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November  2020, 25(11): 4427-4447. doi: 10.3934/dcdsb.2020106

Kink solitary solutions to a hepatitis C evolution model

Tadas Telksnys 1,, , Zenonas Navickas 1, , Miguel A. F. Sanjuán 2,3,4, , Romas Marcinkevicius 5, and  Minvydas Ragulskis 1,

1. 

Research Group for Mathematical, and Numerical Analysis of Dynamical Systems, Kaunas University of Technology, Studentu 50-147, Kaunas LT-51368, Lithuania
2. 

Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain
3. 

Department of Applied Informatics, Kaunas University of Technology, Studentu 50-407, Kaunas LT-51368, Lithuania
4. 

Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, USA
5. 

Department of Software Engineering, Kaunas University of Technology, Studentu 50-415, Kaunas LT-51368, Lithuania

* Corresponding author: Tadas Telksnys (tadas.telksnys@ktu.lt)

Received  July 2017 Published  November 2020 Early access  March 2020

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.

Keywords: Kink solution, nonlinear differential equation, generalized differential operator, hepatitis C model, solitary solution.
Mathematics Subject Classification: Primary: 34A34; Secondary: 35C05, 35C07.
Citation: Tadas Telksnys, Zenonas Navickas, Miguel A. F. Sanjuán, Romas Marcinkevicius, Minvydas Ragulskis. Kink solitary solutions to a hepatitis C evolution model. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4427-4447. doi: 10.3934/dcdsb.2020106
References:
[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. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, Oxford, 2008. 
[4] T. Dauxois and M. Peyrard, Physics 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. Grundland, M. 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. Kelkar, E. 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. López, J. 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. McCallum, N. 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. Navickas, R. Marcinkevicius, T. 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. Navickas, M. 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. Navickas, R. Vilkas, T. 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. Reid, Riccati Differential Equations, Mathematics in Science and Engineering, 86, Academic Press, New York-London, 1972. 
[20]

T. C. Reluga, H. 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.

show all references

References:
[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. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, Oxford, 2008. 
[4] T. Dauxois and M. Peyrard, Physics 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. Grundland, M. 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. Kelkar, E. 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. López, J. 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. McCallum, N. 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. Navickas, R. Marcinkevicius, T. 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. Navickas, M. 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. Navickas, R. Vilkas, T. 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. Reid, Riccati Differential Equations, Mathematics in Science and Engineering, 86, Academic Press, New York-London, 1972. 
[20]

T. C. Reluga, H. 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.

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), $u = -2, v = 0$
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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), $u = -2, v = 0$
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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 $1.09$
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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)
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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)
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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)
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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), $u = -5, v = 2$
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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), $u = -5, v = 2$
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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 $0.39$
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