Abstract
Malaria is a possibly dangerous infection brought about by a parasite. This infection is more normal in nations with heat and humidities. Because of chromosomal changes, the elements of malaria parasites are very mind-boggling to study just as for any predictions. A reaction–diffusion model to characterize the elements inside have Malaria contamination with versatile safe reactions is concentrated in this paper. The aim of the paper is to develop and analyze a spectrally accurate pseudospectral method in time and space to find the approximate solution to the reaction–diffusion model. The approximate solution is represented in terms of basis functions. The spectral coefficients are found in such a way that the residual becomes minimum. Error estimates for interpolating polynomials are derived. The computational experiments are carried out to corroborate the theoretical results and to compare the present method with existing methods in the literature. The registered mathematical outcomes are in great concurrence with those generally accessible in the writing. Simple to apply and accomplish exact arrangement in less time is the solid place of the current strategy.
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References
Anderson RM, May RM (1992) Infectious diseases of humans: dynamics and control. Oxford University Press, Oxford
Anderson RM, May RM, Gupta S (1989) Non-linear phenomena in host-parasite interactions. Parasitology 99(S1):S59–S79
Balyan LK, Mittal AK, Kumar M, Choube M (2020) Stability analysis and highly accurate numerical approximation of fisher’s equations using pseudospectral method. Math Comput Simul 177:86–104
Boyd JP (2001) Chebyshev and Fourier spectral methods. Courier Corporation, North Chelmsford
Cai L, Tuncer N, Martcheva M (2017) How does within host dynamics affect population level dynamics? Insights from an immuno epidemiological model of malaria. Math Methods Appl Sci 40(18):6424–6450
Canuto C, Quarteroni A (1981) Spectral and pseudo-spectral methods for parabolic problems with non periodic boundary conditions. Calcolo 18(3):197–217
Chen H, Wang W, Fu R, Luo J (2015) Global analysis of a mathematical model on malaria with competitive strains and immune responses. Appl Math Comput 259:132–152
Demasse RD, Ducrot A (2013) An age-structured within-host model for multistrain malaria infections. SIAM J Appl Math 73(1):572–593
Elaiw A, Al Agha A (2020) Global analysis of a reaction-diffusion within-host malaria infection model with adaptive immune response. Mathematics 8(4):563
Elaiw AM, Hobiny AD, Al Agha AD (2020) Global dynamics of reaction-diffusion oncolytic M1 virotherapy with immune response. Appl Math Comput 367:124758
Gottlieb D, Hesthaven JS (2001) Spectral methods for hyperbolic problems. J Comput Appl Math 128(1–2):83–131
Gravenor M, Lloyd A (1998) Reply to: Models for the in-host dynamics of malaria revisited: errors in some basic models lead to large over-estimates of growth rates. Parasitology 171:409–410
Hennig J, Ansorg M (2009) A fully pseudospectral scheme for solving singular hyperbolic equations on conformally compactified space–times. J Hyperb Differ Equ 6(01):161–184
Hetzel C, Anderson RM (1996) The within-host cellular dynamics of bloodstage malaria: theoretical and experimental studies. Parasitology 113(1):25–38
Hoshen MB, Heinrich R, Stein WD, Ginsburg H (2000) Mathematical modelling of the within-host dynamics of Plasmodium falciparum. Parasitology 121(3):227–235
Hussaini MY, Streett CL, Zang TA (1983) Spectral methods for partial differential equations (No. NAS 1.26: 172248)
Ierley G, Spencer B, Worthing R (1992) Spectral methods in time for a class of parabolic partial differential equations. J Comput Phys 102(1):88–97
Iggidr A, Kamgang JC, Sallet G, Tewa JJ (2006) Global analysis of new malaria intrahost models with a competitive exclusion principle. SIAM J Appl Math 67(1):260–278
