EDP Sciences logo
Subscriber Authentication Point
Search
Menu
All issues Volume 17 (2022) Math. Model. Nat. Phenom., 17 (2022) 3 References
Issue
Math. Model. Nat. Phenom.
Volume 17, 2022
Fractional Dynamics in Natural Phenomena
Article Number 3
Number of page(s) 21
DOI https://doi.org/10.1051/mmnp/2022003
Published online 03 February 2022
  1. K.K. Ali, M.S. Osman, H.M. Baskonus, N.S. Elazabb and E. İlhan, Analytical and numerical study of the HIV-1 infection of CD4+ T-cells conformable fractional mathematical model that causes acquired immunodeficiency syndrome with the effect of antiviral drug therapy. Math. Methods Appl. Sci. (2020). [Google Scholar]
  2. S. Ariotti, J.B. Beltman, R. Borsje, M.E. Hoekstra, W.P. Halford, J.B. Haanen, R.J. de Boer and T.N. Schumacher, Subtle CXCR3-dependent chemotaxis of CTLs within infected tissue allows efficient target localization. J. Immunol. 195 (2015) 5285–5295. [CrossRef] [PubMed] [Google Scholar]
  3. R. Atkinson, C. Rhodes, D. Macdonald and R. Anderson, Scale-free dynamics in the movement patterns of jackals. Oikos 98 (2002) 134–140. [CrossRef] [Google Scholar]
  4. D. Baleanu, B. Ghanbari, J.H. Asad, A. Jajarmi and H.M. Pirouz, Planar system-masses in an equilateral triangle: numerical study within fractional calculus (2020). [Google Scholar]
  5. D. Baleanu, A. Jajarmi, S.S. Sajjadi and J.H. Asad, The fractional features of a harmonic oscillator with position-dependent mass. Commun. Theor. Phys. 72 (2020) 055002. [Google Scholar]
  6. D. Baleanu, H. Mohammadi and S. Rezapour, Analysis of the model of HIV-1 infection of CD4+ T-cell with a new approach of fractional derivative. Adv. Differ. Equ. 2020 (2020) 1–17. [CrossRef] [Google Scholar]
  7. M. Berenreiterová, J. Flegr, A.A. Kuběna and P. Němec, The distribution of Toxoplasma gondii cysts in the brain of a mouse with latent toxoplasmosis: implications for the behavioral manipulation hypothesis, PloS One 6 (2011) e28925. [CrossRef] [PubMed] [Google Scholar]
  8. G. Bocharov, A. Meyerhans, N. Bessonov, S. Trofimchuk and V. Volpert, Spatiotemporal dynamics of virus infection spreading in tissues. PloS One 11 (2016) e0168576. [CrossRef] [PubMed] [Google Scholar]
  9. D. Boyer, G. Ramos-Fernández, O. Miramontes, J.L. Mateos, G. Cocho, H. Larralde, H. Ramos and F. Rojas, Scale-free foraging by primates emerges from their interaction with a complex environment. Proc. Roy. Soc. B 273 (2006) 1743–1750. [CrossRef] [PubMed] [Google Scholar]
  10. J. Cannon, F. Asperti-Boursin, M. Fricke, K. Letendre and M. Moses, T cell motility within lymph nodes fits a Levy signature (CAM1P. 225) (2014). [Google Scholar]
  11. Á. Cartea and D. del Castillo-Negrete, Fluid limit of the continuous-time random walk with general Lévy jump distribution functions. Phys. Rev. E 76 (2007) 041105. [CrossRef] [PubMed] [Google Scholar]
  12. A. Cartea and D. del Castillo-Negrete, Fractional diffusion models of option prices in markets with jumps. Physica A 374 (2007) 749–763. [CrossRef] [Google Scholar]
  13. J.M. Chambers, C.L. Mallows and B. Stuck, A method for simulating stable random variables. J. Am. Stat. Assoc. 71 (1976) 340–344. [CrossRef] [Google Scholar]
  14. M. Chaplain and A. Matzavinos, Mathematical modelling of spatio-temporal phenomena in tumour immunology, in Tutorials in Mathematical Biosciences III. Springer (2006) 131–183. [CrossRef] [Google Scholar]
  15. D. del Castillo-Negrete, B. Carreras and V. Lynch, Fractional diffusion in plasma turbulence. Phys. Plasmas 11 (2004) 3854–3864. [CrossRef] [Google Scholar]
