Skip to main content

Advertisement

SpringerLink
Log in

A Non-Standard Finite Difference Scheme of a Multiple Infected Compartments Model for Waterborne Disease

  • Original Research
  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

In this paper, we study a multiple infected compartments model for waterborne diseases which is derived from the continuous case by using the well-known Mickensnon-standard discretization. The positivity of solutions with positive initial conditions and the expressions of equilibria are obtained. By applying analytic techniques and constructing discrete Lyapunov functions, we obtain the results that if \(R_0\le 1\), the disease-free equilibrium is globally asymptotically stable, and if \(R_0>1\) the unique endemic equilibrium is also globally asymptotically stable when the system degenerates the fast-slow system. Furthermore, numerical simulations verify our theoretical results. Our numerical results imply that the decay rate of pathogen in the water has no influence on endemic equilibrium, but it has a significant impact on the peak value of infected individuals.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. WHO. Cholera: global surveillance summary. Wkly. Epidemiol. Rec. 2008 84 309-324 (2009)

  2. Liu, L., et al.: Global, regional, and national causes of child mortality: an updated systematic analysis for 2010 with time trends since 2000. Lancet 379, 2151–2161 (2012)

    Article  Google Scholar 

  3. Goh, K., Teo, S., Lam, S., Ling, M.: Person-to-person transmission of cholera in a psychiatric hospital. J. Infect. 20, 193–200 (1990)

    Article  Google Scholar 

  4. Hartley, D., Morris, J., Smith, D.: A critical element in the ability of V. cholerae to cause epidemics. PLOS Med. 3, 63–69 (2006)

    Article  Google Scholar 

  5. Koelle, K., Pascual, M., Yunus, M.: Pathogen adaptation to seasonal forcing and climate change. Proc. R. Soc. Lond. B 272, 971–977 (2005)

    Article  Google Scholar 

  6. Li, M., Ma, J., van den Driessche, P.: Model for disease dynamics of a waterborne pathogen on a random network. J. Math. Biol. 201, 389–394 (2014)

    Google Scholar 

  7. Miller, J.C.: SIR dynamics in random networks. J. Math. Biol. 62, 349–358 (2011)

    Article  MathSciNet  Google Scholar 

  8. Tien, J.H., Earn, D.J.D.: Multiple transmission pathways and disease dynamics in a waterborne pathogen model. Bull. Math. Biol. 72, 1506–33 (2010)

    Article  MathSciNet  Google Scholar 

  9. Wang, Y., Cao, J.: Global stability of a multiple infected compartments model for waterborne diseases. Commun. Nonlinear Sci. Numer. Simul. 19, 3753–3765 (2014)

    Article  MathSciNet  Google Scholar 

  10. Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York (1994)

    Book  Google Scholar 

  11. Hu, Z., Teng, Z., Zhang, L.: Stability and bifurcation analysis in a discrete SIR epidemic model. Math. Comput. Simul. 97, 80–93 (2014)

    Article  MathSciNet  Google Scholar 

  12. Li, X., Wang, W.: A discrete epidemic model with stage structure. Math. Chaos Solitons Fractals 26, 947–958 (2005)

    Article  MathSciNet  Google Scholar 

  13. Mickens, R.E.: Nonstandard Finite Difference Model of Differential Equations. World Scientific, Singapore (1994)

    MATH  Google Scholar 

  14. Mickens, R.E.: Application of Nonstandard Finite Difference Schemes. World Scientific, Singapore (2000)

    Book  Google Scholar 

  15. Mickens, R.E.: Nonstandard finite difference schemes for differential equations. J. Differ. Equ. Appl. 8, 823–847 (2002)

    Article  MathSciNet  Google Scholar 

  16. Anguelov, R., Lubuma, J.M.S.: Contributions to the mathematics of the nonstandard finite difference method and applications. Numer. Methods Partial. Differ. Equ. 17, 518–543 (2001)

    Article  MathSciNet  Google Scholar 

  17. Anguelov, R., Lubuma, J.M.S.: Nonstandard finite difference method by nonlocal approximation. Math. Comput. Simul. 61, 465–475 (2003)

    Article  MathSciNet  Google Scholar 

  18. Segel, L.: On the validity of the steady state assumption of enzyme kinetics. Bull. Math. Biol. 50, 579–93 (1988)

    Article  MathSciNet  Google Scholar 

  19. Ding, D., Ma, Q., Ding, X.: A non-standard finite difference scheme for an epidemic model with vaccination. J. Differ. Equ. Appl. 19, 179–190 (2013)

    Article  MathSciNet  Google Scholar 

  20. Masaki, S., Emiko, I.: Global dynamics of a discretized SIRS epidemic model with time delay. J. Math. Anal. Appl. 371, 195–202 (2010)

    Article  MathSciNet  Google Scholar 

  21. Masaki, S.: Permanence of a discrete SIRS epidemic model with time delays. Appl. Math. Lett. 23, 1280–1285 (2010)

    Article  MathSciNet  Google Scholar 

  22. Wang, W.: Global behavior of an SEIRS epidemic model with time delays. Appl. Math. Lett. 15, 423–428 (2002)

    Article  MathSciNet  Google Scholar 

  23. Hu, Z., Teng, Z., Jia, C., Zhang, L., Chen, X.: Complex dynamical behaviors in a discrete eco-epidemiological model with disease in prey. Adv. Differ. Equ. 2014, 265 (2014)

    Article  MathSciNet  Google Scholar 

  24. Van den Driessche, P., Watmough, J.: Reproductive numbers and subthreshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)

    Article  MathSciNet  Google Scholar 

  25. Han, D., Sun, M., Li, D.: Epidemic process on activity-driven modular networks. Phys. A 432, 354–362 (2015)

    Article  MathSciNet  Google Scholar 

  26. Han, D., Sun, M.: Can memory and conformism resolve the vaccination dilemma? Phys. A 415, 95–104 (2014)

    Article  Google Scholar 

  27. Han, D., Sun, M., Li, D.: The virus variation model by considering the degree-dependent spreading rate. Phys. A 433, 42–50 (2015)

    Article  MathSciNet  Google Scholar 

  28. Enatsu, Y., Nakata, Y., Muroya, Y.: Global stability for a class of discrete SIR epidemic models. Math. Biosci. 7, 347–361 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The research has been supported by the Natural Science Foundation of China (11261004, 11561004), the Natural Science Foundation of Jiangxi Province (20151BAB201016), and the Postgraduate Innovation Fund of Jiangxi Province (YC2014-S410).

Author information

Authors and Affiliations

  1. Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques, Gannan Normal University, Ganzhou, 341000, China

    Lijun Zhang, Shujing Gao & Qin Zou

Authors
  1. Lijun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shujing Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Qin Zou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shujing Gao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, L., Gao, S. & Zou, Q. A Non-Standard Finite Difference Scheme of a Multiple Infected Compartments Model for Waterborne Disease. Differ Equ Dyn Syst 28, 59–73 (2020). https://doi.org/10.1007/s12591-016-0296-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-016-0296-8

Keywords

Search

Navigation