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.

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.

[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.

[1]

Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37

[2]

Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166

[3]

Lin Zhao, Zhi-Cheng Wang, Liang Zhang. Threshold dynamics of a time periodic and two–group epidemic model with distributed delay. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1535-1563. doi: 10.3934/mbe.2017080

[4]

Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595

[5]

Nitu Kumari, Sumit Kumar, Sandeep Sharma, Fateh Singh, Rana Parshad. Basic reproduction number estimation and forecasting of COVID-19: A case study of India, Brazil and Peru. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2021170

[6]

Ling Xue, Caterina Scoglio. Network-level reproduction number and extinction threshold for vector-borne diseases. Mathematical Biosciences & Engineering, 2015, 12 (3) : 565-584. doi: 10.3934/mbe.2015.12.565

[7]

Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455

[8]

Benjamin B. Kennedy. Multiple periodic solutions of state-dependent threshold delay equations. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1801-1833. doi: 10.3934/dcds.2012.32.1801

[9]

Qingwen Hu. A model of regulatory dynamics with threshold-type state-dependent delay. Mathematical Biosciences & Engineering, 2018, 15 (4) : 863-882. doi: 10.3934/mbe.2018039

[10]

Yijun Lou, Xiao-Qiang Zhao. Threshold dynamics in a time-delayed periodic SIS epidemic model. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 169-186. doi: 10.3934/dcdsb.2009.12.169

[11]

Ningning Ye, Zengyun Hu, Zhidong Teng. Periodic solution and extinction in a periodic chemostat model with delay in microorganism growth. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1361-1384. doi: 10.3934/cpaa.2022022

[12]

Tom Burr, Gerardo Chowell. The reproduction number $R_t$ in structured and nonstructured populations. Mathematical Biosciences & Engineering, 2009, 6 (2) : 239-259. doi: 10.3934/mbe.2009.6.239

[13]

Tianhui Yang, Ammar Qarariyah, Qigui Yang. The effect of spatial variables on the basic reproduction ratio for a reaction-diffusion epidemic model. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3005-3017. doi: 10.3934/dcdsb.2021170

[14]

Attila Dénes, Gergely Röst. Single species population dynamics in seasonal environment with short reproduction period. Communications on Pure and Applied Analysis, 2021, 20 (2) : 755-762. doi: 10.3934/cpaa.2020288

[15]

Xinli Hu. Threshold dynamics for a Tuberculosis model with seasonality. Mathematical Biosciences & Engineering, 2012, 9 (1) : 111-122. doi: 10.3934/mbe.2012.9.111

[16]

Gerardo Chowell, Catherine E. Ammon, Nicolas W. Hengartner, James M. Hyman. Estimating the reproduction number from the initial phase of the Spanish flu pandemic waves in Geneva, Switzerland. Mathematical Biosciences & Engineering, 2007, 4 (3) : 457-470. doi: 10.3934/mbe.2007.4.457

[17]

Benjamin B. Kennedy. A periodic solution with non-simple oscillation for an equation with state-dependent delay and strictly monotonic negative feedback. Discrete and Continuous Dynamical Systems - S, 2020, 13 (1) : 47-66. doi: 10.3934/dcdss.2020003

[18]

Dongfeng Zhang, Junxiang Xu. On the reducibility of analytic quasi-periodic systems with Liouvillean basic frequencies. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1417-1445. doi: 10.3934/cpaa.2022024

[19]

Zhenguo Bai, Yicang Zhou. Threshold dynamics of a bacillary dysentery model with seasonal fluctuation. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 1-14. doi: 10.3934/dcdsb.2011.15.1

[20]

Jinhuo Luo, Jin Wang, Hao Wang. Seasonal forcing and exponential threshold incidence in cholera dynamics. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2261-2290. doi: 10.3934/dcdsb.2017095

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (64)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]