  16. D. del Castillo-Negrete, B. Carreras and V. Lynch, Nondiffusive transport in plasma turbulence: a fractional diffusion approach. Phys. Rev. Lett. 94 (2005) 065003. [CrossRef] [PubMed] [Google Scholar]
  17. E.Y. Denkers, T lymphocyte-dependent effector mechanisms of immunity to Toxoplasma gondii. Microbes Infect. 1 (1999) 699–708. [CrossRef] [Google Scholar]
  18. E.Y. Denkers and R.T. Gazzinelli, Regulation and function of T-cell-mediated immunity during Toxoplasma gondii infection. Clin. Microbiol. Rev. 11 (1998) 569–588. [CrossRef] [PubMed] [Google Scholar]
  19. Y. Ding and H. Ye, A fractional-order differential equation model of HIV infection of CD4+ T-cells. Math. Comput. Model. 50 (2009) 386–392. [CrossRef] [MathSciNet] [Google Scholar]
  20. J.P. Dubey, Toxoplasmosis of animals and humans. CRC Press (2016). [CrossRef] [Google Scholar]
  21. D. Ferguson, W. Hutchison and E. Pettersen, Tissue cyst rupture in mice chronically infected with Toxoplasma gondii. Parasitol. Res. 75 (1989) 599–603. [CrossRef] [Google Scholar]
  22. R.T. Gazzinelli, M. Wysocka, S. Hayashi, E.Y. Denkers, S. Hieny, P. Caspar, G. Trinchieri and A. Sher, Parasite-induced IL-12 stimulates early IFN-gamma synthesis and resistance during acute infection with Toxoplasma gondii. J. Immunol. 153 (1994) 2533–2543. [Google Scholar]
  23. S. Halonen, W. Lyman and F. Chiu, Growth and development of Toxoplasma gondii in human neurons and astrocytes. J. Neuropathol. Exp. Neurol. 55 (1996) 1150–1156. [CrossRef] [PubMed] [Google Scholar]
  24. M.M. Hampton, Congenital toxoplasmosis: a review. Neonatal Netw. 34 (2015) 274–278. [CrossRef] [PubMed] [Google Scholar]
  25. E. Hanert, A comparison of three Eulerian numerical methods for fractional-order transport models. Environ. Fluid Mech. 10 (2010) 7–20. [CrossRef] [Google Scholar]
  26. E. Hanert, On the numerical solution of space-time fractional diffusion models. Comput. Fluids 46 (2011) 33–39. [CrossRef] [MathSciNet] [Google Scholar]
  27. E. Hanert, Front dynamics in a two-species competition model driven by Lévy flights. J. Theor. Biol. 300 (2012) 134–142. [CrossRef] [Google Scholar]
  28. E. Hanert and C. Piret, A Chebyshev pseudospectral method to solve the space-time tempered fractional diffusion equation. SIAM J. Sci. Comput. 36 (2014) A1797–A1812. [CrossRef] [Google Scholar]
  29. E. Hanert, E. Schumacher and E. Deleersnijder, Front dynamics in fractional-order epidemic models. J. Theor. Biol. 279 (2011) 9–16. [CrossRef] [Google Scholar]
  30. T.H. Harris, E.J. Banigan, D.A. Christian, C. Konradt, E.D.T. Wojno, K. Norose, E.H. Wilson, B. John, W. Weninger, A.D. Luster et al., Generalized Lévy walks and the role of chemokines in migration of effector CD8+ T cells. Nature 486 (2012) 545–548. [CrossRef] [PubMed] [Google Scholar]
  31. B.T. Hawkins and T.P. Davis, The blood-brain barrier/neurovascular unit in health and disease. Pharmacolog. Rev. 57 (2005) 173–185. [CrossRef] [PubMed] [Google Scholar]
  32. N.E. Humphries, N. Queiroz, J.R. Dyer, N.G. Pade, M.K. Musyl, K.M. Schaefer, D.W. Fuller, J.M. Brunnschweiler, T.K. Doyle, J.D. Houghton et al., Environmental context explains Lévy and Brownian movement patterns of marine predators. Nature 465 (2010) 1066–1069. [CrossRef] [PubMed] [Google Scholar]
  33. A. Jajarmi and D. Baleanu, A new fractional analysis on the interaction of HIV with CD4+ T-cells. Chaos Solitons Fract. 113 (2018) 221–229. [CrossRef] [Google Scholar]
  34. A. Jajarmi and D. Baleanu, A new iterative method for the numerical solution of high-order non-linear fractional boundary value problems. Front. Phys. 8 (2020) 220. [Google Scholar]
