{"user_id":"31496","publication":"European Journal of Applied Mathematics","publisher":"Cambridge University Press (CUP)","issue":"2","page":"301-316","year":"2020","title":"A critical virus production rate for efficiency of oncolytic virotherapy","citation":{"short":"Y. TAO, M. Winkler, European Journal of Applied Mathematics 32 (2020) 301–316.","apa":"TAO, Y., &#38; Winkler, M. (2020). A critical virus production rate for efficiency of oncolytic virotherapy. <i>European Journal of Applied Mathematics</i>, <i>32</i>(2), 301–316. <a href=\"https://doi.org/10.1017/s0956792520000133\">https://doi.org/10.1017/s0956792520000133</a>","chicago":"TAO, YOUSHAN, and Michael Winkler. “A Critical Virus Production Rate for Efficiency of Oncolytic Virotherapy.” <i>European Journal of Applied Mathematics</i> 32, no. 2 (2020): 301–16. <a href=\"https://doi.org/10.1017/s0956792520000133\">https://doi.org/10.1017/s0956792520000133</a>.","ama":"TAO Y, Winkler M. A critical virus production rate for efficiency of oncolytic virotherapy. <i>European Journal of Applied Mathematics</i>. 2020;32(2):301-316. doi:<a href=\"https://doi.org/10.1017/s0956792520000133\">10.1017/s0956792520000133</a>","ieee":"Y. TAO and M. Winkler, “A critical virus production rate for efficiency of oncolytic virotherapy,” <i>European Journal of Applied Mathematics</i>, vol. 32, no. 2, pp. 301–316, 2020, doi: <a href=\"https://doi.org/10.1017/s0956792520000133\">10.1017/s0956792520000133</a>.","mla":"TAO, YOUSHAN, and Michael Winkler. “A Critical Virus Production Rate for Efficiency of Oncolytic Virotherapy.” <i>European Journal of Applied Mathematics</i>, vol. 32, no. 2, Cambridge University Press (CUP), 2020, pp. 301–16, doi:<a href=\"https://doi.org/10.1017/s0956792520000133\">10.1017/s0956792520000133</a>.","bibtex":"@article{TAO_Winkler_2020, title={A critical virus production rate for efficiency of oncolytic virotherapy}, volume={32}, DOI={<a href=\"https://doi.org/10.1017/s0956792520000133\">10.1017/s0956792520000133</a>}, number={2}, journal={European Journal of Applied Mathematics}, publisher={Cambridge University Press (CUP)}, author={TAO, YOUSHAN and Winkler, Michael}, year={2020}, pages={301–316} }"},"_id":"63318","publication_identifier":{"issn":["0956-7925","1469-4425"]},"volume":32,"type":"journal_article","date_updated":"2025-12-18T20:06:35Z","intvolume":"        32","date_created":"2025-12-18T19:33:01Z","publication_status":"published","doi":"10.1017/s0956792520000133","abstract":[{"text":"<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>","lang":"eng"}],"language":[{"iso":"eng"}],"author":[{"full_name":"TAO, YOUSHAN","last_name":"TAO","first_name":"YOUSHAN"},{"first_name":"Michael","id":"31496","last_name":"Winkler","full_name":"Winkler, Michael"}],"status":"public"}