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

Published: 01 January 2018 Publication History

Abstract

Reactiondiffusion 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 reactiondiffusion 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.

References

[1]
R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
[2]
L.J.S. Allen, B.M. Bolker, Y. Lou, A.L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reactiondiffusion model, Disc. Cont. Dyn. Syst., 21 (2007) 1-20.
[3]
V. Anaya, M. Bendahmane, M. Seplveda, Mathematical and numerical analysis for predatorprey system in a polluted environment, Netw. Heterog. Media, 5 (2010) 813-847.
[4]
S. Anita, V. Capasso, A stabilizability problem for a reactiondiffusion system modelling a class of spatially structured epidemic systems, Nonlinear Anal. RWA, 3 (2002) 453-464.
[5]
M. Banerjee, L. Zhang, Influence of discrete delay on pattern formation in a ratio-dependent prey-predator model, chaos, Solitons Fractals, 67 (2014) 73-81.
[6]
F. Brauer, C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, 2012.
[7]
S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element methods, Springer, 2008.
[8]
F. Capone, V. De Cataldis, R. De Luca, On the stability of a SEIR reaction diffusion model for infectious under Neumann boundary conditions, Acta. Appl. Math., 132 (2014) 165-176.
[9]
T. Carco, S. Glavanakov, Stage-structured infection transmission and a spatial epidemic: a model for lyme disease, Am. Nat., 160 (2002) 348-359.
[10]
L.L. Chang, G.Q. Sun, Z. Wang, Z. Jin, Rich dynamics in a spatial predatorprey model with delay, Appl. Math. Comput., 256 (2015) 540-550.
[11]
S.R. Choudhury, Turing instability in competition models with delay I: Linear theory, SIAM J. Appl. Math., 54 (1994) 1425-1450.
[12]
B.S. Choudhury, B. Nasipuri, Spatio-temporal chaos in a HollingTanner predatorprey model with Holling type-IV functional response, Int. J. Ecol. Econ. Stat., 31 (2013) 19-40.
[13]
B.S. Choudhury, B. Nasipuri, Self-organized spatial patterns due to diffusion in a HollingTanner predatorprey model, Comput. Appl. Math., 34 (2015) 177-195.
[14]
P.G. Ciarlet, The Finite Element Methods for Elliptic Problems, North-Holland, Amsterdam, 1978.
[15]
The c++ software library deal.II. http://www.dealii.org/.
[16]
M. Dehghan, M. Sabouri, A legendre spectral element method on a large spatial domain to solve the predatorprey system modeling interacting populations, Appl. Math. Model., 37 (2013) 1028-1038.
[17]
T. Faria, Stability and bifurcation for a delayed predatorprey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001) 433-463.
[18]
Q.T. Gan, R. Xu, P.H. Yang, Travelling waves of a delayed SIRS epidemic model with spatial diffusion, Nonlinear Anal. RWA, 12 (2011) 52-68.
[19]
M.R. Garvie, J.F. Blowey, A reactiondiffusion system of type part II: numerical analysis, Eur. J. Appl. Math., 16 (2005) 621-646.
[20]
M.R. Garvie, Finite difference schemes for reactiondiffusion equations modeling predatorprey interactions in MATLAB, Bull. Math. Biol., 69 (2007) 931-956.
[21]
M.R. Garvie, C. Trenchea, Finite element approximation of spatially extended predatorprey interactions with the Holling type II functional response, Numer. Math., 107 (2007) 641-667.
[22]
M.R. Garvie, C. Trenchea, Spatiotemporal dynamics of two generic predatorprey models, J. Biol. Dyn., 4 (2010) 559-570.
[23]
M.R. Garvie, C. Trenchea, A three level finite element approximation of a pattern formation model in development biology, Numer. Math., 127 (2014) 397-422.
[24]
M.R. Garvie, J. Burkardt, J. Morgan, Simple finite element methods for approximating predatorprey dynamics in two dimensions using MATLAB, Bull. Math. Biol., 77 (2015) 548-578.
[25]
J.P. Keller, L. Gerardo-Giorda, A. Veneziani, Numerical simulation of a susceptible-exposed-infectious space-continuous model for the spread of rabies in raccoons across a realistic landscape, J. Biol. Dyn., 7 (2013) 31-46.