  35. A. Jajarmi and D. Baleanu, On the fractional optimal control problems with a general derivative operator. Asian J. Control 23 (2021) 1062–1071. [Google Scholar]
  36. B. John, T.H. Harris, E.D. Tait, E.H. Wilson, B. Gregg, L.G. Ng, P. Mrass, D.S. Roos, F. Dzierszinski, W. Weninger et al., Dynamic imaging of CD8+ T cells and dendritic cells during infection with Toxoplasma gondii. PLoS Pathog 5 (2009) e1000505. [CrossRef] [PubMed] [Google Scholar]
  37. J.F. Kelly, H. Sankaranarayanan and M.M. Meerschaert, Boundary conditions for two-sided fractional diffusion. J. Comput. Phys. 376 (2019) 1089–1107. [CrossRef] [MathSciNet] [Google Scholar]
  38. J. Klafter and I.M. Sokolov, First steps in random walks: from tools to applications. Oxford University Press (2011). [CrossRef] [Google Scholar]
  39. C. Konradt, N. Ueno, D.A. Christian, J.H. Delong, G.H. Pritchard, J. Herz, D.J. Bzik, A.A. Koshy, D.B. McGavern, M.B. Lodoen et al., Endothelial cells are a replicative niche for entry of Toxoplasma gondii to the central nervous system. Nat. Microbiol. 1 (2016) 1–8. [CrossRef] [Google Scholar]
  40. M.F. Krummel, F. Bartumeus and A. Gérard, T cell migration, search strategies and mechanisms. Nat. Rev. Immunol. 16 (2016) 193. [CrossRef] [PubMed] [Google Scholar]
  41. V.A. Kuznetsov, I.A. Makalkin, M.A. Taylor and A.S. Perelson, Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. Bull. Math. Biol. 56 (1994) 295–321. [Google Scholar]
  42. T.A. Landrith, T.H. Harris and E.H. Wilson, Characteristics and critical function of CD8+ T cells inthe Toxoplasma-infected brain, in vol. 37 of Seminars in immunopathology. Springer (2015) 261–270. [CrossRef] [PubMed] [Google Scholar]
  43. H. Li, S. Qi, H. Jin, Z. Qi, Z. Zhang, L. Fu and Q. Luo, Zigzag generalized levy walk: the in vivo search strategy of immunocytes. Theranostics 5 (2015) 1275. [CrossRef] [PubMed] [Google Scholar]
  44. M. Lihoreau, T.C. Ings, L. Chittka and A.M. Reynolds, Signatures of a globally optimal searching strategy in the three-dimensional foraging flights of bumblebees. Sci. Rep. 6 (2016) 1–13. [CrossRef] [Google Scholar]
  45. C.G. Lüder, M. Giraldo-Velásquez, M. Sendtner and U. Gross, Toxoplasma gondii in primary rat CNS cells: differential contribution of neurons, astrocytes, and microglial cells for the intracerebral development and stage differentiation. Exp. Parasitol. 93 (1999) 23–32. [CrossRef] [Google Scholar]
  46. B.J. Luftand J.S. Remington, Toxoplasmic encephalitis. J. Infect. Dis. 157 (1988) 1–6. [CrossRef] [PubMed] [Google Scholar]
  47. R. McCabe, J. Remington et al., Toxoplasma gondii. Principles and Practice of Infectious Diseases (1990) 2090–2103. [Google Scholar]
  48. R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000) 1–77. [Google Scholar]
  49. M.J. Miller, S.H. Wei, I. Parker and M.D. Cahalan, Two-photon imaging of lymphocyte motility and antigen response in intact lymph node. Science 296 (2002) 1869–1873. [CrossRef] [PubMed] [Google Scholar]
  50. F. Mohammadi, L. Moradi, D. Baleanu and A. Jajarmi, A hybrid functions numerical scheme for fractional optimal control problems: application to nonanalytic dynamic systems. J. Vibr. Control 24 (2018) 5030–5043. [Google Scholar]
  51. E.W. Montroll and G.H. Weiss, Random walks on lattices. II. J. Math. Phys. 6 (1965) 167–181. [CrossRef] [Google Scholar]
  52. M.E. Moses, J.L. Cannon, D.M. Gordon and S. Forrest, Distributed adaptive search in T cells: lessons from ants. Front. Immunol. 10 (2019) 1357. [CrossRef] [Google Scholar]
