Travelling Waves in a Mathematical Model for Oncolytic Virotherapy

TL;DR

This work analyzes a three-component, non-cooperative reaction–diffusion model for oncolytic virotherapy to understand how viral fronts invade tumor tissue. The authors transform the system to a travelling-wave problem, derive a dispersion relation and a minimal invasion speed ${c_m}$, and construct explicit upper and lower solutions to apply Schauder's fixed-point theorem, proving existence of positive travelling waves for all $c\ge\bar{c}$ with $\bar{c}$ derived from a speed curve. They also present a limiting argument to obtain a wave at the minimal speed ${c=\bar{c}}$ and verify the results with numerical simulations of the full PDE system, highlighting regimes where wave existence remains uncertain. The findings illuminate the conditions under which oncolytic viruses can propagate through tumor tissue and provide a rigorous mathematical framework that can inform parameter optimization for virotherapy front propagation.

Abstract

Oncolytic virotherapy (OVT) is a promising cancer treatment strategy in which engineered viruses selectively infect and destroy tumor cells. Motivated by the biological mechanisms underlying viral spread and tumor invasion into the tissue, we analyze a non-cooperative reaction-diffusion model capturing the invasion of tumor tissue by oncolytic viruses. Using carefully constructed upper and lower solutions together with Schauder's fixed point theorem, we establish the existence of positive travelling-wave solutions. In particular, we identify a minimal wave speed value $\bar c$ such that positive travelling waves exist for all $c \geq \bar c$ . Our analysis also highlights parameter regions where the existence of travelling waves remains ambiguous, suggesting new mathematical questions about the propagation of viral treatments through tumor environments.

Figures (4)

• Figure 1: Bifurcation diagram of the model \ref{['eqn:base model']} with bifurcation parameter $\theta$. The equilibrium $E_{1}$ loses stability at $\theta \approx 12.7$, while $E_{2}$ becomes biologically relevant and remains stable up to the Hopf bifurcation at $\theta_{H} \approx 220$. Beyond this point, the system exhibits oscillatory dynamics. The parameter values used here are $a = 0.96$ and $\gamma = \frac{40}{3}$. The circles mark the $\theta$ values selected in Section \ref{['sec:numerics']} for numerical simulations.
• Figure 2: Numerical illustration of $S(\rho)$, and its minimum value indicated by the orange ball, and $H_1(\rho),\ H_2(\rho), \ H_3(\rho)$, and $H_4(\rho)$ for different parameter values. The parameter values have been chosed to illustrate each of the regimes and are given by: (a) $a= \frac{10}{3},\ \gamma= \frac{40}{3},\ \theta=250,\ D=0.025$ (b) $a=0.96,\ \gamma= \frac{40}{3}, \theta=50, D=0.025$, (c) $a=5 ,\ \gamma=2,\ \theta=17,\ D=6$ (d) $a= 0.96,\ \gamma= 1.25,\ \theta=17,\ D=0.025$. Note that case (c) may not arise in biologically relevant cases.
• Figure 3: Left: Wave–speed function $S(\rho)$ together with the values $\bar{c}$, $c$, and $c_{\epsilon}$. The point $(\bar{\rho},\bar{c})$ is marked in red. The points $\rho=\bar{\rho}-\tau$ and $\rho_{\epsilon}=\rho+\epsilon$, with $\tau=1$ and $\epsilon=0.5$, are also shown, along with their corresponding wave speeds $(c,c_{\epsilon})$. Right: Sketch of the upper solutions and their transition points $\bar{\xi}_{1}$, $\bar{\xi}_{2}$, and $\bar{\xi}_{3}$.
• Figure 4: Solutions of the full model \ref{['eqn:main_PDE']} for two different values of the parameter $a$. We are showing the solutions as $(B,I,V)$ where $B=1-C$. The left column displays simulations for $a=\frac{10}{3}>1$ at times $t = 8$ (dotted lines) and $t = 34$ (solid lines). Panel (a) ($\theta = 150$) exhibits no oscillations, while Panel (e) ($\theta = 350$) shows oscillatory behaviour has started Panel (c) ($\theta = 500$) illustrates the dynamics in a oscillatory regime with longer oscillations. The right column shows numerical solutions for the case $a<1$ at times $t = 8$ (dotted lines) and $t = 54$ (solid lines), illustrating the formation of oscillatory dynamics behind the travelling wave. For visual clarity, the variables $B$ and $I$ have been multiplied by a factor of $10$. Panel (b) ($\theta = 25$) exhibits no oscillations, whereas Panel (d) ($\theta = 150$) shows oscillations converging toward the coexistence equilibrium. Panel (f) ($\theta = 250$) corresponds to a regime beyond the Hopf bifurcation, where oscillatory patterns arise.

Theorems & Definitions (23)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5
• proof