A critical virus production rate for efficiency of oncolytic virotherapy.

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 $\beta = 1$. [ABSTRACT FROM AUTHOR]