\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Kink solitary solutions to a hepatitis C evolution model

Abstract / Introduction Full Text(HTML) Figure(9) Related Papers Cited by
  • The standard nonlinear hepatitis C evolution model described in (Reluga et al. 2009) is considered in this paper. The generalized differential operator technique is used to construct analytical kink solitary solutions to the governing equations coupled with multiplicative and diffusive terms. Conditions for the existence of kink solitary solutions are derived. It appears that kink solitary solutions are either in a linear or in a hyperbolic relationship. Thus, a large perturbation in the population of hepatitis infected cells does not necessarily lead to a large change in uninfected cells. Computational experiments are used to illustrate the evolution of transient solitary solutions in the hepatitis C model.

    Mathematics Subject Classification: Primary: 34A34; Secondary: 35C05, 35C07.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Kink solutions $ \widehat{x}, \widehat{y}$ to (112) with $ \widehat{c} = 1$. The black line denotes $ \widehat{x}\left(t\right)$; the gray line denotes $ \widehat{y}\left(t\right)$. In (a), $u = 10, v = -4$; in (b), $u = -2, v = 0$

    Figure 2.  Kink solutions $x, y$ to (112) with $c = 0$. The black line denotes $x\left(\tau\right)$; the gray line denotes $y\left(\tau\right)$. In (a), $u = 10, v = -4$; in (b), $u = -2, v = 0$

    Figure 3.  Phase plot of (112). Black lines denote kink solution trajectories. The gray circle denotes the unstable node (110). The gray dashed line denotes the equilibrium line (111). Gray arrows denote the direction field. The dotted line illustrates that perturbations in infected cell population $y$ lead to proportional changes in uninfected cell population $x$. As the solution evolves from point $A$ to $B$, $y$ increases by $0.46$, while $x$ decreases by $1.09$

    Figure 4.  Plot of error (128) for $c = 0, u = 5, v = 1$. Conditions (126) and (127) hold true. The step size $h$ is $10^{-4}$; error is estimated over $N = 100$ steps. Errors higher than 10 are truncated to 10 for clarity. Note that the error is almost zero on the curve defined by (125)

    Figure 5.  Plot of error (128) for $c = 0, u = 5, v = 1$. Conditions (125) and (127) hold true. The step size $h$ is $10^{-3}$; error is estimated over $N = 30$ steps. Errors higher than 2 are truncated to 2 for clarity. Note that the error is almost zero on the line defined by (126)

    Figure 6.  Plot of error (128) for $c = 0, u = 5, v = 1$. Conditions (125), (126) hold true. The step size $h$ is $10^{-3}$; error is estimated over $N = 30$ steps. Errors higher than 100 are truncated to 100 for clarity. Note that the error is almost zero on the hyperbola defined by (127)

    Figure 7.  Kink solutions to (132) with $ \widehat{c} = 1$. The black line denotes $ \widehat{x}\left(t\right)$; the gray line denotes $ \widehat{y}\left(t\right)$. In (a), $u = 4, v = 1$; in (b), $u = -5, v = 2$

    Figure 8.  Kink solutions to (131) with $c = 0$. The black line denotes $x\left(\tau\right)$; the gray line denotes $y\left(\tau\right)$. In (a), $u = 4, v = 1$; in (b), $u = -5, v = 2$

    Figure 9.  Phase plot of (131). Black lines denote kink solution trajectories. The gray diamond denotes the saddle point (129). The gray dashed line denotes the equilibrium line (130). Gray arrows denote direction field. The dotted line illustrates that large perturbations in infected cell population $y$ lead to small changes in uninfected cell population $x$. As the solution evolves from point $A$ to $B$, $y$ decreases by $5.19$, while $x$ increases by $0.39$

