# American Institute of Mathematical Sciences

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

## Global stability for epidemic model with constant latency and infectious periods

 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.
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. [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. [3] E. Beretta and Y. Takeuchi, Global stability of an SIR model with time delays, J. Math. Biol., 33 (1995), 250-260. [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. [5] J. M. Cushing, "An Introduction to Structured Population Dynamics," CBMS-NSF Regional Conference Series in Applied Mathematics, 71, SIAM, 1998. [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. [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. [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. [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. [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. [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. [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. [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. [14] A. V. Melnik and A. Korobeinikov, Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility, (2011), preprint. [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. [16] C. C. McCluskey, Delay versus age-of-infection-global stability, Appl. Math. Comput., 217 (2010), 3046-3049. [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. [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. [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. [20] G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics," Marcel Dekker, New York, 1985. [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. [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.

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##### 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. [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. [3] E. Beretta and Y. Takeuchi, Global stability of an SIR model with time delays, J. Math. Biol., 33 (1995), 250-260. [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. [5] J. M. Cushing, "An Introduction to Structured Population Dynamics," CBMS-NSF Regional Conference Series in Applied Mathematics, 71, SIAM, 1998. [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. [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. [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. [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. [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. [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. [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. [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. [14] A. V. Melnik and A. Korobeinikov, Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility, (2011), preprint. [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. [16] C. C. McCluskey, Delay versus age-of-infection-global stability, Appl. Math. Comput., 217 (2010), 3046-3049. [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. [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. [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. [20] G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics," Marcel Dekker, New York, 1985. [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. [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.
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