Kantorovich LV (1934) On a new method of approximate solution of partial differential equations. Dokl Akad Nauk SSSR 4:532–536
Khoury DS, Aogo R, Randriafanomezantsoa-Radohery G, McCaw JM, Simpson JA, McCarthy JS, Davenport MP (2018) With in host modeling of blood stage malaria. Immunol Rev 285(1):168–193
Miao H, Teng Z, Abdurahman X, Li Z (2018) Global stability of a diffusive and delayed virus infection model with general incidence function and adaptive immune response. Comput Appl Math 37(3):3780–3805
Mittal AK (2020) A stable time–space Jacobi pseudospectral method for two-dimensional sine-Gordon equation. J Appl Math Comput 63(1):239–264
Mittal AK, Balyan LK (2019) A highly accurate time–space pseudospectral approximation and stability analysis of two dimensional brusselator model for chemical systems. Int J Appl Comput Math 5(5):140
Mittal AK, Balyan LK (2020) Chebyshev pseudospectral approximation of two dimensional fractional Schrödinger equation on a convex and rectangular domain. AIMS Math 5(3):1642–1662
Mittal RC, Goel R, Ahlawat N (2021) An efficient numerical simulation of a reaction–diffusion malaria infection model using B-splines collocation. Chaos Solitons Fractals 143:110566
Nowak M, May RM (2000) Virus dynamics: mathematical principles of immunology and virology: mathematical principles of immunology and virology. Oxford University Press, Oxford
Orwa T, Mbogo R, Luboobi L (2018) Mathematical model for the in-host malaria dynamics subject to malaria vaccines. Lett Biomath 5(1):222–251
Saralamba S, Pan-Ngum W, Maude R, Lee TJ, Lindegårdh N, Chotivanich K, Nosten F, Day N, Socheat D, White N, Dondorp A, White L (2011a) Intrahost modeling of artemisinin resistance in Plasmodium falciparum. Proc Natl Acad Sci 108:397–402
Saralamba S, Pan-Ngum W, Maude RJ, Lee SJ, Tarning J, Lindegårdh N, White LJ (2011b) Intrahost modeling of artemisinin resistance in Plasmodium falciparum. Proc Natl Acad Sci 108(1):397–402
Saul A (1998) Models for the in-host dynamics of malaria revisited: errors in some basic models lead to large over-estimates of growth rates. Parasitology 117(5):405–407
Song T, Wang C, Tian B (2019) Mathematical models for within-host competition of malaria parasites. Math Biosci Eng 16(6):6623–6653
Slater JC (1934) Electronic energy bands in metal. Phys Rev 45:794–801
Takoutsing E, Temgoua A, Yemele D, Bowong S (2019) Dynamics of an intra-host model of malaria with periodic antimalarial treatment. Int J Nonlinear Sci 27:148–164
Tewa JJ, Fokouop R, Mewoli B, Bowong S (2012) Mathematical analysis of a general class of ordinary differential equations coming from within-hosts models of malaria with immune effectors. Appl Math Comput 218(14):7347–7361
Thieme H R (2003) Princeton series in theoretical and computational biology. In: Rosenberg NA, Thieme HR (eds) Mathematics in population biology. Princeton University Press
Tumwiine J, Mugisha JYT, Luboobi LS (2008) On global stability of the intra-host dynamics of malaria and the immune system. J Math Anal Appl 341(2):855–869
Uddin M, Ali H (2018) The space–time kernel-based numerical method for Burgers’ equations. Mathematics 6(10):212
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Communicated by Carla M.A. Pinto.
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Mittal, A.K. A spectrally accurate time–space pseudospectral method for reaction–diffusion Malaria infection model. Comp. Appl. Math. 41, 390 (2022). https://doi.org/10.1007/s40314-022-02094-9
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DOI: https://doi.org/10.1007/s40314-022-02094-9
Keywords
- Reaction–diffusion infection model
- Malaria
- Pseudospectral method
- Sobolev norm
- Chebyshev–Gauss–Lobbato (CGL) points
- RBC (red blood cells)
- Error estimates