**1. Introduction**

Oncolytic viruses are a form of immunotherapy that uses viruses to infect and destroy cancer cells. These viruses can selectively replicate in cancer cells but leave healthy normal cells largely intact. In oncolytic virotherapy, the free viruses infect tumor cells and replicate themselves in tumor cells; upon analysis of infected tumor cells, new virion particles burst out and proceed to infect additional tumor cells. This idea was initially tested in the middle of the last century and merged with renewed ones over the last three decades due to the technological advances in virology and in the use of viruses as vectors for gene transfer. Over the last decade, great efforts have been made for understanding dynamics and molecular mechanics of viral cytotoxicity of oncolytic viruses. Those efforts provided an interesting possible alternative therapeutic approach to help cure cancer patients. However, the outcomes of virotherapy depends in a complex way on interactions between viruses and tumor cells [1]. One of the main advantages of applying the oncolytic virotherapy is that it can selectively damage cancerous tissues leaving normal cells unharmed. In addition, oncolytic viruses can mediate the killing of the normal cells by indirect mechanisms such as the destruction of tumor blood vessels, the amplification of specific anticancer immune responses or through the specific activities of transgene-encoded proteins expressed from engineered viruses.

During the last two decades, several mathematical models have been applied to understanding oncolytic virotherapy. For example, Wu et al. [2] and Wein et al. [3] proposed and analyzed some partial differential equations models to study some aspects of cancer virotherapy. For ordinary differential equations models, Wodarz in [4, 5], Komarova and Wodarz [6], Novozhilov et al. [7], and Bajzer et al. [8], Tian in [9, 10], and others. Wodarz and Komarova [11] have modeled the dynamics of the oncolytic virus replication by ordinary differential equations that describe the development of the average population sizes of cells and viruses over time. For this purpose, they used a generalized approach and considered a class of models instead of a specific model and took into account two populations: uninfected tumor cells, denoted by *x* and infected tumor cells, denoted *y*. The general model is based on the law of mass action and is given by

$$\begin{aligned} \frac{d\mathbf{x}}{dt} &= \mathbf{x}F(\mathbf{x}, \mathbf{y}) - \beta \mathbf{y}G(\mathbf{x}, \mathbf{y})\\ \frac{d\mathbf{y}}{dt} &= \beta \mathbf{y}G(\mathbf{x}, \mathbf{y}) - a\mathbf{y}, \end{aligned} \tag{1}$$

Considering this last model as a starting point of our discussion of mathematical

virotherapy represented by the stable positive equilibrium solution. Since the tumor load is a decreasing function of the burst size, the minimum tumor load can be reached by genetically increasing the burst size of the virus up to the second threshold value. If the set in which the positive equilibrium solution is stable has more than one open intervals, we can increase the burst size up to the supreme value of this set, and still have stable partial therapeutic success with even lower tumor load. Once the burst size is greater than the second threshold value, there are one or three families of stable periodic solutions to the system of virotherapy

For simplicity, the above system can be non-dimensionalized by setting *τ* ¼ *δt*,

*dt* <sup>¼</sup> *rx*ð Þ� <sup>1</sup> � *<sup>x</sup>* � *<sup>y</sup> axv*

(4)

*<sup>a</sup>* ; and unstable when

models of oncolytic virotherapy, we first show some analytical results. In this model, there are two threshold values for the burst size. When the burst size is smaller than the first threshold value, virotherapy always fails. When the burst size

is in the between of the two threshold values, we have a partial success of

*Mathematical Modeling and Dynamics of Oncolytic Virotherapy*

*DOI: http://dx.doi.org/10.5772/intechopen.96963*

*x* ¼ *Kx*^, *y* ¼ *K*^*y*, *v* ¼ *K*^*v*, and rename parameters *r* ¼ *λ*, *a* ¼ *βK*, and *c* ¼ *γ:* Then dropping all hats over the variables and write *τ* as *t*, we have