  53. P. Mrass, S.R. Oruganti, G.M. Fricke, J. Tafoya, J.R. Byrum, L. Yang, S.L. Hamilton, M.J. Miller, M.E. Moses and J.L. Cannon, ROCK regulates the intermittent mode of interstitial T cell migration in inflamed lungs, Nat. Commun. 8 (2017) 1–14. [CrossRef] [Google Scholar]
  54. T. Okada, M.J. Miller, I. Parker, M.F. Krummel, M. Neighbors, S.B. Hartley, A. O’Garra, M.D. Cahalan and J.G. Cyster, Antigen-engaged B cells undergo chemotaxis toward the T zone and form motile conjugates with helper T cells. PLoS Biol. 3 (2005) e150. [CrossRef] [PubMed] [Google Scholar]
  55. A. Okubo and S.A. Levin, Vol. 14 of Diffusion and ecological problems: modern perspectives. Springer Science & Business Media (2013). [Google Scholar]
  56. K. Oldham and J. Spanier, The fractional calculus theory and applications of differentiation and integration to arbitrary order. Elsevier (1974). [Google Scholar]
  57. W.E. Paul, Fundamental immunology. New York (2003). [Google Scholar]
  58. I. Podlubny, Fractional differential equations, vol. 198 of Mathematics in Science and Engineering. Academic Press, San Diego, Calif, USA (1999). [Google Scholar]
  59. S. Preston, S. Waters, O. Jensen, P. Heaton and D. Pritchard, T-cell motility in the early stages of the immune response modeled as a random walk amongst targets. Phys. Rev. E 74 (2006) 011910. [CrossRef] [PubMed] [Google Scholar]
  60. J.R. Radke, B. Striepen, M.N. Guerini, M.E. Jerome, D.S. Roos and M.W. White, Defining the cell cycle for the tachyzoite stage of Toxoplasma gondii. Mol. Biochem. Parasitol. 115 (2001) 165–175. [CrossRef] [Google Scholar]
  61. M. Raghib, S.A. Levin and I.G. Kevrekidis, Multiscale analysis of collective motion and decision-making in swarms: an advection–diffusion equation with memory approach. J. Theor. Biol. 264 (2010) 893–913. [CrossRef] [Google Scholar]
  62. A.M. Reynolds and M.A. Frye, Free-flight odor tracking in Drosophila is consistent with an optimal intermittent scale-free search. PloS One 2 (2007) e354. [CrossRef] [PubMed] [Google Scholar]
  63. A.M. Reynolds, A.D. Smith, R. Menzel, U. Greggers, D.R. Reynolds and J.R. Riley, Displaced honey bees perform optimal scale-free search flights. Ecology 88 (2007) 1955–1961. [CrossRef] [PubMed] [Google Scholar]
  64. A.M. Reynolds, A.D. Smith, D.R. Reynolds, N.L. Carreck and J.L. Osborne, Honeybees perform optimal scale-free searching flights when attempting to locate a food source. J. Exp. Biol. 210 (2007) 3763–3770. [CrossRef] [PubMed] [Google Scholar]
  65. T. Riggs, A. Walts, N. Perry, L. Bickle, J.N. Lynch, A. Myers, J. Flynn, J.J. Linderman, M.J. Miller and D.E. Kirschner, A comparison of random vs. chemotaxis-driven contacts of T cells with dendritic cells during repertoire scanning. J. Theor. Biol. 250 (2008) 732–751. [Google Scholar]
  66. S.S. Sajjadi, D. Baleanu, A. Jajarmi and H.M. Pirouz, A new adaptive synchronization and hyperchaos control of a biological snap oscillator. Chaos Solitons Fractals 138 (2020) 109919. [Google Scholar]
  67. E. Scalas, R. Gorenflo and F. Mainardi, Fractional calculus and continuous-time finance. Physica A 284 (2000) 376–384. [CrossRef] [MathSciNet] [Google Scholar]
  68. D. Schlüter, M. Deckert, H. Hof and K. Frei, Toxoplasma gondii infection of neurons induces neuronal cytokine and chemokine production, but gamma interferon-and tumor necrosis factor-stimulated neurons fail to inhibit the invasion and growth of T. gondii. Infect. Immun. 69 (2001) 7889–7893. [CrossRef] [PubMed] [Google Scholar]
