Computational and Mathematical Methods in Medicine

Volume 2019, Article ID 1732815, 12 pages

https://doi.org/10.1155/2019/1732815

## Analysis of a Multiple Delays Model for Treatment of Cancer with Oncolytic Virotherapy

Correspondence should be addressed to Adil El Alami laaroussi; moc.liamg@issuoraalimalalelida

Received 24 February 2019; Revised 8 July 2019; Accepted 29 July 2019; Published 30 September 2019

Academic Editor: Maria N. D.S. Cordeiro

Copyright © 2019 Adil El Alami laaroussi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Despite advanced discoveries in cancerology, conventional treatments by surgery, chemotherapy, or radiotherapy remain ineffective in some situations. Oncolytic virotherapy, i.e., the involvement of replicative viruses targeting specific tumor cells, opens new perspectives for better management of this disease. Certain viruses naturally have a preferential tropism for the tumor cells; others are genetically modifiable to present such properties, as the lytic cycle virus, which is a process that represents a vital role in oncolytic virotherapy. In the present paper, we present a mathematical model for the dynamics of oncolytic virotherapy that incorporates multiple time delays representing the multiple time periods of a lytic cycle. We compute the basic reproductive ratio , and we show that there exist a disease-free equilibrium point (DFE) and an endemic equilibrium point (DEE). By formulating suitable Lyapunov function, we prove that the disease-free equilibrium (DFE) is globally asymptotically stable if and unstable otherwise. We also demonstrate that under additional conditions, the endemic equilibrium is stable. Also, a Hopf bifurcation analysis of our dynamic system is used to understand how solutions and their stability change as system parameters change in the case of a positive delay. To illustrate the effectiveness of our theoretical results, we give numerical simulations for several scenarios.

#### 1. Introduction

The continuous improvement of conventional treatments in cancerology (surgery, chemotherapy, and radiotherapy) allows for a major progress in the fight against cancer. Nevertheless, in some situations, these modes of therapy may be ineffective. The development of new therapeutic strategies, therefore, appears essential in order to improve the healing of this disease. Thus, in the last decades, virotherapy of cancers appears to be a credible alternative to some situations, due to advanced discoveries and accurate informations about viruses and also the production of recombinant viral vectors which can be used in cancer gene therapy. The use of replicative viruses as antitumor therapeutic agents (oncolytic viruses) is based on the idea that they reproduce preferentially within the tumor cells; however, normal cells remain immune to infection. These viruses (oncolytic viruses) are either virus with a natural ability to replicate preferentially within tumor cells or viruses genetically modified to hold this property. Genetic modifications are primarily used to improve the specificity of viruses against tumor cells, by targeting a particular surface molecule, by deleting specific viral genes required for replication in healthy cells, or by using activatable viral promoters only in tumor cells [1]. Using viruses to treat cancer is not a new concept. Viruses have attracted interest as anticancer therapeutics since the beginning of the 20th century. However, for several years, research in this field was limited due to technological limitations. In the last 30 years, by increasing understanding of the nature of viruses, their mechanisms of oncolytic activity and their ability to manipulate and exploit genetically has prompted a new wave of oncolytic virotherapy. Today, there is extensive literature describing progress in both theoretical and clinical trials of oncolytic viruses. For more details, we refer the interested reader to [2–7].

Mathematical modeling of oncolytic virotherapy can illuminate the underlying dynamics of treatment systems and lead to optimal treatment strategies. Several studies have been the subject of the study of virotherapy. The first mathematical models of oncolytic viral therapy used ordinary differential equations to describe the fundamental interactions between two types of tumor cells (infected cells and uninfected cells) [8, 9]. Other works consider spatial representation of tumors [10, 11, 12], multiscale effects [13], and stochastic processes [14]. For a review of different mathematical modeling approaches ranging from ordinary differential equations to spatially explicit agent-based models, see [15–17].

