A critical virus production rate for efficiency of oncolytic virotherapy

<jats:p>In a planar smoothly bounded domain<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline1.png" /><jats:tex-math>$\Omega$</jats:tex-math></jats:alternatives></jats:inline-formula>, we consider the model for oncolytic virotherapy given by<jats:disp-formula id="S0956792520000133_udisp1"><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0956792520000133_eqnu1.png" /><jats:tex-math>$$\left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v) - uz, \\[1mm] v_t = - (u+w)v, \\[1mm] w_t = d_w \Delta w - w + uz, \\[1mm] z_t = d_z \Delta z - z - uz + \beta w, \end{array} \right.$$</jats:tex-math></jats:alternatives></jats:disp-formula>with positive parameters<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline2.png" /><jats:tex-math>$ D_w $</jats:tex-math></jats:alternatives></jats:inline-formula>,<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline3.png" /><jats:tex-math>$ D_z $</jats:tex-math></jats:alternatives></jats:inline-formula>and<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline4.png" /><jats:tex-math>$\beta$</jats:tex-math></jats:alternatives></jats:inline-formula>. It is firstly shown that whenever<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline5.png" /><jats:tex-math>$\beta \lt 1$</jats:tex-math></jats:alternatives></jats:inline-formula>, for any choice of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline6.png" /><jats:tex-math>$M \gt 0$</jats:tex-math></jats:alternatives></jats:inline-formula>, one can find initial data such that the solution of an associated no-flux initial-boundary value problem, well known to exist globally actually for any choice of<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline7.png" /><jats:tex-math>$\beta \gt 0$</jats:tex-math></jats:alternatives></jats:inline-formula>, satisfies<jats:disp-formula id="S0956792520000133_udisp2"><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0956792520000133_eqnu2.png" /><jats:tex-math>$$u\ge M \qquad \mbox{in } \Omega\times (0,\infty).$$</jats:tex-math></jats:alternatives></jats:disp-formula>If<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline8.png" /><jats:tex-math>$\beta \gt 1$</jats:tex-math></jats:alternatives></jats:inline-formula>, however, then for arbitrary initial data the corresponding is seen to have the property that<jats:disp-formula id="S0956792520000133_udisp3"><jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0956792520000133_eqnu3.png" /><jats:tex-math>$$\liminf_{t\to\infty} \inf_{x\in\Omega} u(x,t)\le \frac{1}{\beta-1}.$$</jats:tex-math></jats:alternatives></jats:disp-formula>This may be interpreted as indicating that<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline9.png" /><jats:tex-math>$\beta$</jats:tex-math></jats:alternatives></jats:inline-formula>plays the role of a critical virus replication rate with regard to efficiency of the considered virotherapy, with corresponding threshold value given by<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0956792520000133_inline10.png" /><jats:tex-math>$\beta = 1$</jats:tex-math></jats:alternatives></jats:inline-formula>.</jats:p>