  69. D.W. Sims, E.J. Southall, N.E. Humphries, G.C. Hays, C.J. Bradshaw, J.W. Pitchford, A. James, M.Z. Ahmed, A.S. Brierley, M.A. Hindell et al., Scaling laws of marine predator search behavior. Nature 451 (2008) 1098–1102. [CrossRef] [PubMed] [Google Scholar]
  70. S. Skariah, M.K. McIntyre and D.G. Mordue, Toxoplasma gondii: determinants of tachyzoite to bradyzoite conversion. Parasitol. Res. 107 (2010) 253–260. [CrossRef] [PubMed] [Google Scholar]
  71. A. Sullivan, F. Agusto, S. Bewick, C. Su, S. Lenhart and X. Zhao, A mathematical model for within-host Toxoplasma gondii invasion dynamics. Math. Biosci. Eng. 9 (2012) 647. [CrossRef] [MathSciNet] [Google Scholar]
  72. A.M. Sullivan, X. Zhao, Y. Suzuki, E. Ochiai, S. Crutcher and M.A. Gilchrist, Evidence for finely-regulated asynchronous growth of Toxoplasma gondii cysts based on data-driven model selection. PLoS Comput. Biol. 9 (2013) e1003283. [CrossRef] [Google Scholar]
  73. Y. Suzuki, X. Wang, B.S. Jortner, L. Payne, Y. Ni, S.A. Michie, B. Xu, T. Kudo and S. Perkins, Removal of Toxoplasma gondii cysts from the brain by perforin-mediated activity of CD8+ T cells. Am. J. Pathol. 176 (2010) 1607–1613. [CrossRef] [Google Scholar]
  74. A. Tiwari, R. Hannah, J. Lutshumba, E. Ochiai, L.M. Weiss and Y. Suzuki, Penetration of CD8+ cytotoxic T cells into large target, tissue cysts of Toxoplasma gondii, leads to its elimination. Am. J. Pathol. 189 (2019) 1594–1607. [CrossRef] [Google Scholar]
  75. E.F. Torrey and R.H. Yolken, Toxoplasma gondii and schizophrenia. Emerg. Infect. Dis. 9 (2003) 1375. [Google Scholar]
  76. V. Vallaeys, R.C. Tyson, W.D. Lane, E. Deleersnijder and E. Hanert, A Lévy-flight diffusion model to predict transgenic pollen dispersal. J. Royal Soc. Interface 14 (2017) 20160889. [CrossRef] [PubMed] [Google Scholar]
  77. C. Von Economo, Cellular structure of the human cerebral cortex. Karger Medical and Scientific Publishers (2009). [Google Scholar]
  78. X. Wang, S.A. Michie, B. Xu and Y. Suzuki, Importance of IFN-γ-mediated expression of endothelial VCAM-1 on recruitment of CD8+ T cells into the brain during chronic infection with Toxoplasma gondii. J. Interf. Cytokine Res. 27 (2007) 329–338. [Google Scholar]
  79. S.H. Wei, I. Parker, M.J. Miller and M.D. Cahalan, A stochastic view of lymphocyte motility and trafficking within the lymph node. Immunolog. Rev. 195 (2003) 136–159. [CrossRef] [Google Scholar]
  80. L.M. Weiss and K. Kim, Toxoplasma gondii: the model apicomplexan. Perspectives and methods. Elsevier (2011). [Google Scholar]
  81. A. Weron and R. Weron, Computer simulation of Lévy α-stable variables and processes, in Chaos–The interplay between stochastic and deterministic behavior. Springer (1995) 379–392. [Google Scholar]
  82. E.H. Wilson, W. Weninger, C.A. Hunter et al., Trafficking of immune cells in the central nervous system. J. Clin. Invest. 120 (2010) 1368–1379. [CrossRef] [PubMed] [Google Scholar]
  83. C.M. Witt, S. Raychaudhuri, B. Schaefer, A.K. Chakraborty and E.A. Robey, Directed migration of positively selected thymocytes visualized in real time. PLoS Biol. 3 (2005) e160. [CrossRef] [PubMed] [Google Scholar]
  84. D. Wodarz, J.P. Christensen and A.R. Thomsen, The importance of lytic and nonlytic immune responses in viral infections. Trends Immunol. 23 (2002) 194–200. [CrossRef] [PubMed] [Google Scholar]
  85. E.A. Wohlfert, I.J. Blader and E.H. Wilson, Brains and brawn: Toxoplasma infections of the central nervous system and skeletal muscle. Trends Parasitol. 33 (2017) 519–531. [CrossRef] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.