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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (1995). Google Scholar

[19]

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

[20]

H. R. Thieme, Convergence results and Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755. 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. 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. 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. 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. doi: 10.1016/j.mbs.2006.09.025. Google Scholar

[25]

X.-Q. Zhao, Dynamical Systems in Population Biology,, Springer-Verlag, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (1995). Google Scholar

[19]

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

[20]

H. R. Thieme, Convergence results and Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations,, J. Math. Biol., 30 (1992), 755. 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. 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. 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. 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. doi: 10.1016/j.mbs.2006.09.025. Google Scholar

[25]

X.-Q. Zhao, Dynamical Systems in Population Biology,, Springer-Verlag, (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. doi: 10.1016/j.jmaa.2006.01.085. Google Scholar

[1]

Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & 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 & 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]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

Benjamin B. Kennedy. A periodic solution with non-simple oscillation for an equation with state-dependent delay and strictly monotonic negative feedback. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 47-66. doi: 10.3934/dcdss.2020003

[14]

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

[15]

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

[16]

Aleksa Srdanov, Radiša Stefanović, Aleksandra Janković, Dragan Milovanović. "Reducing the number of dimensions of the possible solution space" as a method for finding the exact solution of a system with a large number of unknowns. Mathematical Foundations of Computing, 2019, 2 (2) : 83-93. doi: 10.3934/mfc.2019007

[17]

Evelyn Sander, E. Barreto, S.J. Schiff, P. So. Dynamics of noninvertibility in delay equations. Conference Publications, 2005, 2005 (Special) : 768-777. doi: 10.3934/proc.2005.2005.768

[18]

N. Romero, A. Rovella, F. Vilamajó. Dynamics of vertical delay endomorphisms. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 409-422. doi: 10.3934/dcdsb.2003.3.409

[19]

Michael Hochman. Lectures on dynamics, fractal geometry, and metric number theory. Journal of Modern Dynamics, 2014, 8 (3&4) : 437-497. doi: 10.3934/jmd.2014.8.437

[20]

Jan Sieber, Matthias Wolfrum, Mark Lichtner, Serhiy Yanchuk. On the stability of periodic orbits in delay equations with large delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3109-3134. doi: 10.3934/dcds.2013.33.3109

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]