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A Diffusive Sveir Epidemic Model with Time Delay and General Incidence

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Abstract

In this paper, we consider a delayed diffusive SVEIR model with general incidence. We first establish the threshold dynamics of this model. Using a Nonstandard Finite Difference (NSFD) scheme, we then give the discretization of the continuous model. Applying Lyapunov functions, global stability of the equilibria are established. Numerical simulations are presented to validate the obtained results. The prolonged time delay can lead to the elimination of the infectiousness.

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References

  1. Kribs-Zaleta C M, Velasco-Hernández J X. A simple vaccination model with multiple endemic states. Mathematical Biosciences, 2000, 164(2): 183–201

    Article  Google Scholar 

  2. Arino J, McCluskey C C, van den Driess P. Global results for an epidemic model with vaccination that exhibits backward bifurcation. SIAM Journal on Applied Mathematics, 2003, 64(1): 260–276

    Article  MathSciNet  Google Scholar 

  3. Li J, Ma Z, Zhou Y. Global analysis of SIS epidemic model with a simple vaccination and multiple endemic equilibria. Acta Mathematica Scientia, 2006, 26B(1): 83–93

    Article  MathSciNet  Google Scholar 

  4. Liu X, Takeuchi Y, Iwami S. SVIR epidemic models with vaccination strategies. Journal of Theoretical Biology, 2008, 253(1): 1–11

    Article  MathSciNet  Google Scholar 

  5. Li J, Yang Y. SIR-SVS epidemic models with continuous and impulsive vaccination strategies. Journal of Theoretical Biology, 2011, 280(1): 108–116

    Article  MathSciNet  Google Scholar 

  6. Zhu Q, Hao Y, Ma J, Yu S, Wang Y. Surveillance of hand, foot, and mouth disease in mainland china (2008–2009). Biomedical and Environmental Sciences, 2011, 24(4): 349–356

    Google Scholar 

  7. Martcheva M. Avian flu: modeling and implications for control. Journal of Biological Systems, 2014, 22(1): 151–175

    Article  MathSciNet  Google Scholar 

  8. Wang W, Ruan S. Simulating the SARS outbreak in beijing with limited data. Journal of Theoretical Biology, 2004, 227(3): 369–379

    Article  MathSciNet  Google Scholar 

  9. Liljeros F, Edling C R, Amaral L A N. Sexual networks: implications for the transmission of sexually transmitted infections. Microbes and Infection, 2003, 5(2): 189–196

    Article  Google Scholar 

  10. Liu W M, Levin S A, Iwasa Y. Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological model. Journal of Mathematical Biology, 1986, 23(2): 187–204. 18

    Article  MathSciNet  Google Scholar 

  11. Capasso V, Serio G. A generalization of the Kermack-McKendrick deterministic epidemic model. Mathematical Biosciences, 1978, 42(1/2): 43–61

    Article  MathSciNet  Google Scholar 

  12. Hattaf K, Lashari A A, Louartassi Y, Yousfi N. A delayed SIR epidemic model with general incidence rate. Electronic Journal of Qualitative Theory of Differential Equations, 2013(3): 1–9

  13. Wang L, Liu Z, Zhang X. Global dynamics of an SVEIR epidemic model with distributed delay and nonlinear incidence. Applied Mathematics and Computation, 2016, 284: 47–65

    Article  MathSciNet  Google Scholar 

  14. Hattaf K. Global stability and Hopf bifurcation of a generalized viral infection model with multi-delays and humoral immunity. Physica A: Statal Mechanics and its Applications, 2020, 545: 123689

    Article  MathSciNet  Google Scholar 

  15. Wang J, Huang G, Takeuchi Y, Liu S. SVEIR epidemiological model with varying infectivity and distributed delays. Mathematical Biosciences and Engineering, 2011, 8(3): 875–888

    Article  MathSciNet  Google Scholar 

  16. Xu R. Global stability of a delayed epidemic model with latent period and vaccination strategy. Applied Mathematical Modelling, 2012, 36(11): 5293–5300

    Article  MathSciNet  Google Scholar 

  17. Duan X, Yuan S, Li X. Global stability of an SVIR model with age of vaccination. Applied Mathematics and Computation, 2014, 226: 528–540

    Article  MathSciNet  Google Scholar 

  18. Meng X, Chen L, Wu B. A delay SIR epidemic model with pulse vaccination and incubation times. Nonlinear Analysis: Real World Applications, 2010, 11(1): 88–98

    Article  MathSciNet  Google Scholar 

  19. Gao S, Chen L, Nieto J J, Torres A. Analysis of a delayed epidemic model with pulse vaccination and saturation incidence. Vaccine, 2006, 24(35/36): 6037–6045

    Article  Google Scholar 

  20. Zhang T, Zhang T Q, Meng X. Stability analysis of a chemostat model with maintenance energy. Applied Mathematics Letters, 2017, 68: 1–7

    Article  MathSciNet  Google Scholar 

  21. Zhang T, Liu X, Meng X, Zhang T Q. Spatio-temporal dynamics near the steady state of a planktonic system. Computers and Mathematics with Applications, 2018, 75(12): 4490–4504

