992), 541. doi: 10.1137/0152030. Google Scholar [39] P. van den Driessche, Some epidemiological models with delays,, in, (1996), 507. Google Scholar [40] P. van den Driessche, L. Wang and X. Zou, Modeling diseases with latency and relapse,, Math. Biosci. Eng., 4 (2007), 205. Google Scholar [41] P. van den Driessche, L. Wang and X. Zou, Impact of group mixing on disease dynamics,, Math. Biosci., 228 (2010), 71. doi: 10.1016/j.mbs.2010.08.008. Google Scholar [42] 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 [43] 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 [44] Z. Yuan and X. Zou, Global threshold property in an epidemic model for disease with latency spreading in a heterogeneous host population,, Nonlinear Anal. Real World Appl., 11 (2010), 3479. doi: 10.1016/j.nonrwa.2009.12.008. Google Scholar

show all references

##### References:
 [1] H. Andersson and T. Britton, Heterogeneity in epidemic models and its effect on the spread of infection,, J. App. Prob., 35 (1998), 651. Google Scholar [2] R. M. Anderson and R. M. May, Spatial, temporal, and genetic heterogeneity in host populations and the design of immunization programmes,, IMA J. Math. Appl. Med. Biol., 1 (1984), 233. doi: 10.1093/imammb/1.3.233. Google Scholar [3] R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control,", Oxford University Press, (1992). 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] A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Sciences,", Computer Science and Applied Mathematics, (1979). Google Scholar [6] K. L. Cooke and P. van den Driessche, Analysis of an SEIRS epidemic model with two delays,, J. Math. Biol., 35 (1996), 240. doi: 10.1007/s002850050051. Google Scholar [7] 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 [8] Z. Feng, D. Xu and H. Zhao, Epidemiological models with non-exponentially distributed disease stages and applications to disease control,, Bull. Math. Biol., 69 (2007), 1511. doi: 10.1007/s11538-006-9174-9. Google Scholar [9] H. Guo and M. Y. Li, Global dynamics of a staged-progression model with amelioration for infectious diseases,, J. Biol. Dyn., 2 (2008), 154. doi: 10.1080/17513750802120877. Google Scholar [10] 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 [11] H. Guo, M. Y. Li and Z. 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 [12] T. J. Hagenaars, C. A. Donnelly and N. M. Ferguson, Spatial heterogeneity and the persistence of infectious diseases,, J. Theor. Biol., 229 (2004), 349. doi: 10.1016/j.jtbi.2004.04.002. Google Scholar [13] J. K. Hale and S. M. V. Lunel, "Introduction to Functional-Differential Equations,", Appl. Math. Sci., (1993). Google Scholar [14] H. W. Hethcote, H. W. Stech, and P. van den Driessche, Stability analysis for models of diseases without immunity,, J. Math. Biol., 13 (1981), 185. doi: 10.1007/BF00275213. Google Scholar [15] H. W. Hethcote and D. W. Tudor, Integral equation models for endemic infectious diseases,, J. Math. Biol., 9 (1980), 37. doi: 10.1007/BF00276034. Google Scholar [16] Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay,", Lectures Notes in Math., (1473). Google Scholar [17] G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence,, J. Math. Biol., 63 (2011), 125. doi: 10.1007/s00285-010-0368-2. Google Scholar [18] G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections,, SIAM J. App. Math., 70 (2010), 2693. doi: 10.1137/090780821. Google Scholar [19] J. M. Hyman, J. Li and E. A. Stanley, The differential infectivity and staged progression models for the transmission of HIV,, Math. Biosci., 155 (1999), 77. doi: 10.1016/S0025-5564(98)10057-3. Google Scholar [20] A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models,, Math. Med. Biol., 21 (2004), 75. doi: 10.1093/imammb/21.2.75. Google Scholar [21] A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence,, Math. Biosci. Eng., 1 (2004), 57. doi: 10.3934/mbe.2004.1.57. Google Scholar [22] 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 [23] J. P. LaSalle, "The Stability of Dynamical Systems,", With an appendix: \emph{Limiting equations and stability of nonautonomous ordinary differential equations}, (1976). Google Scholar [24] M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology,, Math. Biosci., 125 (1995), 155. doi: 10.1016/0025-5564(95)92756-5. Google Scholar [25] M. Y. Li and J. S. Muldowney, A geometric approach to global stability,, SIAM J. Math. Anal., 27 (1996), 1070. Google Scholar [26] M. Y. Li, J. S. Muldowney and P. van den Driessche, Global stability of SEIRS models in epidemiology,, Can. Appl. Math. Q., 7 (1999), 409. Google Scholar [27] 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 [28] M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, Bull. Math. Bio., 72 (2010), 1429. doi: 10.1007/s11538-010-9503-x. Google Scholar [29] M. Y. Li and Z. 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 [30] 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 [31] 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 [32] W. M. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates,, J. Math. Biol., 25 (1987), 359. doi: 10.1007/BF00277162. Google Scholar [33] A. L. Lloyd and R. M. May, Spatial heterogeneity in epidemic models,, J. Theor. Biol., 179 (1996), 1. doi: 10.1006/jtbi.1996.0042. Google Scholar [34] C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 6 (2009), 603. doi: 10.3934/mbe.2009.6.603. Google Scholar [35] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay--distributed or discrete,, Nonlinear Anal. Real World Appl., 11 (2010), 55. doi: 10.1016/j.nonrwa.2008.10.014. Google Scholar [36] R. K. Miller, "Nonlinear Volterra Integral Equations,", Mathematics Lecture Note Series, (1971). Google Scholar [37] M. E. J. Newman, The spread of epidemic disease on networks,, Phys. Rev. E, 66 (2002). doi: 10.1103/PhysRevE.66.016128. Google Scholar [38] C. P. Simon and J. A. Jacquez, Reproduction numbers and the stability of equilibria of SI models for heterogeneous populations,, SIAM J. Appl. Math., 52 (1992), 541. doi: 10.1137/0152030. Google Scholar [39] P. van den Driessche, Some epidemiological models with delays,, in, (1996), 507. Google Scholar [40] P. van den Driessche, L. Wang and X. Zou, Modeling diseases with latency and relapse,, Math. Biosci. Eng., 4 (2007), 205. Google Scholar [41] P. van den Driessche, L. Wang and X. Zou, Impact of group mixing on disease dynamics,, Math. Biosci., 228 (2010), 71. doi: 10.1016/j.mbs.2010.08.008. Google Scholar [42] 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 [43] 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 [44] Z. Yuan and X. Zou, Global threshold property in an epidemic model for disease with latency spreading in a heterogeneous host population,, Nonlinear Anal. Real World Appl., 11 (2010), 3479. doi: 10.1016/j.nonrwa.2009.12.008. Google Scholar
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