September  2015, 20(7): 1971-1981. doi: 10.3934/dcdsb.2015.20.1971

Lyapunov functions and global stability for a discretized multigroup SIR epidemic model

1. 

Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, China, China, China

Received  January 2014 Revised  January 2015 Published  July 2015

In this paper, a discretized multigroup SIR epidemic model is constructed by applying a nonstandard finite difference schemes to a class of continuous time multigroup SIR epidemic models. This discretization scheme has the same dynamics with the original differential system independent of the time step, such as positivity of the solutions and the stability of the equilibria. Discrete-time analogue of Lyapunov functions is introduced to show that the global asymptotic stability is fully determined by the basic reproduction number $R_0$.
Citation: Deqiong Ding, Wendi Qin, Xiaohua Ding. Lyapunov functions and global stability for a discretized multigroup SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1971-1981. doi: 10.3934/dcdsb.2015.20.1971
References:
[1]

J. Bruggeman, H. Burchard, B. W. Kooi and B. Sommeijer, A second-order, unconditionally positive, mass-conserving integration scheme for biochemical systems,, Appl. Numer. Math., 57 (2007), 36.  doi: 10.1016/j.apnum.2005.12.001.  Google Scholar

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D. Ding and X. Ding, Global stability of multi-group vaccination epidemic models with delays,, Nonlinear Anal.-Real World Appl., 12 (2011), 1991.  doi: 10.1016/j.nonrwa.2010.12.015.  Google Scholar

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D. Ding, X. Wang and X. Ding, Global Stability of Multigroup Dengue Disease Transmission Model,, Journal of Applied Mathematics., 2012 (2012).  doi: 10.1155/2012/342472.  Google Scholar

[4]

D. Ding and X. Ding, A non-standard finite difference scheme for an epidemic model with vaccination,, J. Differ. Equ. Appl., 19 (2013), 179.  doi: 10.1080/10236198.2011.614606.  Google Scholar

[5]

Y. Enatsu, Y. Nakata, Y. Muroya, G. Izzo and A. Vecchio, Global dynamics of difference equations for SIR epidemic models with a class of nonlinear incidence rates,, J. Differ. Equ. Appl., 18 (2012), 1163.  doi: 10.1080/10236198.2011.555405.  Google Scholar

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Y. Enatsu, Y. Nakata and Y. Muroya, Global stability for a class of discrete SIR epidemic models,, Math. Biosci. Eng., 7 (2010), 347.  doi: 10.3934/mbe.2010.7.347.  Google Scholar

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H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases,, Math. Biosci. Eng., 3 (2006), 513.  doi: 10.3934/mbe.2006.3.513.  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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H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review., 42 (2000), 599.  doi: 10.1137/S0036144500371907.  Google Scholar

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R. A. Horn and C. R. Johnson, Martrix Analysis,, Post and Telecom Press, (2005).   Google Scholar

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S. Jang and N. Elaydi, Difference equations from discretization of a continuous epidemic model with immigration of infectives,, Can. Appl. Math. Q., 11 (2003), 93.   Google Scholar

[12]

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

[13]

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. 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

[15]

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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R. E. Mickens, In: Applications of Nonstandard Finite Difference Schemes,, World Scientific, (2000).  doi: 10.1142/9789812813251.  Google Scholar

[17]

R. E. Mickens, Advances in the Applications of Nonstandard Finite Diffference Schemes,, World Scientific, (2005).  doi: 10.1142/9789812703316.  Google Scholar

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R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations,, World Scientific, (1994).   Google Scholar

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S. M. Moghadas and A. B. Gumel, A mathematical study of a model for childhood diseases with non-permanent immunity,, J. Comput. Appl. Math., 157 (2003), 347.  doi: 10.1016/S0377-0427(03)00416-3.  Google Scholar

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K. C. Patidar, On the use of nonstandard finite difference methods,, J. Differ. Equ. Appl. 11 (2005), 11 (2005), 735.  doi: 10.1080/10236190500127471.  Google Scholar

[21]

L.-I. W. Roeger, Nonstandard finite-difference schemes for the Lotka-Volterra systems: Generalization of Mickens's method,, J. Differ. Equ. Appl., 12 (2006), 937.  doi: 10.1080/10236190600909380.  Google Scholar

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L.-I. W. Roeger, Dynamically consistent discrete Lotka-Volterra competition models derived from nonstandard finite-difference schemes,, Discret. Contin. Dyn. Syst. B., 9 (2008), 415.  doi: 10.3934/dcdsb.2008.9.415.  Google Scholar

[23]

L.-I. W. Roeger and R. Gelca, Dynamically consistent discrete-time Lotka-Volterra competition models,, Discret. Contin. Dyn. Syst., (2009), 650.   Google Scholar

[24]

L.-I. W. Roeger, Exact nonstandard finite-difference methods for a linear system-the case of centers,, J. Differ. Equ. Appl., 14 (2008), 381.  doi: 10.1080/10236190701607669.  Google Scholar

[25]

L.-I. W. Roeger, A nonstandard discretization method for Lotka-Volterra models that preserves periodic solutions,, J. Differ. Equ. Appl., 11 (2005), 721.  doi: 10.1080/10236190500127612.  Google Scholar

[26]

M. Sekiguchi and E. Ishiwata, Global dynamics of a discretized SIRS epidemic model with time delay,, J. Math. Anal. Appl., 371 (2010), 195.  doi: 10.1016/j.jmaa.2010.05.007.  Google Scholar

[27]

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

[28]

