American Institute of Mathematical Sciences

Advanced
2012, 9(2): 297-312. doi: 10.3934/mbe.2012.9.297

Global stability for epidemic model with constant latency and infectious periods

Gang Huang 1, , Edoardo Beretta 2, and  Yasuhiro Takeuchi 3,

1. 

School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074, China
2. 

CIMAB, University of Milano, via C. Saldini 50, I20133 Milano
3. 

Graduate School of Science and Technology, Shizuoka University, Hamamatsu, 4328561, Japan

Received  August 2011 Revised  December 2011 Published  March 2012

In recent years many delay epidemiological models have been proposed to study at which stage of the epidemics the delays can destabilize the disease free equilibrium, or the endemic equilibrium, giving rise to stability switches. One of these models is the SEIR model with constant latency time and infectious periods [2], for which the authors have proved that the two delays are harmless in inducing stability switches. However, it is left open the problem of the global asymptotic stability of the endemic equilibrium whenever it exists. Even the Lyapunov functions approach, recently proposed by Huang and Takeuchi to study many delay epidemiological models, fails to work on this model. In this paper, an age-infection model is presented for the delay SEIR epidemic model, such that the properties of global asymptotic stability of the equilibria of the age-infection model imply the same properties for the original delay-differential epidemic model. By introducing suitable Lyapunov functions to study the global stability of the disease free equilibrium (when $\mathcal{R}_0\leq 1$) and of the endemic equilibria (whenever $ \mathcal{R}_0>1$) of the age-infection model, we can infer the corresponding global properties for the equilibria of the delay SEIR model in [2], thus proving that the endemic equilibrium in [2] is globally asymptotically stable whenever it exists.
    Furthermore, we also present a review of the SIR, SEIR epidemic models, with and without delays, appeared in literature, that can be seen as particular cases of the approach presented in the paper.
Keywords: epidemic model, infectious period, Global stability, age-structure..
Mathematics Subject Classification: Primary: 92D30, 34A34; Secondary: 34D20, 34D2.
Citation: Gang Huang, Edoardo Beretta, Yasuhiro Takeuchi. Global stability for epidemic model with constant latency and infectious periods. Mathematical Biosciences & Engineering, 2012, 9 (2) : 297-312. doi: 10.3934/mbe.2012.9.297
References:
[1]

J. Arino and P. van den Driessche, Time delays in epidemic models: Modeling and numerical considerations, in "Delay Differential Equations and Applications" (eds. O. Arino, M. L. Hbid and E. Ait Dads), Springer, (2006), 539-578. Google Scholar

[2]

E. Beretta and D. Breda, An SEIR epidemic model with constant latency time and infectious period, Math. Biosci. Eng., 8 (2011), 931-952. doi: 10.3934/mbe.2011.8.931.  Google Scholar

[3]

E. Beretta and Y. Takeuchi, Global stability of an SIR model with time delays, J. Math. Biol., 33 (1995), 250-260.  Google Scholar

[4]

E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Anal., 47 (2001), 4107-4115. doi: 10.1016/S0362-546X(01)00528-4.  Google Scholar

[5]

J. M. Cushing, "An Introduction to Structured Population Dynamics," CBMS-NSF Regional Conference Series in Applied Mathematics, 71, SIAM, 1998. Google Scholar

[6]

Y. Enatsu, Y. Nakata and Y. Muroya, Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays, Discrete and Continuous Dynamical Systems Ser. B, 15 (2011), 61-74. doi: 10.3934/dcdsb.2011.15.61.  Google Scholar

[7]

G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207. doi: 10.1007/s11538-009-9487-6.  Google Scholar

[8]

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.  Google Scholar

[9]

G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection model, SIAM J. Appl. Math., 72 (2012), 25-38. doi: 10.1137/110826588.  Google Scholar

[10]

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.  Google Scholar

[11]

A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626. doi: 10.1007/s11538-005-9037-9.  Google Scholar

[12]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1889. doi: 10.1007/s11538-007-9196-y.  Google Scholar

[13]

M. Iannelli, "Mathematical Theory of Age-Structured Population Dynamics," Applied Mathematical Monographs, 7, Consiglio Nazionale delle Ricerche, CNR, Giardini Editori e Stampatori in Pisa, Aprile, 1995. Google Scholar

[14]

A. V. Melnik and A. Korobeinikov, Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility, (2011), preprint. Google Scholar

[15]

P. Magal, C. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140. doi: 10.1080/00036810903208122.  Google Scholar

[16]

C. C. McCluskey, Delay versus age-of-infection-global stability, Appl. Math. Comput., 217 (2010), 3046-3049.  Google Scholar

[17]

