# American Institute of Mathematical Sciences

2015, 12(3): 555-564. doi: 10.3934/mbe.2015.12.555

## Threshold dynamics of a periodic SIR model with delay in an infected compartment

 1 School of Mathematics and Statistics, Xidian University, Xi'an, Shaanxi 710071, China

Received  March 2014 Revised  December 2014 Published  January 2015

Threshold dynamics of epidemic models in periodic environments attract more attention. But there are few papers which are concerned with the case where the infected compartments satisfy a delay differential equation. For this reason, we investigate the dynamical behavior of a periodic SIR model with delay in an infected compartment. We first introduce the basic reproduction number $\mathcal {R}_0$ for the model, and then show that it can act as a threshold parameter that determines the uniform persistence or extinction of the disease. Numerical simulations are performed to confirm the analytical results and illustrate the dependence of $\mathcal {R}_0$ on the seasonality and the latent period.
Citation: Zhenguo Bai. Threshold dynamics of a periodic SIR model with delay in an infected compartment. Mathematical Biosciences & Engineering, 2015, 12 (3) : 555-564. doi: 10.3934/mbe.2015.12.555
##### References:
 [1] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436. doi: 10.1007/s00285-006-0015-0. [2] Z. Bai and Y. Zhou, Threshold dynamics of a delayed SEIRS model with pulse vaccination and general nonlinear incidence,, 2014, (). [3] N. Bacaër and E. Ait Dads, Genealogy with seasonality, the basic reproduction number, and the influenza pandemic, J. Math. Biol., 62 (2011), 741-762. doi: 10.1007/s00285-010-0354-8. [4] N. Bacaër and R. Ouifki, Growth rate and basic reproduction number for population models with a simple periodic factor, Math. Biosci., 210 (2007), 647-658. doi: 10.1016/j.mbs.2007.07.005. [5] Z. Bai and Y. Zhou, Existence of multiple periodic solutions for an SIR model with seasonality, Nonlinear Anal., 74 (2011), 3548-3555. doi: 10.1016/j.na.2011.03.008. [6] K. L. Cooke, Stability analysis for a vector disease model, Rocky Mountain J. Math., 9 (1979), 31-42. doi: 10.1216/RMJ-1979-9-1-31. [7] N. C. Grassly and C. Fraser, Seasonal infectious disease epidemiology, Proc. R. Soc. B., 273 (2006), 2541-2550. doi: 10.1098/rspb.2006.3604. [8] 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. [9] J. Lou, Y. Lou and J. Wu, Threshold virus dynamics with impulsive antiretroviral drug effects, J. Math. Biol., 65 (2012), 623-652. doi: 10.1007/s00285-011-0474-9. [10] Y. Lou and X.-Q. Zhao, A climate-based malaria transmission model with structured vector population, SIAM J. Appl. Math., 70 (2010), 2023-2044. doi: 10.1137/080744438. [11] 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. [12] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal. RWA, 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014. [13] W. Ma, M. Song and Y. Takeuchi, Global stability of an SIR epidemic model with time delay, Appl. Math. Lett., 17 (2004), 1141-1145. doi: 10.1016/j.aml.2003.11.005. [14] P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173. [15] Y. Nakata and T. Kuniya, Global dynamics of a class of SEIRS epidemic models in a periodic environment, J. Math. Anal. Appl., 363 (2010), 230-237. doi: 10.1016/j.jmaa.2009.08.027. [16] C. Rebelo, A. Margheri and N. Bacaër, Persistence in seasonally forced epidemiological models, J. Math. Biol., 64 (2012), 933-949. doi: 10.1007/s00285-011-0440-6. [17] C. Rebelo, A. Margheri and N. Bacaër, Persistence in some periodic epidemic models with infection age or constant periods of infection, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1155-1170. doi: 10.3934/dcdsb.2014.19.1155. [18] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995. [19] H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations, J. Integral Equations, 7 (1984), 253-277. [20] H. R. Thieme, Convergence results and Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763. doi: 10.1007/BF00173267. [21] Y. Tang, D. Huang, S. Ruan and W. Zhang, Coexistence of limit cycles and homoclinic loops in a SIRS model wiht a nonlinear incidence rate, SIAM J. Appl. Math., 69 (2008), 621-639. doi: 10.1137/070700966. [22] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Diff. Equat., 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8. [23] 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. [24] D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429. doi: 10.1016/j.mbs.2006.09.025. [25] X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1. [26] F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), 496-516. doi: 10.1016/j.jmaa.2006.01.085.

