Article
Kyungpook Mathematical Journal 2022; 62(1): 119132
Published online March 31, 2022
Copyright © Kyungpook Mathematical Journal.
Hopfbifurcation Analysis of a Delayed Model for the Treatment of Cancer using Virotherapy
Maharajan Rajalakshmi and Mini Ghosh*
Division of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Chennai Campus, Chennai600127, India
email : rajalakshmi.m2016@vitstudent.ac.in and minighosh@vit.ac.in
Received: February 23, 2020; Revised: October 7, 2020; Accepted: November 16, 2020
Abstract
Virotherapy is an effective method for the treatment of cancer. The oncolytic virus specifically infects the lyse cancer cell without harming normal cells. There is a time delay between the time of interaction of the virus with the tumor cells and the time when the tumor cells become infectious and produce new virus particles. Several types of viruses are used in virotherapy and the delay varies with the type of virus. This delay can play an important role in the success of virotherapy. Our present study is to explore the impact of this delay in cancer virotherapy through a mathematical model based on delay differential equations. The partial success of virotherapy is guarenteed when one gets a stable nontrivial equilibrium with a low level of tumor cells. There exits Hopfbifurcation by considering the delay as bifurcation parameter. We have estimated the length of delay which preserves the stability of the nontrivial equilibrium point. So when the delay is less than a threshold value, we can predict partial success of virotherapy for suitable sets of parameters. Here numerical simulations are also performed to support the analytical findings.
Keywords: Cancer, Virotherapy, Delay model, Hopfbifurcation
1. Introduction
Cancer is a disease which is caused by unusual cell growth. It is one of the leading cause of death in the world. There are several types of cancer, and they are generally described by the body part they originate in. Most cancers are curable if detected early enough. Cancer is a very complex disease to treat and treatment of cancer varies. The choice of treatment depends on the location of the tumor, size of the tumor, cell type and most importantly the overall health of the patient. For the small size malignant tumors, surgery is often recommended. Chemotherapy, radiotherapy, immunotherapy, gene therapy, virotherapy, hormone therapy etc. are treatment methods which are used alone or in combination depending upon the size of the cancer and patient's health. Mathematical modeling is an important tool which is being used in treatment of several diseases including cancer. As our human body is very complex, the success of any therapy not only depends upon the types of treatment but also at the duration of treatment, frequency of treatment and effectiveness of a specific therapy. Mathematical modeling and simulation help in designing suitable personalized therapy for a patient. There are several research findings based on mathematical models on cancer growth and treatment of cancer [19, 21, 9, 6, 2, 22, 18, 12, 11]. In [19], the authors formulated and analyzed a mathematical model for a malignant tumor system by considering preypredator type interactions. Here the authors investigated both deterministic and stochastic models and obtained thresholds which play a key role in the control of malignant tumor growth. In [21], the author investigated a mathematical model for tumor growth with quiescence by incorporating delay. In [9], the authors considered a kinetic theory approach to model the growth of tumors and macrophages heterogeneity and plasticity by incorporating diffusion. A mathematical model for tumorimmune interaction with delay is formulated and analyzed in [2]. Here the authors have estimated the length of the delay parameter which preserve the stability of the system. In [22], the authors presented a competition model for the growth of a malignant tumor by incorporating immune response. Here the authors demonstrated the presence of a Hopfbifurcation. A delayed model for tumorimmune response with chemoimmunotherapy and optimal control is investigated in [18]. Here the authors have emphasized the importance of combination therapy and optimal control. In [12], the authors described fundamentals of modeling of tumor growth and discussed different approaches to model tumor growth. Modeling tumorimmune dynamics with treatment is investigated in [11]. Mathematical modeling of the treatment of cancer using virotherapy by considering a system of ordinary differential equations is reported in [1, 4, 7, 20, 15, 17]. In [16], the authors have formulated and analyzed a delay differential equation model for the treatment of cancer using virotherapy. Here the authors have considered the delay in tumor cells to become infectious after getting infected with viruses. This work is an extension of the author's own work in [15] where the ordinary differential equation model was formulated and analyzed. In both papers, the authors considered logistic growth of the tumor cells whereas a generalized logistic growth model is more suitable to describe the dynamics of the tumor cells. Keeping this in mind in [17], the authors formulated and analyzeda mathematical model for the treatment of cancer using virotherapy by considering generalized logistic growth of the tumor cells. Some existing models are derived using the framework of population dynamics and others include space dynamics which is an important feature. With these two approaches of population dynamics and system dynamics, existing mathematical models can be further extended by including space dynamics through diffusion [10] as it shows a key role in delaying the dynamics, or through reactiondiffusion [3] as it relates to the dynamic linked with angiogenesis.