C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. RWA, 11 (2010), 3106-3109. doi: 10.1016/j.nonrwa.2009.11.005.  Google Scholar

[18]

C. C. McCluskey, Global stability of an SIR epidemic model with delay and general nonlinear incidence, Math. Biosci. Eng., 7 (2010), 837-850. doi: 10.3934/mbe.2010.7.837.  Google Scholar

[19]

P. van den Driessche, Some epidemiological models with delays, in "Differential Equations and Applications to Biology and To Industry" (eds. M. Martell, K. Cooke, E. Cumberbatch, R. Tang and H. Thieme), World Scientific, (1994), 507-520. Google Scholar

[20]

G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics," Marcel Dekker, New York, 1985. Google Scholar

[21]

R. Xu and Z. Ma, Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear Anal. RWA, 10 (2009), 3175-3189. doi: 10.1016/j.nonrwa.2008.10.013.  Google Scholar

[22]

R. Xu and Y. Du, A delayed SIR epidemic model with saturation incidence and a constant infectious period, J. Appl. Math. Comput., 35 (2011), 229-250. doi: 10.1007/s12190-009-0353-3.  Google Scholar

show all references

References:
[1]

J. Arino and P. van den Driessche, Time delays in epidemic models: Modeling and numerical considerations, in "Delay Differential Equations and Applications" (eds. O. Arino, M. L. Hbid and E. Ait Dads), Springer, (2006), 539-578. Google Scholar

[2]

E. Beretta and D. Breda, An SEIR epidemic model with constant latency time and infectious period, Math. Biosci. Eng., 8 (2011), 931-952. doi: 10.3934/mbe.2011.8.931.  Google Scholar

[3]

E. Beretta and Y. Takeuchi, Global stability of an SIR model with time delays, J. Math. Biol., 33 (1995), 250-260.  Google Scholar

[4]

E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Anal., 47 (2001), 4107-4115. doi: 10.1016/S0362-546X(01)00528-4.  Google Scholar

[5]

J. M. Cushing, "An Introduction to Structured Population Dynamics," CBMS-NSF Regional Conference Series in Applied Mathematics, 71, SIAM, 1998. Google Scholar

[6]

Y. Enatsu, Y. Nakata and Y. Muroya, Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays, Discrete and Continuous Dynamical Systems Ser. B, 15 (2011), 61-74. doi: 10.3934/dcdsb.2011.15.61.  Google Scholar

[7]

G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207. doi: 10.1007/s11538-009-9487-6.  Google Scholar

[8]

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.  Google Scholar

[9]

G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection model, SIAM J. Appl. Math., 72 (2012), 25-38. doi: 10.1137/110826588.  Google Scholar

[10]

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.  Google Scholar

[11]

A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626. doi: 10.1007/s11538-005-9037-9.  Google Scholar

[12]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1889. doi: 10.1007/s11538-007-9196-y.  Google Scholar

[13]

M. Iannelli, "Mathematical Theory of Age-Structured Population Dynamics," Applied Mathematical Monographs, 7, Consiglio Nazionale delle Ricerche, CNR, Giardini Editori e Stampatori in Pisa, Aprile, 1995. Google Scholar

[14]

A. V. Melnik and A. Korobeinikov, Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility, (2011), preprint. Google Scholar

[15]

P. Magal, C. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140. doi: 10.1080/00036810903208122.  Google Scholar

[16]

C. C. McCluskey, Delay versus age-of-infection-global stability, Appl. Math. Comput., 217 (2010), 3046-3049.  Google Scholar

[17]

C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. RWA, 11 (2010), 3106-3109. doi: 10.1016/j.nonrwa.2009.11.005.  Google Scholar

[18]

C. C. McCluskey, Global stability of an SIR epidemic model with delay and general nonlinear incidence, Math. Biosci. Eng., 7 (2010), 837-850. doi: 10.3934/mbe.2010.7.837.  Google Scholar

[19]

P. van den Driessche, Some epidemiological models with delays, in "Differential Equations and Applications to Biology and To Industry" (eds. M. Martell, K. Cooke, E. Cumberbatch, R. Tang and H. Thieme), World Scientific, (1994), 507-520. Google Scholar

[20]

G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics," Marcel Dekker, New York, 1985. Google Scholar

[21]

R. Xu and Z. Ma, Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear Anal. RWA, 10 (2009), 3175-3189. doi: 10.1016/j.nonrwa.2008.10.013.  Google Scholar

[22]

R. Xu and Y. Du, A delayed SIR epidemic model with saturation incidence and a constant infectious period, J. Appl. Math. Comput., 35 (2011), 229-250. doi: 10.1007/s12190-009-0353-3.  Google Scholar

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