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##### References:
 [1] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436. doi: 10.1007/s00285-006-0015-0. [2] Z. Bai and Y. Zhou, Threshold dynamics of a delayed SEIRS model with pulse vaccination and general nonlinear incidence,, 2014, (). [3] N. Bacaër and E. Ait Dads, Genealogy with seasonality, the basic reproduction number, and the influenza pandemic, J. Math. Biol., 62 (2011), 741-762. doi: 10.1007/s00285-010-0354-8. [4] N. Bacaër and R. Ouifki, Growth rate and basic reproduction number for population models with a simple periodic factor, Math. Biosci., 210 (2007), 647-658. doi: 10.1016/j.mbs.2007.07.005. [5] Z. Bai and Y. Zhou, Existence of multiple periodic solutions for an SIR model with seasonality, Nonlinear Anal., 74 (2011), 3548-3555. doi: 10.1016/j.na.2011.03.008. [6] K. L. Cooke, Stability analysis for a vector disease model, Rocky Mountain J. Math., 9 (1979), 31-42. doi: 10.1216/RMJ-1979-9-1-31. [7] N. C. Grassly and C. Fraser, Seasonal infectious disease epidemiology, Proc. R. Soc. B., 273 (2006), 2541-2550. doi: 10.1098/rspb.2006.3604. [8] 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. [9] J. Lou, Y. Lou and J. Wu, Threshold virus dynamics with impulsive antiretroviral drug effects, J. Math. Biol., 65 (2012), 623-652. doi: 10.1007/s00285-011-0474-9. [10] Y. Lou and X.-Q. Zhao, A climate-based malaria transmission model with structured vector population, SIAM J. Appl. Math., 70 (2010), 2023-2044. doi: 10.1137/080744438. [11] 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. [12] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal. RWA, 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014. [13] W. Ma, M. Song and Y. Takeuchi, Global stability of an SIR epidemic model with time delay, Appl. Math. Lett., 17 (2004), 1141-1145. doi: 10.1016/j.aml.2003.11.005. [14] P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173. [15] Y. Nakata and T. Kuniya, Global dynamics of a class of SEIRS epidemic models in a periodic environment, J. Math. Anal. Appl., 363 (2010), 230-237. doi: 10.1016/j.jmaa.2009.08.027. [16] C. Rebelo, A. Margheri and N. Bacaër, Persistence in seasonally forced epidemiological models, J. Math. Biol., 64 (2012), 933-949. doi: 10.1007/s00285-011-0440-6. [17] C. Rebelo, A. Margheri and N. Bacaër, Persistence in some periodic epidemic models with infection age or constant periods of infection, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1155-1170. doi: 10.3934/dcdsb.2014.19.1155. [18] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995. [19] H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations, J. Integral Equations, 7 (1984), 253-277. [20] H. R. Thieme, Convergence results and Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763. doi: 10.1007/BF00173267. [21] Y. Tang, D. Huang, S. Ruan and W. Zhang, Coexistence of limit cycles and homoclinic loops in a SIRS model wiht a nonlinear incidence rate, SIAM J. Appl. Math., 69 (2008), 621-639. doi: 10.1137/070700966. [22] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Diff. Equat., 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8. [23] 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. [24] D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429. doi: 10.1016/j.mbs.2006.09.025. [25] X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1. [26] F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), 496-516. doi: 10.1016/j.jmaa.2006.01.085.
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