Time delay induces a back to back Hopf bifurcation on oncolytic virotherapy

This study analyzes a basic mathematical model for the dynamic interactions among tumor cells, infected tumor cells and viruses population, focusing on the viral lytic cycle for oncolytic virotherapy. I study the time delay effect of viral infection on tumor cell populations by identifying bifurcation thresholds in both the burst rate and time delay of viral infection in oncolytic virus therapy. Time delay plays an important role in changing the structure of tumor cell populations in a dynamical system. The multi-bifurcation thresholds of the time delay are observed and also dependent on the bursting rate. This study demonstrates a strong relationship between viral burst rates and time delays in population dynamics. The results of this study show that time delay affects oscillation generation and results in back-to-back Hopf bifurcation. This study provides insight into understanding the relationship between the two control parameters, in which tumor cell populations pattern from equilibrium steady-state solutions to periodic solutions and from periodic solutions to equilibrium-state solutions .


Introduction
Oncolytic viruses are genetically modified viruses that can infect and multiply cancerous cells, but leave normal, healthy cells intact. Oncolytic viruses can be divided into two types: oncolytic wild viruses, which occur naturally and preferentially in human cancer cells, and genetically modified viruses engineered to achieve selective oncolysis. Wild-type viruses have shown limited oncolytic potency in some preclinical trials, whereas transgenic viruses appear to have large oncolytic potency (Kirn & McCormick(1996), Kaplan(2005), Roberts, et. al.(2006)). Prior to the 1990s, case studies and small-scale experiments with various viruses in cancer treatment were reported (Chiocca(2002)). Genetic engineering began to be used for oncolytic viruses in the 1990s (Martuza, et. al.(1991)). To date, many types of viruses have been modified for experiments (Lawler et.al.(2017)), and some oncolytic viruses have been approved for human clinical trials (Maroun, et. al.(2017)). However, the potential of oncolytic viruses does not seem to have been reached yet (Chiocca & Rabkin(2014)). One major challenge is how to fully spread the virus into solid tumors (Mok, et. al.(2009)). An understanding of the dynamics of spread of oncolytic viruses through tumors can help overcome these difficulties and develop strategies for clinical application. Mathematical modeling can explore the full spectrum of possible outcomes and provide a basis for optimizing treatment. Several attempts have been made to understand and characterize viral dynamics with mathematical models. See Bajzer, et. al.(2008), Friedman, et. al.(2006), Wodarz(2001) and Wu, et. al.(2001) for example. These mathematical models can be roughly divided into two classes. One class uses Ordinary Differential Equations (ODEs), including Delay Differential Equations (DDEs) , Karev, et. al.(2006), Wodarz & Komarova (2009), and Wang, et. al.(2013)) and the other class uses Partial Differential Equations (PDEs) (Friedman, et. al.(2006), Wein, et. al.(2003) and Wu, et. al.(2001)). For PDE models of oncolytic virus therapy, most use the idea of fluid dynamics to model solid tumor growth in which tumor cells convene in fluid velocity fields within the tumor and the virus simply spreads within the tumor. All these modeling studies have provided specific insights into viral treatment. A study by Jain and colleagues in particular highlights the importance of the spreading properties of viruses (Mok, et. al.(2009)). However, it is well known that the growth of solid tumors, especially brain tumor gliomas, also exhibits characteristics of cell proliferation (Harpold, et. al.(2007)). The viral lytic cycle is the duration of the viral life cycle within a cell, starting from the moment the virus enters the cell and ending when a certain number (viral burst size) of newly replicated virus emerges upon cell lysis. It is an important parameter of viral dynamics. Wang, et. al.(2013) is the first to incorporate the viral lysis cycle as a delay parameter in a mathematical model for oncolytic virus therapy. In this paper, we aim to analyze the effect of time delay of the virus infection on oncolytic virotherapy. Our numerical results show there is a strong relationship between the virus bursting size and the time delay in the generation of oscillations. This paper is structured as follows. Section 2 reviews models of the dynamics of oncolytic viruses with basic equations and introduces common basic models. In addition, equilibrium analysis and stability are reviewed, and conditions of numerical simulation are checked. In Section 3, we investigate the effect of time delay on the dynamics of the tumor cell population through a basic model. We can know that the bifurcation value in the bursting rate depends on a time delay. Finally, in conclusion, we summarize our results and we highlight that time delay affects oscillation generation.

Model
The OV model is a three dimensional is the infection rate of the virus and the term 2 describes the rate of infected tumor cells by free viruses (t). τ is a time delay 1 represents the death rate of infected tumor cells. The 3 is the bursting size of free virus particles. The term 2 is the clearance rate of free virus particles.
For non-dimensionalization, we set τ = 1 , =̂, y =̂, z =. Then We have the following model by setting the parameters; All parameters are described in Table 1.   ,

Numerical Simulation
We

Time delay induces a back to back Hopf bifurcation on oncolytic virotherapy.
International Journal for Innovation Education and Research Vol. 11 No. 5 (2023), pg. 69

The bifurcation value in the bursting rate depends on a time delay.
The changes in the stability of an equilibrium depend on parameters. This qualitative changes in a dynamic structure is called bifurcation. Figure 2 shows bifurcation diagrams of equilibrium tumor cell population over the bursting rate. We calculated the equilibrium tumor cell population over time for each bursting rate ranged from 0 to 40 with step size 0.01. Our model exhibits two bifurcation values in the bursting rate at = 1 * , 2 * .