The present work is an extension of the work in [17] which does not incorporate latent delay. It is a proven fact that delay is capable of changing the whole dynamics of the system under consideration [21, 14, 13]. The remaining part of this paper is organized as follows: Section 2 describes the proposed mathematical model; Section 3 discusses the reproduction numbers and equilibria of the model; Section 4 demonstrates the stability analysis of the nontrivial equilibrium point of the model. Section 5 demonstrates the Hopfbifurcation analysis. Section 6 exhibits the numerical simulation results. Finally, we summarize our results in Section 7.
2. Mathematical Model
As delay plays an important role in the dynamics of cancer growth and treatment, here we extend the model by Rajalakshmi and Ghosh [17] by incorporating latent delay. It is assumed that after the interaction of tumor cells with viruses, tumor cells get infected and produce new viruses. We refer to the time between interaction and production of new viruses as 'delay', The tumor cells are divided into two disjoint classes: uninfected tumor cells with population
Let
Here
3. The Reproduction Number and Equilibria of the Model
As delay does not change the equilibria of the system, we get, for this delay model, the same equilibria as obtained for the model discussed in [17]. The basic reproduction number
The largest eigenvalue of
Next we compute the immune response reproduction number
where
Now the largest eigenvalue of
where
(i)
(ii)
(iii)
(iv)
which exists for
It is easy to observe that for
where
For more details about the existence of these equilibria one can refer [17].
Our aim is to investigate the success or failure of virotherapy. Hence we need to concentrate on the stability of the partial success equilibrium point
4. Stability Analysis
We linearize the system (2.1) about the equilibrium point
where
The characteristic equation can be written as,
where
and
When
Using RouthHurwitz criteria, all the roots of the above biquadratic equation will have negative real parts provided following conditions hold:
Hence in absence of delay the equilibrium point
5. Hopfbifurcation
Now let us discuss the case when delay is not zero. When
Now squaring and adding equations (5.1) and (5.2), we get the following trancendental equation:
Assuming
where
Now we have the following theorems based on the roots of the equation
Theorem 5.1. If the coefficients
Theorem 5.2. If the coefficients
If
Theorem 5.3. The endemic equilibrium point
Here it can be noted that if the equilibrium point
5.1. Analysis of Hopfbifurcation
Now, we shall investigate the Hopf bifurcation of the model system (2.1), for which we need to verify the transversality condition
On differentiating (4.3) with respect to τ, we get
which leads to
Since
expressions involved in the above derivative at
where
Now
Using the equations in (5.1) and (5.2), we can rewrite above expression as follows:
Therefore
As
We already have
6. Numerical Simulations
Here first we simulate our model system (2.1) for the following set of parameters without delay i.e. when
For this set of parameters we get our nontrivial equilibrium

Figure 1. Variation of state variables with time with delay
τ=0 .

Figure 2. Variation of state variables with time with delay
τ=1.0 .

Figure 3. Variation of state variables with time with delay
τ=2.0 .

Figure 4. Variation of state variables with time with delay
τ=2.0465 .
7. Conclusions
This paper aims to study the impact of delay on the cancer virotherapy. Here we formulate a mathematical model by incorporating the time gap between the time of interaction of virus with tumor cells and the time when the tumor cells become infectious to produce new virus particles. We analyze the model by keeping in mind the partial success of virotherapy. There exists a threshold value of delay below which the equilibrium point corresponding to partial success of virotherapy is stable. This threshold value is derived analytically and computed numerically. It is also observed that this threshold value changes with change in the other parameters such as β and δ etc. We also verify the transversality condition for Hopfbifurcation. Numerical simulation is performed to support the analytical findings.
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