    Article  MathSciNet  Google Scholar 

  22. Hattaf K, Yousfi N. Global stability for reaction-diffusion equations in biology. Computers Mathematics with Applications, 2013, 66(8): 1488–1497

    Article  MathSciNet  Google Scholar 

  23. Webby R J, Webster R G. Are we ready for pandemic influenza? Science, 2003, 302: 1519–1522

    Article  Google Scholar 

  24. Xu Z, Ai C. Traveling waves in a diffusive influenza epidemic model with vaccination. Applied Mathematical Modelling, 2016, 40(15/16): 7265–7280

    Article  MathSciNet  Google Scholar 

  25. Abdelmalek S, Bendoukha S. Global asymptotic stability of a diffusive SVIR epidemic model with immigration of individuals. Electronic Journal of Differential Equations, 2016, 2016(129/324): 1–14

    MathSciNet  MATH  Google Scholar 

  26. Xu Z, Xu Y, Huang Y. Stability and traveling waves of a vaccination model with nonlinear incidence. Computers and Mathematics with Applications, 2018, 75(2): 561–581

    Article  MathSciNet  Google Scholar 

  27. Mickens R E. Nonstandard Finite Difference Models of Differential Equations. World Scientific, December 1993

  28. Arenas A J, Morano J A, Cortés J C. Non-standard numerical method for a mathematical model of RSV epidemiological transmission. Computers and Mathematics with Applications, 2008, 56(3): 670–678

    Article  MathSciNet  Google Scholar 

  29. Hattaf K, Yousfi N. Global properties of a discrete viral infection model with general incidence rate. Mathematical Methods in the Applied Sciences, 2016, 39(5): 998–1004

    Article  MathSciNet  Google Scholar 

  30. Hattaf K, Yousfi N. A numerical method for delayed partial differential equations describing infectious diseases. Computers and Mathematics with Applications, 2016, 72(11): 2741–2750

    Article  MathSciNet  Google Scholar 

  31. Liu J, Peng B, Zhang T. Effect of discretization on dynamical behavior of SEIR and SIR models with nonlinear incidence. Applied Mathematics Letters, 2015, 39: 60–66

    Article  MathSciNet  Google Scholar 

  32. Muroya Y, Bellen A, Enatsu Y, Nakata Y. Global stability for a discrete epidemic model for disease with immunity and latency spreading in a heterogeneous host population. Nonlinear Analysis: Real World Applications, 2013, 13(1): 258–274

    Article  MathSciNet  Google Scholar 

  33. Qin W, Wang L, Ding X. A non-standard finite difference method for a hepatitis B virus infection model with spatial diffusion. Journal of Difference Equations and Applications, 2014, 20(12): 1641–1651

    Article  MathSciNet  Google Scholar 

  34. Geng Y, Xu J. Stability preserving NSFD scheme for a multi-group SVIR epidemic model. Mathematical Methods in the Applied Sciences, 2017, 40(13): 4917–4927

    MathSciNet  MATH  Google Scholar 

  35. Ding D, Ma Q, Ding X. A non-standard finite difference scheme for an epidemic model with vaccination. Journal of Difference Equations and Applications, 2013, 19(2): 179–190

    Article  MathSciNet  Google Scholar 

  36. Yang Y, Zhou J, Ma X, Zhang T. Nonstandard finite difference scheme for a diffusive within-host virus dynamics model with both virus-to-cell and cell-to-cell transmissions. Computers and Mathematics with Applications, 2016, 72(4): 1013–1020

    Article  MathSciNet  Google Scholar 

  37. Zhou J, Yang Y, Zhang T. Global dynamics of a reaction-diffusion waterborne pathogen model with general incidence rate. Journal of Mathematical Analysis and Applications, 2018, 466(1): 835–859

    Article  MathSciNet  Google Scholar 

  38. Martin R H, Smith H L. Abstract functional differential equations and reaction-diffusion systems. Transactions of the American Mathematical Society, 1990, 321(1): 1–44

    MathSciNet  MATH  Google Scholar 

  39. Fujimoto T, Ranade R R. Two characterizations of inverse-positive matrices: the Hawkins-Simon condition and the Le Chatelier-Braun principle. The Electronic Journal of Linear Algebra, 2004, 11: 59–65

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang International Studies University, Hangzhou, 310023, China

    Jinling Zhou  (周金玲) & Xinsheng Ma  (马新生)

  2. School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai, 201209, China

    Yu Yang  (杨瑜)

  3. Department of Mathematics, Swinburne University of Technology, Hawthorn, Victoria, 3122, Australia

    Tonghua Zhang  (张同华)

Authors
  1. Jinling Zhou  (周金玲)
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  2. Xinsheng Ma  (马新生)
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  3. Yu Yang  (杨瑜)
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  4. Tonghua Zhang  (张同华)
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Corresponding author

Correspondence to Yu Yang  (杨瑜).

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Zhou, J., Ma, X., Yang, Y. et al. A Diffusive Sveir Epidemic Model with Time Delay and General Incidence. Acta Math Sci 41, 1385–1404 (2021). https://doi.org/10.1007/s10473-021-0421-9

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  • DOI: https://doi.org/10.1007/s10473-021-0421-9

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