*dt* <sup>¼</sup> *axv* � *<sup>y</sup>*

At *b* ¼ *μ*1, the positive equilibrium *E*<sup>þ</sup> moves into the domain *D* ¼ f g ð Þ *x*, *y*, *v* : *x*≥0, *y*≥0, *v*≥0, 0 ≤*x* þ *y*≤1 , a type of transcritical bifurcation

is locally asymptotically stable. When *b*>*μ*<sup>1</sup> and *b*∈*In*, *E*<sup>þ</sup> is unstable. Hopf bifurcations occur for some *b*≥ *μ*2, and these bifurcations give rise to one or three families of periodic solutions. Here, *μ*<sup>1</sup> and *μ*<sup>2</sup> are thresholds, *Ip* ¼ *b*>*μ*<sup>1</sup> f g : *H b*ð Þ> 0 ,

*In* ¼ *b*>*μ*<sup>1</sup> f g : *H b*ð Þ<0 , and *H b*ð Þ is defined next in formula (10).

• *<sup>E</sup>*<sup>1</sup> is globally asymptotically stable when *<sup>b</sup>*<<sup>1</sup> <sup>þ</sup> *<sup>c</sup>*

*dt* <sup>¼</sup> *by* � *axv* � *cv*

Model (4) has three equilibrium points, *E*<sup>0</sup> ¼ ð Þ 0, 0, 0 , *E*<sup>1</sup> ¼ ð Þ 1, 0, 0 , and the positive equilibrium *<sup>E</sup>*<sup>þ</sup> <sup>¼</sup> *<sup>x</sup>*<sup>∗</sup> , *<sup>y</sup>* <sup>∗</sup> , *<sup>v</sup>* <sup>∗</sup> ð Þ. The equilibrium *<sup>E</sup>*<sup>0</sup> is always unstable for all positive values of the burst size *b*. The equilibrium *E*<sup>1</sup> is globally asymptotically

occurs with *E*<sup>1</sup> and *E*þ. As the parameter *b* increases, while *μ*<sup>1</sup> <*b*<*μ*<sup>2</sup> and *b*∈ *Ip*, *E*<sup>þ</sup>

• When *μ*<sup>1</sup> <*b*<*μ*2, the equilibrium solution *E*<sup>þ</sup> is locally asymptotically stable

Two types of bifurcations occur in the system as the parameter *b* varies. A transcritical bifurcation at *b* ¼ *μ*<sup>1</sup> introduces the equilibrium point *E*<sup>þ</sup> into the positive invariant domain *D*. The Hopf bifurcation at some value *b*>*μ*<sup>1</sup> gives rise

to the periodic solutions. The system (4) is a basic model of the oncplytic virotherapy. Three equilibrium points can be found: the trivial equilibrium

*dx*

*dy*

*dv*

It is assumed that all parameters are nonnegative.

stable when 0 <*b*<*μ*1, and it is unstable when *b*≥*μ*1.

• *E*<sup>0</sup> is unstable.

*<sup>b</sup>*≥<sup>1</sup> <sup>þ</sup> *<sup>c</sup> a* .

**141**

dynamics.

where the function *F* describes the growth properties of the uninfected tumor cells, and the function *G* describes the rate at which tumor cells become infected by the virus. The two functions can take several forms depending on the biological content and meaning that we may want to incorporate into the model. The parameter *β* represents the infectivity of the virus, and the death rate *ay* represents the virus-infected cells die.

Then, a three populations model was introduced by Wodarz [12] as

$$\begin{aligned} \frac{d\mathbf{x}}{dt} &= r\mathbf{x}\left(1 - \frac{\mathbf{x} + \mathbf{y}}{C}\right) - d\mathbf{x} - \beta \mathbf{x}v\\ \frac{d\mathbf{y}}{dt} &= \beta \mathbf{x}v - (d + a)\mathbf{y} \\ \frac{dv}{dt} &= a\mathbf{y} - \gamma v, \end{aligned} \tag{2}$$

in which *v* stands for the free virus population and *C* is maximal tumor size. The term *αy* models the release of virions by infected tumor cells, and *γv* is the clearance rate of free virus particles by various causes including non-specific binding and generation of defective interfering particles. The death rate of tumor cells *dx* seems redundant, since it is included in the logistic model.
