March  2018, 23(2): 525-541. doi: 10.3934/dcdsb.2017206

Asymptotic behaviour of the solutions to a virus dynamics model with diffusion

1. 

Graduate School of Environmental and Life Science, Okayama University, Okayama, 700-8530, Japan

2. 

Graduate School of Engineering Science, Osaka University, Osaka, 560-8531, Japan

Received  February 2017 Revised  May 2017 Published  March 2018 Early access  December 2017

Asymptotic behaviour of the solutions to a basic virus dynamics model is discussed. We consider the population of uninfected cells, infected cells, and virus particles. Diffusion effect is incorporated there. First, the Lyapunov function effective to the spatially homogeneous part (ODE model without diffusion) admits the $L^1$ boundedness of the orbit. Then the pre-compactness of this orbit in the space of continuous functions is derived by the semigroup estimates. Consequently, from the invariant principle, if the basic reproductive number $R_0$ is less than or equal to 1, each orbit converges to the disease free spatially homogeneous equilibrium, and if $R_0>1$, each orbit converges to the infected spatially homogeneous equilibrium, which means that the simple diffusion does not affect the asymptotic behaviour of the solutions.

Citation: Toru Sasaki, Takashi Suzuki. Asymptotic behaviour of the solutions to a virus dynamics model with diffusion. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 525-541. doi: 10.3934/dcdsb.2017206
References:
[1]

S. BonhoefferR. M. MayG. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. USA, 94 (1997), 6971-6976. 

[2]

J. A. CarrilloA. JüngelP. A. MarkowichG. Toscani and A. Unterriter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 133 (2001), 1-82. 

[3]

C. L. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998.

[4]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, New York, 1983. doi: 10.1007/978-3-642-61798-0.

[5]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Springer-Verlag, New York, 1981.

[6]

A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883.  doi: 10.1016/j.bulm.2004.02.001.

[7]

E. LatosT. Suzuki and Y. Yamada, Transient and asymptotic dynamics of a prey-predator system with diffusion, Math. Methods Appl. Sci., 35 (2012), 1101-1109.  doi: 10.1002/mma.2524.

[8]

M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. 

[9]

J. PrüssR. Zacher and R. Schnaubelt, Global asymptotic stability of equilibria in models for virus dynamics, Math. Model. Nat. Phenom., 3 (2008), 126-142. 

[10]

F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Mathematics, Springer-Verlag, New York, 1984. doi: 10.1007/BFb0099278.

[11]

G. D. Smith, Numerical Solution of Partial Differential Equations–Finite Difference Methods, 3rd edition, Oxford University press, Oxford, 1985.

[12]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[13]

H. Tanabe, Equations of Evolution, Pitman, London, 1979.

[14]

J. WangJ. Yang and T. Kuniya, Dynamics of a PDE viral infection model incorporating cell-to-cell transmission, J. Math. Anal. Appl., 444 (2016), 1542-1564. 

[15]

A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer-Verlag, New York, 2010. doi: 10.1007/978-3-642-04631-5.

show all references

References:
[1]

S. BonhoefferR. M. MayG. M. Shaw and M. A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. USA, 94 (1997), 6971-6976. 

[2]

J. A. CarrilloA. JüngelP. A. MarkowichG. Toscani and A. Unterriter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 133 (2001), 1-82. 

[3]

C. L. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998.

[4]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer-Verlag, New York, 1983. doi: 10.1007/978-3-642-61798-0.

[5]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Springer-Verlag, New York, 1981.

[6]

A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883.  doi: 10.1016/j.bulm.2004.02.001.

[7]

E. LatosT. Suzuki and Y. Yamada, Transient and asymptotic dynamics of a prey-predator system with diffusion, Math. Methods Appl. Sci., 35 (2012), 1101-1109.  doi: 10.1002/mma.2524.

[8]

M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. 

[9]

J. PrüssR. Zacher and R. Schnaubelt, Global asymptotic stability of equilibria in models for virus dynamics, Math. Model. Nat. Phenom., 3 (2008), 126-142. 

[10]

F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Mathematics, Springer-Verlag, New York, 1984. doi: 10.1007/BFb0099278.

[11]

G. D. Smith, Numerical Solution of Partial Differential Equations–Finite Difference Methods, 3rd edition, Oxford University press, Oxford, 1985.

[12]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[13]

H. Tanabe, Equations of Evolution, Pitman, London, 1979.

[14]

J. WangJ. Yang and T. Kuniya, Dynamics of a PDE viral infection model incorporating cell-to-cell transmission, J. Math. Anal. Appl., 444 (2016), 1542-1564. 

[15]

A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer-Verlag, New York, 2010. doi: 10.1007/978-3-642-04631-5.

Figure 1.  The graphs of $u_i$'s: In the case $R_0 <1$. (a) $u_1$. (b) $u_2$. (c) $u_3$. The oscillation of the initial functions decays with the passage of time, and the solutions tend to the spatially homogeneous disease free steady states.
Figure 2.  The graphs of $u_i$'s: In the case $R_0>1$. (a, d) $u_1$. (b, e) $u_2$. (c, f) $u_3$. The view angle of the graphs in the upper row is the same as that of Figure 1.
Figure 3.  The graphs of $u_3$: In the case $R_0>1$. We divide Figure 2(c) into three parts: (a) $0\leqq t \leqq 2.1$, (b) $2.1\leqq t \leqq 6.0$, and (c) $6.0 \leqq t \leqq 15.0$.
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