Skip to main content Accessibility help
×
Home
Hostname: page-component-7f7b94f6bd-wzgmz Total loading time: 0.479 Render date: 2022-06-30T21:04:26.195Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

Asymptotic behaviour in a doubly haptotactic cross-diffusion model for oncolytic virotherapy

Published online by Cambridge University Press:  19 April 2022

Yifu Wang
Affiliation:
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, P.R. China (wangyifu@bit.edu.cn; XuChi1993@126.com)
Chi Xu
Affiliation:
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, P.R. China (wangyifu@bit.edu.cn; XuChi1993@126.com)

Abstract

This paper considers a model for oncolytic virotherapy given by the doubly haptotactic cross-diffusion system

\[ \left\{\begin{array}{@{}ll} u_t=D_u\Delta u-\xi_u\nabla\cdot(u\nabla v)+\mu_u u(1-u)-\rho uz,\\ v_t={-} (\alpha_u u+\alpha_w w)v,\\ w_t=D_w\Delta w-\xi_w\nabla\cdot(w\nabla v)- w+\rho uz,\\ z_t=D_z\Delta z-\delta_z z- \rho uz+\beta w, \end{array}\right. \]
with positive parameters $D_u,D_w,D_z,\xi _u,\xi _w,\delta _z,\rho$, $\alpha _u,\alpha _w,\mu _u,\beta$. When posed under no-flux boundary conditions in a smoothly bounded domain $\Omega \subset {\mathbb {R}}^{2}$, and along with initial conditions involving suitably regular data, the global existence of classical solution to this system was asserted in Tao and Winkler (2020, J. Differ. Equ. 268, 4973–4997). Based on the suitable quasi-Lyapunov functional, it is shown that when the virus replication rate $\beta <1$, the global classical solution $(u,v,w,z)$ is uniformly bounded and exponentially stabilizes to the constant equilibrium $(1, 0, 0, 0)$ in the topology $(L^{\infty }(\Omega ))^{4}$ as $t\rightarrow \infty$.

