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Efficient numerical methods for spatially extended population and epidemic models with time delay

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  • Chang, Lili
  • Jin, Zhen

Abstract

Reaction–diffusion models with time delay have been widely applied in population biology as well as epidemiology. This type of models can possibly exhibit complex dynamical behaviors such as traveling wave, self-organized spatial pattern, or chaos. Numerical methods play an essential role in the study of these dynamical behaviors. This paper concerns the finite element approximation for reaction–diffusion models with time delay. Two fully discrete schemes and corresponding a priori error estimates are derived. Generally, the research on evolution of population and epidemic needs to survey long-time dynamical behaviors of these models, so that it is important to improve the speed of numerical simulation. To this end, interpolation technique is used in our schemes to avoid numerical integration of reaction term. An outstanding advantage of using interpolation of reaction term is that it improves the operation speed greatly, meanwhile does not reduce convergence order. Applications are given to some model problems arising from population biology and epidemiology. From these simulations some interesting phenomena can be found and we try to explain them in biological significance.

Suggested Citation

  • Chang, Lili & Jin, Zhen, 2018. "Efficient numerical methods for spatially extended population and epidemic models with time delay," Applied Mathematics and Computation, Elsevier, vol. 316(C), pages 138-154.
    • Handle: RePEc:eee:apmaco:v:316:y:2018:i:c:p:138-154
    DOI: 10.1016/j.amc.2017.08.028
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    References listed on IDEAS

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    1. Banerjee, Malay & Zhang, Lai, 2014. "Influence of discrete delay on pattern formation in a ratio-dependent prey–predator model," Chaos, Solitons & Fractals, Elsevier, vol. 67(C), pages 73-81.
    2. Fabio Milner & Ruijun Zhao, 2008. "S-I-R Model with Directed Spatial Diffusion," Mathematical Population Studies, Taylor & Francis Journals, vol. 15(3), pages 160-181.
    3. Chang, Lili & Sun, Gui-Quan & Wang, Zhen & Jin, Zhen, 2015. "Rich dynamics in a spatial predator–prey model with delay," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 540-550.
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    Cited by:

    1. Barman, Madhab & Mishra, Nachiketa, 2024. "Hopf bifurcation analysis for a delayed nonlinear-SEIR epidemic model on networks," Chaos, Solitons & Fractals, Elsevier, vol. 178(C).
    2. Zhang, Wenbo & Gu, Wei, 2024. "Machine learning for a class of partial differential equations with multi-delays based on numerical Gaussian processes," Applied Mathematics and Computation, Elsevier, vol. 467(C).
    3. Sajjadi, Samaneh Sadat & Baleanu, Dumitru & Jajarmi, Amin & Pirouz, Hassan Mohammadi, 2020. "A new adaptive synchronization and hyperchaos control of a biological snap oscillator," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).

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