Biological experiments helped to understand and explain the lytic cycle, which takes place in six stages; the six stages are as follows: attachment, penetration, transcription, biosynthesis, maturation, and lysis. To infect a new cell, a virus must penetrate inside the cell through the plasma membrane; the virus attacks a receptor on the cell membrane and then releases its genetic material in the cell. In the third step, the host cell’s DNA is degraded and the cell’s metabolism is directed to initiate the fourth step; biosynthesis, here the virus uses cellular mechanisms to constitute a large amount of viral components and, in the meantime, destroys the DNA of the host cell. Then, it enters the last two stages, maturation and lysis. When many copies of viral are manufactured, they are assembled into complete formed viruses. About 25 minutes after initial infection, approximately 200 new bacteriophages (virions) are formed. Once enough virions have matured and accumulated, specialized viral proteins are used to dissolve the bacterial cell wall, where they can go on to infect other cells and another lytic cycle begins (for more details on the lytic cycle, see [5, 18]). In this work, the dynamics of oncolytic virotherapy are studied by incorporating the viral lytic cycle time. The duration of the intracellular viral life cycle is an essential factor in viral therapy. For example, some viruses require only 30 minutes, some viruses take several hours to complete this process, and some may take days [19]. Therefore, it is necessary and realistic to consider and taking into account the cycle time in modeling the oncolytic virotherapy which allows us to better predict its dynamics. We construct a mathematical model of virotherapy with multiple delays representing the six time periods of the lytic cycle; it is assumed that the time of each stage of the lytic cycle is constant.

Several studies have studied and analyzed systems of delayed differential equations that model virotherapy. In the paper entitled “Hopf Bifurcation Analysis in a Delayed System for Cancer Virotherapy” [20], the authors consider a delayed differential equation system. In [19], Wang et al. propose a mathematical model for oncolytic virotherapy where they consider the time period of the viral lytic cycle as a delay parameter. The novelty of our work is modeling the variation of duration in the intracellular viral life cycle by adding multiple delays; each one represents the time period of each stage of the lytic cycle. We compute the basic reproductive ratio , and we show that there exist a disease-free equilibrium point (DFE) and an endemic equilibrium point (DEE). By formulating suitable Lyapunov function, we prove that the DFE is globally asymptotically stable if and unstable otherwise. We also demonstrate that under additional conditions, the DEE is stable. Furthermore, a bifurcation analysis of our dynamical system is used to understand how solutions and their stability change as the parameters in the system vary. To illustrate our theoretical results, numerical simulations are also presented for several scenarios.

This paper is organized as follows. In Section 2, we present our mathematical model. In Section 3, we compute the equilibrium of our model and investigate its stability. Following that, in Section 4, a bifurcation analysis of the dynamical system is used to understand how the solutions and their stability change as the parameters change. Numerical studies are shown, in Section 5, to validate the analytical results. Finally, we conclude the paper in Section 6.

#### 2. The Basic Mathematical Model

In a previous work [21], we analyzed the stability of a nonlinear system of differential equations based on the models proposed by [22, 23]. Our model contains three variables, which are, uninfected tumor cell population , infected tumor cell population , and free virus particles which are outside cell , and it has the following form:where , , and are given.

The term describes the logistic growth rate of an uninfected tumor cell population . The constant is the growth rate, with *K* being the carrying capacity or maximal tumor size so that . The term represents the rate of infected cells by free virus , with being the corresponding constant rate. The term models infection from an encounter between an infected cell and an uninfected cell resulting in cell fusion that produces a syncytium, with being the constant rate describing cell to cell fusion with the formation of syncytia. Infected cells die at a rate of , and is the rate of elimination of free virus particles by various causes including nonspecific binding and generation of defective interfering particles. Its burst size models the virus replication ability, the burst size of a virus, which is an essential parameter of virus reproduction. So, our model includes also a parameter *b* that models the burst size.

As we mentioned in Introduction, the model (1) did not take into account the time needed to complete the lytic cycle. As a reminder, the lytic cycle is the process where a virus overtakes a cell and uses the cellular machinery of its host to reproduce. Copies of the virus fill the cell to bursting, killing the cell and releasing viruses to infect more cells. The duration of this process varies from virus and more cells. Wang et al. [19] proposed a model of virotherapy with a single delay time; the originality of our work is to make a generalization by introducing 6 delays representing each period of stages of the lytic cycle in order to describe a more realistic situation because the virus goes through 6 stages of life and each one of them may have a different delay. We denote , the different times period of the lytic cycle. The rate of change of infected tumor cells at time *t* will be determined by the tumor cell population and free virus at time , namely, ; for more details about the lytic cycle, see Figure 1. Therefore, the model we propose is given as follows:where , , and .