American Institute of Mathematical Sciences

Advanced
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

Zhenguo Bai 1,

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.
Keywords: periodic solution., Basic reproduction number, delay, threshold dynamics.
Mathematics Subject Classification: Primary: 34K13, 92D30; Secondary: 37N2.
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.  Google Scholar

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Z. Bai and Y. Zhou, Threshold dynamics of a delayed SEIRS model with pulse vaccination and general nonlinear incidence,, 2014, ().   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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N. C. Grassly and C. Fraser, Seasonal infectious disease epidemiology, Proc. R. Soc. B., 273 (2006), 2541-2550. doi: 10.1098/rspb.2006.3604.  Google Scholar

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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

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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.  Google Scholar

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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.  Google Scholar

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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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations, J. Integral Equations, 7 (1984), 253-277.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

Z. Bai and Y. Zhou, Threshold dynamics of a delayed SEIRS model with pulse vaccination and general nonlinear incidence,, 2014, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

H. R. Thieme, Renewal theorems for linear periodic Volterra integral equations, J. Integral Equations, 7 (1984), 253-277.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

[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.  Google Scholar

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