[26]
A.B. Medvinsky, S.V. Petrovskii, I.A. Tikhonova, H. Malchow, B.L. Li, Spatiotemporal complexity of plankton and fish dynamics, SIAM Rev., 44 (2002) 311-370.
[27]
F.A. Milner, R.J. Zhao, S-I-R model with directed spatial diffusion, Math. Popul. Stud., 15 (2008) 160-181.
[28]
A. Morozov, B.L. Li, On the importance of dimensionality of space in models of space-mediated population persistence, Theor. Popu. Biol., 71 (2007) 278-289.
[29]
J.D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer, 2003.
[30]
C.V. Pao, Coupled nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 196 (1995) 237-265.
[31]
C.V. Pao, Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 198 (1996) 751-779.
[32]
C.V. Pao, Numerical methods for systems of nonlinear parabolic equations with time delays, J. Math. Anal. Appl., 240 (1999) 249-279.
[33]
C.V. Pao, Existence and dynamics of quasilinear parabolic systems with time delays, J. Differ. Equ., 258 (2015) 3248-3285.
[34]
R. Peng, S.Q. Liu, Global stability of the steady states of an SIS epidemic reactiondiffusion model, Nonlinear Anal., 71 (2009) 239-247.
[35]
S. Sen, P. Ghosh, S.S. Riaz, D.S. Ray, Time-delay induced instabilities in reactiondiffusion systems, Phys. Rev. E, 80 (2009) 046212.
[36]
G.Q. Sun, Pattern formation of an epidemic model with diffusion, Nonlinear Dyn., 69 (2012) 1097-1104.
[37]
V. Thome, Galerkin Finite Element Methods for Parabolic Problems, Springer, 2006.
[38]
D. Tilman, P.M. Kareiva, Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions, Princeton University Press, 1997.
[39]
N. Tuncer, M. Martcheva, Analytical and numerical approaches to coexistence of strains in a two strain SIS model with diffusion, J. Biol. Dyn., 6 (2012) 406-439.
[40]
A.M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. B, 237 (1952) 37-72.
[41]
W.D. Wang, X.Q. Zhao, A nonlocal and time-delayed reactiondiffusion model of dengue transmission, SIAM J. Appl. Math, 71 (2011) 147-168.
[42]
G.F. Webb, A reactiondiffusion model for a deterministic diffusive epidemic, J. Math. Anal. Appl., 84 (1981) 150-161.
[43]
A.G. Xiao, G.G. Zhang, J. Zhou, Implicitexplicit time discretization coupled with finite element methods for delayed predatorprey competition reactiondiffusion system, Comput. Math. Appl., 71 (2016) 2106-2123.
[44]
X.C. Zhang, G.Q. Sun, Z. Jin, Spatial dynamics in a predatorprey model with BeddingtonDeangelis functional response, Phys. Rev. E, 85 (2012) 021924.
[45]
T.H. Zhang, H. Zang, Delay-induced turing instability in reactiondiffusion equations, Phys. Rev. E, 90 (2014) 052908.
[46]
W.S. Zhang, Finite Difference Methods for Partial Differential Equations in Science Computation, Higher Education Press, 2006.

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

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        Published In

        cover image Applied Mathematics and Computation
        Applied Mathematics and Computation  Volume 316, Issue C
        January 2018
        431 pages

        Publisher

        Elsevier Science Inc.

        United States

        Publication History

        Published: 01 January 2018

        Author Tags

        1. 35K57
        2. 65M60
        3. 92D25
        4. 92D30
        5. Epidemic model
        6. Finite element
        7. Population model
        8. Reactiondiffusion equation
        9. Time delay

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        • (2024)Maximum-Norm Error Estimates of Fourth-Order Compact and ADI Compact Finite Difference Methods for Nonlinear Coupled Bacterial SystemsJournal of Scientific Computing10.1007/s10915-024-02588-0100:2Online publication date: 18-Jun-2024
        • (2021)Superconvergence analysis of two-grid methods for bacteria equationsNumerical Algorithms10.1007/s11075-020-00882-086:1(123-152)Online publication date: 1-Jan-2021

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