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The Malthusian parameter and $R_0$ for heterogeneous populations in periodic environments
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 
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.
References:
[1] 
J. Arino and P. van den Driessche, Time delays in epidemic models: Modeling and numerical considerations,, in, (2006), 539. 
[2] 
E. Beretta and D. Breda, An SEIR epidemic model with constant latency time and infectious period,, Math. Biosci. Eng., 8 (2011), 931. 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. 
[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. doi: 10.1016/S0362546X(01)005284. 
[5] 
J. M. Cushing, "An Introduction to Structured Population Dynamics,", CBMSNSF Regional Conference Series in Applied Mathematics, 71 (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. 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. doi: 10.1007/s1153800994876. 
[8] 
G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence,, J. Math. Biol., 63 (2011), 125. doi: 10.1007/s0028501003682. 
[9] 
G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for agestructured HIV infection model,, SIAM J. Appl. Math., 72 (2012), 25. 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. doi: 10.3934/mbe.2004.1.57. 
[11] 
A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with nonlinear transmission,, Bull. Math. Biol., 68 (2006), 615. doi: 10.1007/s1153800590379. 
[12] 
A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence,, Bull. Math. Biol., 69 (2007), 1871. doi: 10.1007/s115380079196y. 
[13] 
M. Iannelli, "Mathematical Theory of AgeStructured Population Dynamics,", Applied Mathematical Monographs, 7 (1995). 
[14] 
A. V. Melnik and A. Korobeinikov, Lyapunov functions and global stability for SIR and SEIR models with agedependent susceptibility,, (2011), (2011). 
[15] 
P. Magal, C. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infectionage model,, Appl. Anal., 89 (2010), 1109. doi: 10.1080/00036810903208122. 
[16] 
C. C. McCluskey, Delay versus ageofinfectionglobal stability,, Appl. Math. Comput., 217 (2010), 3046. 
[17] 
C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence,, Nonlinear Anal. RWA, 11 (2010), 3106. 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. doi: 10.3934/mbe.2010.7.837. 
[19] 
P. van den Driessche, Some epidemiological models with delays,, in, (1994), 507. 
[20] 
G. F. Webb, "Theory of Nonlinear AgeDependent Population Dynamics,", Marcel Dekker, (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. 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. doi: 10.1007/s1219000903533. 
show all references
References:
[1] 
J. Arino and P. van den Driessche, Time delays in epidemic models: Modeling and numerical considerations,, in, (2006), 539. 
[2] 
E. Beretta and D. Breda, An SEIR epidemic model with constant latency time and infectious period,, Math. Biosci. Eng., 8 (2011), 931. 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. 
[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. doi: 10.1016/S0362546X(01)005284. 
[5] 
J. M. Cushing, "An Introduction to Structured Population Dynamics,", CBMSNSF Regional Conference Series in Applied Mathematics, 71 (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. 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. doi: 10.1007/s1153800994876. 
[8] 
G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence,, J. Math. Biol., 63 (2011), 125. doi: 10.1007/s0028501003682. 
[9] 
G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for agestructured HIV infection model,, SIAM J. Appl. Math., 72 (2012), 25. 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. doi: 10.3934/mbe.2004.1.57. 
[11] 
A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with nonlinear transmission,, Bull. Math. Biol., 68 (2006), 615. doi: 10.1007/s1153800590379. 
[12] 
A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence,, Bull. Math. Biol., 69 (2007), 1871. doi: 10.1007/s115380079196y. 
[13] 
M. Iannelli, "Mathematical Theory of AgeStructured Population Dynamics,", Applied Mathematical Monographs, 7 (1995). 
[14] 
A. V. Melnik and A. Korobeinikov, Lyapunov functions and global stability for SIR and SEIR models with agedependent susceptibility,, (2011), (2011). 
[15] 
P. Magal, C. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infectionage model,, Appl. Anal., 89 (2010), 1109. doi: 10.1080/00036810903208122. 
[16] 
C. C. McCluskey, Delay versus ageofinfectionglobal stability,, Appl. Math. Comput., 217 (2010), 3046. 
[17] 
C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence,, Nonlinear Anal. RWA, 11 (2010), 3106. 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. doi: 10.3934/mbe.2010.7.837. 
[19] 
P. van den Driessche, Some epidemiological models with delays,, in, (1994), 507. 
[20] 
G. F. Webb, "Theory of Nonlinear AgeDependent Population Dynamics,", Marcel Dekker, (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. 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. doi: 10.1007/s1219000903533. 
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