Eur. Phys. J. Plus (2020) 135: 117
https://doi.org/10.1140/epjp/s13360-020-00188-z

Regular Article

A reaction–diffusion model for oncolytic M1 virotherapy with distributed delays

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

^* e-mail: a_m_elaiw@yahoo.com

Received: 26 September 2019
Accepted: 28 October 2019
Published online: 23 January 2020

Abstract

Oncolytic virotherapy (OVT) is a promising treatment for cancer which can replace or support the traditional treatments like chemotherapy and radiotherapy. Mathematical models have been considered as a powerful tool to develop oncolytic viral treatments and predict the possible outcomes. We study a spatial model for OVT with cytotoxic T lymphocyte immune response and distributed delays. This model is an extended version of the model studied by Wang et al. (Math Biosci 276:19–27, 2016). We study the basic properties of the model including the existence, nonnegativity, and boundedness of solutions. We carefully analyze all equilibrium points and determine the conditions for their existence. We show the global stability of each one of these points by constructing suitable Lyapunov functionals. We use the characteristic equations to confirm the corresponding instability conditions. We carry out some numerical simulations to support the theoretical results and draw some important conclusions. The results show that the distributed delay can have a large impact on the efficacy and amount of OVT. When the immune response is present, the concentration of oncolytic viruses is decreased and the efficiency of treatment is reduced. Changing the diffusion coefficients does not affect the long-time behavior of solutions.