  • [1] N. N. Akhmediev and A. Ankiewicz, Dissipative Solitons: From Optics to Biology and Medicine, Lecture Notes in Physics, 751, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-78217-9.
    [2] E. G. Charalampidis, P. G. Kevrekidis, D. J. Frantzeskakis and B. A. Malomed, Dark-bright solitons in coupled nonlinear Schrödinger equations with unequal dispersion coefficients, Phys. Rev. E, 91 (2015). doi: 10.1103/PhysRevE.91.012924.
    [3] F. CourchampL. Berec and  J. GascoigneAllee Effects in Ecology and Conservation, Oxford University Press, Oxford, 2008. 
    [4] T. Dauxois and  M. PeyrardPhysics of Solitons, Cambridge University Press, Cambridge, 2010. 
    [5] W. Feng, J. Q. Li and Y. Kishimoto, Theory on bright and dark soliton formation in strongly magnetized plasmas, Physics Plasmas, 23 (2016). doi: 10.1063/1.4962846.
    [6] F. Genoud, Extrema of the dynamic pressure in a solitary wave, Nonlinear Anal., 155 (2017), 65-71.  doi: 10.1016/j.na.2017.01.009.
    [7] A. M. GrundlandM. Kovalyov and M. Sussman, Interaction of kink-type solutions of the harmonic map equations, J. Math. Phys., 35 (1994), 6774-6783.  doi: 10.1063/1.530642.
    [8] Y.-H. Hu and S.-Y. Lou, Analytical descriptions of dark and gray solitons in nonlocal nonlinear media, Commun. Theor. Phys. (Beijing), 64 (2015), 665-670.  doi: 10.1088/0253-6102/64/6/665.
    [9] A. KelkarE. Yomba and R. Djeloulli, Solitary wave solutions and modulational instability in a system of coupled complex Newell-Segel-Whitehead equations, Commun. Nonlinear Sci. Numer. Simul., 41 (2016), 118-139.  doi: 10.1016/j.cnsns.2016.04.034.
    [10] J. D. Lambert, Numerical Methods for Ordinary Differential Systems. The Initial Value Problem, John Wiley & Sons, Ltd., Chichester, 1991.
    [11] A. G. LópezJ. M. Seoane and M. A. F. Sanjuán, A validated mathematical model of tumor growth including tumor-host interaction, cell-mediated immune response and chemotherapy, Bull. Math. Biol., 76 (2014), 2884-2906.  doi: 10.1007/s11538-014-0037-5.
    [12] H. McCallumN. Barlow and J. Hone, How should pathogen transmission be modelled?, Trends Ecol. Evol., 16 (2001), 295-300.  doi: 10.1016/S0169-5347(01)02144-9.
    [13] Z. Navickas and L. Bikulciene, Expressions of solutions of ordinary differential equations by standard functions, Math. Model. Anal., 11 (2006), 399-412.  doi: 10.3846/13926292.2006.9637327.
    [14] Z. NavickasR. MarcinkeviciusT. Telksnys and M. Ragulskis, Existence of second order solitary solutions to Riccati differential equations coupled with a multiplicative term, IMA. J. Appl. Math., 81 (2016), 1163-1190.  doi: 10.1093/imamat/hxw050.
    [15] Z. NavickasM. Ragulskis and T. Telksnys, Existence of solitary solutions in a class of nonlinear differential equations with polynomial nonlinearity, Appl. Math. Comput., 283 (2016), 333-338.  doi: 10.1016/j.amc.2016.02.049.
    [16] Z. NavickasR. VilkasT. Telksnys and M. Ragulskis, Direct and inverse relationships between Riccati systems coupled with multiplicative terms, J. Biol. Dyn., 10 (2016), 297-313.  doi: 10.1080/17513758.2016.1181801.
    [17] A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Chapman & Hall/CRC, Boca Raton, FL, 2003.
    [18] Z. Qiao and J. Li, Negative-order KdV equation with both solitons and kink wave solutions, Europhys. Lett., 94 (2011), 50003. doi: 10.1209/0295-5075/94/50003.
    [19] W. T. ReidRiccati Differential Equations, Mathematics in Science and Engineering, 86, Academic Press, New York-London, 1972. 
    [20] T. C. RelugaH. Dahari and A. S. Perelson, Analysis of hepatitis C virus infection models with hepatocyte homeostasis, SIAM J. Appl. Math., 69 (2009), 999-1023.  doi: 10.1137/080714579.
    [21] M. Remoissenet, Waves Called Solitons. Concepts and Experiments, Advanced Texts in Physics, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-03790-4.
    [22] W. H. Renninger and P. T. Rakich, Closed-form solutions and scaling laws for Kerr frequency combs, Scientific Reports, 6 (2016). doi: 10.1038/srep24742.
    [23] K. Sakkaravarthi and T. Kanna, Bright solitons in coherently coupled nonlinear Schrödinger equations with alternate signs of nonlinearities, J. Math. Phys., 54 (2013), 14pp. doi: 10.1063/1.4772611.
    [24] A. Scott, Encyclopedia of Nonlinear Science, Routledge, New York, 2005. doi: 10.4324/9780203647417.
    [25] V. A. Vladimirov, E. V. Kutafina and A. Pudelko, Constructing soliton and kink solutions of PDE models in transport and biology, Symmetry Integrability Geom. Methods Appl., 2 (2006), 15pp. doi: 10.3842/SIGMA.2006.061.
    [26] Y. Yang, Solitons in Field Theory and Nonlinear Analysis, Springer Monographs in Mathematics, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-6548-9.
    [27] G.-A. Zakeri and E. Yomba, Dissipative solitons in a generalized coupled cubic-quintic Ginzburg-Landau equations, J. Phys. Soc. Jpn., 82 (2013). doi: 10.7566/JPSJ.82.084002.
    [28] S. Zdravković and G. Gligorić, Kinks and bell-type solitons in microtubules, Chaos, 35 (2016), 7pp. doi: 10.1063/1.4953011.
  • 加载中

Figures(9)

SHARE

Article Metrics

HTML views(1685) PDF downloads(270) Cited by(0)

Access History

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return