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A critical virus production rate for efficiency of oncolytic virotherapy

Part of: Physiological, cellular and medical topics Equations of mathematical physics and other areas of application Partial differential equations Parabolic equations and systems

Published online by Cambridge University Press:  08 May 2020

YOUSHAN TAO  and
MICHAEL WINKLER
YOUSHAN TAO
Affiliation:
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, P.R. China. email: taoys@sjtu.edu.cn
MICHAEL WINKLER
Affiliation:
Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany. email: michael.winkler@math.uni-paderborn.de
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Abstract

In a planar smoothly bounded domain $\Omega$ , we consider the model for oncolytic virotherapy given by

$$\left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v) - uz, \\[1mm] v_t = - (u+w)v, \\[1mm] w_t = d_w \Delta w - w + uz, \\[1mm] z_t = d_z \Delta z - z - uz + \beta w, \end{array} \right.$$
with positive parameters $ D_w $ , $ D_z $ and $\beta$ . It is firstly shown that whenever $\beta \lt 1$ , for any choice of $M \gt 0$ , one can find initial data such that the solution of an associated no-flux initial-boundary value problem, well known to exist globally actually for any choice of $\beta \gt 0$ , satisfies
$$u\ge M \qquad \mbox{in } \Omega\times (0,\infty).$$
If $\beta \gt 1$ , however, then for arbitrary initial data the corresponding is seen to have the property that
$$\liminf_{t\to\infty} \inf_{x\in\Omega} u(x,t)\le \frac{1}{\beta-1}.$$
This may be interpreted as indicating that $\beta$ plays the role of a critical virus replication rate with regard to efficiency of the considered virotherapy, with corresponding threshold value given by $\beta = 1$ .

Keywords

Oncolytic virotherapyhaptotaxiscritical parameter

MSC classification

Primary: 35B40: Asymptotic behavior of solutions
Secondary: 35K57: Reaction-diffusion equations 35Q92: PDEs in connection with biology and other natural sciences 92C17: Cell movement (chemotaxis, etc.)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Issue 2 , April 2021 , pp. 301 - 316
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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