Y. Wang, Dynamics of a nonstandard finite-difference scheme for delay differential equations with unimodal feedback,, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 3967.  doi: 10.1016/j.cnsns.2012.02.023.  Google Scholar

[29]

Z. Yuan and L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates,, Nonlinear Anal.-Real World Appl., 11 (2010), 995.  doi: 10.1016/j.nonrwa.2009.01.040.  Google Scholar

[30]

D. Zwillinger, Handbook of Differential Equations,, Academic Press, (1989).   Google Scholar

show all references

References:
[1]

J. Bruggeman, H. Burchard, B. W. Kooi and B. Sommeijer, A second-order, unconditionally positive, mass-conserving integration scheme for biochemical systems,, Appl. Numer. Math., 57 (2007), 36.  doi: 10.1016/j.apnum.2005.12.001.  Google Scholar

[2]

D. Ding and X. Ding, Global stability of multi-group vaccination epidemic models with delays,, Nonlinear Anal.-Real World Appl., 12 (2011), 1991.  doi: 10.1016/j.nonrwa.2010.12.015.  Google Scholar

[3]

D. Ding, X. Wang and X. Ding, Global Stability of Multigroup Dengue Disease Transmission Model,, Journal of Applied Mathematics., 2012 (2012).  doi: 10.1155/2012/342472.  Google Scholar

[4]

D. Ding and X. Ding, A non-standard finite difference scheme for an epidemic model with vaccination,, J. Differ. Equ. Appl., 19 (2013), 179.  doi: 10.1080/10236198.2011.614606.  Google Scholar

[5]

Y. Enatsu, Y. Nakata, Y. Muroya, G. Izzo and A. Vecchio, Global dynamics of difference equations for SIR epidemic models with a class of nonlinear incidence rates,, J. Differ. Equ. Appl., 18 (2012), 1163.  doi: 10.1080/10236198.2011.555405.  Google Scholar

[6]

Y. Enatsu, Y. Nakata and Y. Muroya, Global stability for a class of discrete SIR epidemic models,, Math. Biosci. Eng., 7 (2010), 347.  doi: 10.3934/mbe.2010.7.347.  Google Scholar

[7]

H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases,, Math. Biosci. Eng., 3 (2006), 513.  doi: 10.3934/mbe.2006.3.513.  Google Scholar

[8]

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

[9]

H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review., 42 (2000), 599.  doi: 10.1137/S0036144500371907.  Google Scholar

[10]

R. A. Horn and C. R. Johnson, Martrix Analysis,, Post and Telecom Press, (2005).   Google Scholar

[11]

S. Jang and N. Elaydi, Difference equations from discretization of a continuous epidemic model with immigration of infectives,, Can. Appl. Math. Q., 11 (2003), 93.   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

R. E. Mickens, In: Applications of Nonstandard Finite Difference Schemes,, World Scientific, (2000).  doi: 10.1142/9789812813251.  Google Scholar

[17]

R. E. Mickens, Advances in the Applications of Nonstandard Finite Diffference Schemes,, World Scientific, (2005).  doi: 10.1142/9789812703316.  Google Scholar

[18]

R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations,, World Scientific, (1994).   Google Scholar

[19]

S. M. Moghadas and A. B. Gumel, A mathematical study of a model for childhood diseases with non-permanent immunity,, J. Comput. Appl. Math., 157 (2003), 347.  doi: 10.1016/S0377-0427(03)00416-3.  Google Scholar

[20]

K. C. Patidar, On the use of nonstandard finite difference methods,, J. Differ. Equ. Appl. 11 (2005), 11 (2005), 735.  doi: 10.1080/10236190500127471.  Google Scholar

[21]

L.-I. W. Roeger, Nonstandard finite-difference schemes for the Lotka-Volterra systems: Generalization of Mickens's method,, J. Differ. Equ. Appl., 12 (2006), 937.  doi: 10.1080/10236190600909380.  Google Scholar

[22]

L.-I. W. Roeger, Dynamically consistent discrete Lotka-Volterra competition models derived from nonstandard finite-difference schemes,, Discret. Contin. Dyn. Syst. B., 9 (2008), 415.  doi: 10.3934/dcdsb.2008.9.415.  Google Scholar

[23]

L.-I. W. Roeger and R. Gelca, Dynamically consistent discrete-time Lotka-Volterra competition models,, Discret. Contin. Dyn. Syst., (2009), 650.   Google Scholar

[24]

L.-I. W. Roeger, Exact nonstandard finite-difference methods for a linear system-the case of centers,, J. Differ. Equ. Appl., 14 (2008), 381.  doi: 10.1080/10236190701607669.  Google Scholar

[25]

L.-I. W. Roeger, A nonstandard discretization method for Lotka-Volterra models that preserves periodic solutions,, J. Differ. Equ. Appl., 11 (2005), 721.  doi: 10.1080/10236190500127612.  Google Scholar

[26]

M. Sekiguchi and E. Ishiwata, Global dynamics of a discretized SIRS epidemic model with time delay,, J. Math. Anal. Appl., 371 (2010), 195.  doi: 10.1016/j.jmaa.2010.05.007.  Google Scholar

[27]

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

[28]

Y. Wang, Dynamics of a nonstandard finite-difference scheme for delay differential equations with unimodal feedback,, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 3967.  doi: 10.1016/j.cnsns.2012.02.023.  Google Scholar

[29]

Z. Yuan and L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates,, Nonlinear Anal.-Real World Appl., 11 (2010), 995.  doi: 10.1016/j.nonrwa.2009.01.040.  Google Scholar

[30]

D. Zwillinger, Handbook of Differential Equations,, Academic Press, (1989).   Google Scholar

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