Type
Research Article
Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alzahrani, T., Eftimie, R. and Trucu, D.. Multiscale modelling of cancer response to oncolytic viral therapy. Math. Bioci. 310 (2019), 7695.Google ScholarPubMed
Anderson, A. R., Chaplain, M. A. J., Newman, E. L., Steele, R. J. C. and Thompson, A. M.. Mathematical modelling of tumour invasion and metastasis. J. Theor. Med. 2 (2000), 129154.CrossRefGoogle Scholar
Cao, X.. Boundedness in a three-dimensional chemotaxis–haptotaxis model. Z. Angew. Math. Phys. 67 (2016), 11.CrossRefGoogle Scholar
Chen, Z.. Dampening effect of logistic source in a two-dimensional haptotaxis system with nonlinear zero-order interaction. J. Math. Anal. Appl. 492 (2020), 124435.CrossRefGoogle Scholar
Chaplain, M. A. J. and Lolas, G.. Mathematical modelling of cancer cell invasion of tissue: the role of the urokinase plasminogen activation system. Math. Mod. Meth. Appl. Sci. 18 (2005), 16851734.CrossRefGoogle Scholar
Fontelos, M. A., Friedman, A. and Hu, B.. Mathematical analysis of a model for the initiation of angiogenesis. SIAM J. Math. Anal. 33 (2002), 13301355.CrossRefGoogle Scholar
Fuest, M.. Global solutions near homogeneous steady states in a multi-dimensional population model with both predator-and prey-taxis. SIAM J. Math. Anal. 52 (2020), 58635891.CrossRefGoogle Scholar
Fukuhara, H., Ino, Y. and Todo, T.. Oncolytic virus therapy: a new era of cancer treatment at dawn. Cancer Sci. 107 (2016), 13731379.CrossRefGoogle ScholarPubMed
Gujar, S., Pol, J. G., Kim, Y., Lee, P. W. and Kroemer, G.. Antitumor benefits of antiviral immunity: an underappreciated aspect of oncolytic virotherapies. Trends Immunol. 39 (2018), 209221.CrossRefGoogle ScholarPubMed
Ganly, I. and Kirn, D.. A phase I study of Onyx-015, an E1B-attenuated adenovirus, administered intratumorally to patients with recurrent head and neck cancer. Clin. Cancer Res. 6 (2000), 798806.Google ScholarPubMed
Jin, C.. Global classical solutions and convergence to a mathematical model for cancer cells invasion and metastatic spread. J. Differ. Equ. 269 (2020), 39874021.CrossRefGoogle Scholar
Jin, H. Y. and Xiang, T.. Negligibility of haptotaxis effect in a chemotaxis–haptotaxis model. Math. Mod. Meth. Appl. Sci. 31 (2021), 13731417.CrossRefGoogle Scholar
Komarova, N. L.. Viral reproductive strategies: how can lytic viruses be evolutionarily competitive?. J. Theor. Biol. 249 (2007), 766784.CrossRefGoogle ScholarPubMed
Li, Y. and Lankeit, J.. Boundedness in a chemotaxis–haptotaxis model with nonlinear diffusion. Nonlinearity 29 (2016), 15641595.CrossRefGoogle Scholar
Li, J. and Wang, Y.. Boundedness in a haptotactic cross-diffusion system modeling oncolytic virotherapy. J. Differ. Equ. 270 (2021), 94113.CrossRefGoogle Scholar
Nemunaitis, J. and Ganly, I.. Selective replication and oncolysis in p53 mutant tumors with ONYX-015, an E1B-55kD gene-deleted adenovirus, in patients with advanced head and neck cancer: a phase II trial. Cancer Res. 60 (2000), 63596366.Google ScholarPubMed
Pang, P. Y. H. and Wang, Y.. Global boundedness of solutions to a chemotaxis–haptotaxis model with tissue remodeling. Math. Models Methods Appl. Sci. 28 (2018), 22112235.CrossRefGoogle Scholar
Pang, P. Y. H. and Wang, Y.. Asymptotic behavior of solutions to a tumor angiogenesis model with chemotaxis–haptotaxis. Math. Models Methods Appl. Sci. 29 (2019), 13871412.CrossRefGoogle Scholar
Prüss, J., Zacher, R. and Schnaubelt, R.. Global asymptotic stability of equilibria in models for virus dynamics. Math. Model. Nat. Phenom. 3 (2008), 126142.CrossRefGoogle Scholar
Ren, G. and Liu, B.. Global classical solvability in a three-dimensional haptotaxis system modeling oncolytic virotherapy. Math. Methods Appl. Sci. 44 (2021), 92759291.CrossRefGoogle Scholar
Stinner, C., Surulescu, C. and Winkler, M.. Global weak solutions in a PDE–ODE system modeling multiscale cancer cell invasion. SIAM J. Math. Anal. 46 (2014), 19692007.CrossRefGoogle Scholar
Tao, Y. and Winkler, M.. Global classical solutions to a doubly haptotactic cross-diffusion system modeling oncolytic virotherapy. J. Differ. Equ. 268 (2020), 49734997.CrossRefGoogle Scholar
Tao, Y. and Winkler, M.. Asymptotic stability of spatial homogeneity in a haptotaxis model for oncolytic virotherapy. Proc. R. Soc. Edinburgh Sect. A Math. 52 (2022), 81101.CrossRefGoogle Scholar
Tao, Y. and Winkler, M.. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete Contin. Dyn. Syst. A 41 (2021), 439454.CrossRefGoogle Scholar
Tao, Y. and Winkler, M.. A critical virus production rate for efficiency of oncolytic virotherapy. European J. Appl. Math. 32 (2021), 301316.CrossRefGoogle Scholar
Tao, Y. and Winkler, M.. A critical virus production rate for blow-up suppression in a haptotaxis model for oncolytic virotherapy. Nonlinear Anal. 198 (2020), 111870.CrossRefGoogle Scholar
Tao, Y. and Winkler, M.. Large time behavior in a multidimensional chemotaxis–haptotaxis model with slow signal diffusion. SIAM J. Math. Anal. 47 (2015), 42294250.CrossRefGoogle Scholar
Tao, Y. and Winkler, M.. Energy-type estimates and global solvability in a two-dimensional chemotaxis–haptotaxis model with remodeling of non-diffusible attractant. J. Differ. Equ. 257 (2014), 784815.CrossRefGoogle Scholar
Walker, C. and Webb, G. F.. Global existence of classical solutions for a haptotaxis model. SIAM J. Math. Anal. 38 (2007), 16941713.CrossRefGoogle Scholar
Wang, Y.. Boundedness in the higher-dimensional chemotaxis–haptotaxis model with nonlinear diffusion. J. Differ. Equ. 260 (2016), 19751989.CrossRefGoogle Scholar
Winkler, M.. Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model. J. Differ. Equ. 12 (2010), 28892905.CrossRefGoogle Scholar
Zheng, J. and Ke, Y.. Large time behavior of solutions to a fully parabolic chemotaxis–haptotaxis model in $N$ dimensions. J. Differ. Equ. 266 (2019), 19692018.CrossRefGoogle Scholar
Zhigun, A., Surulescu, C. and Uatay, A.. Global existence for a degenerate haptotaxis model of cancer invasion. Z. Angew. Math. Phys. 67 (2016), 146.CrossRefGoogle Scholar

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Asymptotic behaviour in a doubly haptotactic cross-diffusion model for oncolytic virotherapy
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Asymptotic behaviour in a doubly haptotactic cross-diffusion model for oncolytic virotherapy
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Asymptotic behaviour in a doubly haptotactic cross-diffusion model for oncolytic virotherapy
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *