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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

Zhisheng Shuai 1, and  P. van den Driessche 1,

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.
Keywords: Lyapunov functional., multi-group model, multi-stage model, global stability, SEIR model, heterogeneity.
Mathematics Subject Classification: Primary: 92D30; Secondary: 34K2.
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), 651-661.

[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), 233-266. doi: 10.1093/imammb/1.3.233.

[3]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, Oxford, 1992.

[4]

F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations, Funkcial. Ekvac., 31 (1988), 331-347.

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A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Sciences," Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979.

[6]

K. L. Cooke and P. van den Driessche, Analysis of an SEIRS epidemic model with two delays, J. Math. Biol., 35 (1996), 240-260. doi: 10.1007/s002850050051.

[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), 365-382. doi: 10.1007/BF00178324.

[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), 1511-1536. doi: 10.1007/s11538-006-9174-9.

[9]

H. Guo and M. Y. Li, Global dynamics of a staged-progression model with amelioration for infectious diseases, J. Biol. Dyn., 2 (2008), 154-168. doi: 10.1080/17513750802120877.

[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), 259-284.

[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), 2793-2802. doi: 10.1090/S0002-9939-08-09341-6.

[12]

T. J. Hagenaars, C. A. Donnelly and N. M. Ferguson, Spatial heterogeneity and the persistence of infectious diseases, J. Theor. Biol., 229 (2004), 349-359. doi: 10.1016/j.jtbi.2004.04.002.

[13]

J. K. Hale and S. M. V. Lunel, "Introduction to Functional-Differential Equations," Appl. Math. Sci., Vol. 99, Springer-Verlag, New York, 1993.

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H. W. Hethcote, H. W. Stech, and P. van den Driessche, Stability analysis for models of diseases without immunity, J. Math. Biol., 13 (1981), 185-198. doi: 10.1007/BF00275213.

[15]

H. W. Hethcote and D. W. Tudor, Integral equation models for endemic infectious diseases, J. Math. Biol., 9 (1980), 37-47. doi: 10.1007/BF00276034.

[16]

Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay," Lectures Notes in Math., Vol. 1473, Springer-Verlag, Berlin, 1991.

[17]

G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2011), 125-139. doi: 10.1007/s00285-010-0368-2.

[18]

G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. App. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821.

[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), 77-109. doi: 10.1016/S0025-5564(98)10057-3.

[20]

A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Math. Med. Biol., 21 (2004), 75-83. doi: 10.1093/imammb/21.2.75.

[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), 57-60. doi: 10.3934/mbe.2004.1.57.

[22]

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.

[23]

J. P. LaSalle, "The Stability of Dynamical Systems," With an appendix: Limiting equations and stability of nonautonomous ordinary differential equations, by Z. Artstein, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA, 1976.

[24]

M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology, Math. Biosci., 125 (1995), 155-164. doi: 10.1016/0025-5564(95)92756-5.

[25]

M. Y. Li and J. S. Muldowney, A geometric approach to global stability, SIAM J. Math. Anal., 27 (1996), 1070-1083.

[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), 409-425.

[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), 2434-2448. doi: 10.1137/090779322.

[28]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Bio., 72 (2010), 1429-1505. doi: 10.1007/s11538-010-9503-x.

[29]

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

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

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

[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), 359-380. doi: 10.1007/BF00277162.

[33]

A. L. Lloyd and R. M. May, Spatial heterogeneity in epidemic models, J. Theor. Biol., 179 (1996), 1-11. doi: 10.1006/jtbi.1996.0042.

[34]

C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610. doi: 10.3934/mbe.2009.6.603.

[35]

C. C. McCluskey, Complete global stability for an SIR epidemic model with delay--distributed or discrete, Nonlinear Anal. Real World Appl., 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014.

[36]

R. K. Miller, "Nonlinear Volterra Integral Equations," Mathematics Lecture Note Series, W. A. Benjamin, Inc., Menlo Park, California, 1971.

[37]

M. E. J. Newman, The spread of epidemic disease on networks, Phys. Rev. E, 66 (2002), 016128. doi: 10.1103/PhysRevE.66.016128.

[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), 541-576. doi: 10.1137/0152030.

[39]

P. van den Driessche, Some epidemiological models with delays, in "Differential Equations and Applications to Biology and to Industry" (eds. M. Martelli et al.) (Claremont, CA, 1994), World Sci. Publ., River Edge, (1996), 507-520.

[40]

P. van den Driessche, L. Wang and X. Zou, Modeling diseases with latency and relapse, Math. Biosci. Eng., 4 (2007), 205-219.

[41]

P. van den Driessche, L. Wang and X. Zou, Impact of group mixing on disease dynamics, Math. Biosci., 228 (2010), 71-77. doi: 10.1016/j.mbs.2010.08.008.

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

[43]

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

[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), 3479-3490. doi: 10.1016/j.nonrwa.2009.12.008.

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), 651-661.

[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), 233-266. doi: 10.1093/imammb/1.3.233.

[3]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, Oxford, 1992.

[4]

F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations, Funkcial. Ekvac., 31 (1988), 331-347.

[5]

A. Berman and R. J. Plemmons, "Nonnegative Matrices in the Mathematical Sciences," Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979.

[6]

K. L. Cooke and P. van den Driessche, Analysis of an SEIRS epidemic model with two delays, J. Math. Biol., 35 (1996), 240-260. doi: 10.1007/s002850050051.

[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), 365-382. doi: 10.1007/BF00178324.

[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), 1511-1536. doi: 10.1007/s11538-006-9174-9.

[9]

H. Guo and M. Y. Li, Global dynamics of a staged-progression model with amelioration for infectious diseases, J. Biol. Dyn., 2 (2008), 154-168. doi: 10.1080/17513750802120877.

[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), 259-284.

[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), 2793-2802. doi: 10.1090/S0002-9939-08-09341-6.

[12]

T. J. Hagenaars, C. A. Donnelly and N. M. Ferguson, Spatial heterogeneity and the persistence of infectious diseases, J. Theor. Biol., 229 (2004), 349-359. doi: 10.1016/j.jtbi.2004.04.002.

[13]

J. K. Hale and S. M. V. Lunel, "Introduction to Functional-Differential Equations," Appl. Math. Sci., Vol. 99, Springer-Verlag, New York, 1993.

[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), 185-198. doi: 10.1007/BF00275213.

[15]

H. W. Hethcote and D. W. Tudor, Integral equation models for endemic infectious diseases, J. Math. Biol., 9 (1980), 37-47. doi: 10.1007/BF00276034.

[16]

Y. Hino, S. Murakami and T. Naito, "Functional Differential Equations with Infinite Delay," Lectures Notes in Math., Vol. 1473, Springer-Verlag, Berlin, 1991.

[17]

G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2011), 125-139. doi: 10.1007/s00285-010-0368-2.

[18]

G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. App. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821.

[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), 77-109. doi: 10.1016/S0025-5564(98)10057-3.

[20]

A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Math. Med. Biol., 21 (2004), 75-83. doi: 10.1093/imammb/21.2.75.

[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), 57-60. doi: 10.3934/mbe.2004.1.57.

[22]

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.

[23]

J. P. LaSalle, "The Stability of Dynamical Systems," With an appendix: Limiting equations and stability of nonautonomous ordinary differential equations, by Z. Artstein, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA, 1976.

[24]

M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology, Math. Biosci., 125 (1995), 155-164. doi: 10.1016/0025-5564(95)92756-5.

[25]

M. Y. Li and J. S. Muldowney, A geometric approach to global stability, SIAM J. Math. Anal., 27 (1996), 1070-1083.

[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), 409-425.

[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), 2434-2448. doi: 10.1137/090779322.

[28]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Bio., 72 (2010), 1429-1505. doi: 10.1007/s11538-010-9503-x.

[29]

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

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

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

[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), 359-380. doi: 10.1007/BF00277162.

[33]

A. L. Lloyd and R. M. May, Spatial heterogeneity in epidemic models, J. Theor. Biol., 179 (1996), 1-11. doi: 10.1006/jtbi.1996.0042.

[34]

C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610. doi: 10.3934/mbe.2009.6.603.

[35]

C. C. McCluskey, Complete global stability for an SIR epidemic model with delay--distributed or discrete, Nonlinear Anal. Real World Appl., 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014.

[36]

R. K. Miller, "Nonlinear Volterra Integral Equations," Mathematics Lecture Note Series, W. A. Benjamin, Inc., Menlo Park, California, 1971.

[37]

M. E. J. Newman, The spread of epidemic disease on networks, Phys. Rev. E, 66 (2002), 016128. doi: 10.1103/PhysRevE.66.016128.

[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), 541-576. doi: 10.1137/0152030.

[39]

P. van den Driessche, Some epidemiological models with delays, in "Differential Equations and Applications to Biology and to Industry" (eds. M. Martelli et al.) (Claremont, CA, 1994), World Sci. Publ., River Edge, (1996), 507-520.

[40]

P. van den Driessche, L. Wang and X. Zou, Modeling diseases with latency and relapse, Math. Biosci. Eng., 4 (2007), 205-219.

[41]

P. van den Driessche, L. Wang and X. Zou, Impact of group mixing on disease dynamics, Math. Biosci., 228 (2010), 71-77. doi: 10.1016/j.mbs.2010.08.008.

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

[43]

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

[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), 3479-3490. doi: 10.1016/j.nonrwa.2009.12.008.

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