**5. Optimization by control theory**

In this section, we develop a model for the controlled infected brain tumor cells. optimal control theory is applied to the cost functional and is supposed to achieve the ultimate goal of optimizing that functional and find a best strategy for minimizing the cost of the virotherapy. The goal here is to model, analyze, and explore optimal ways that can minimize a tumor and the cost of the virotherapy.

Optimal control theory is a branch of the applied mathematics that deals with finding the best possible control that can take a dynamical system from one state to another. The Hamiltonian of optimal control theory was developed by the Russian mathematician Lev Pontryagin as a part of his investigation into the maximum principle. Pontryagin proved that the necessary condition for solving certain optimal control problems is that the control should be chosen in such a way that minimizes the Hamiltonian, [17].

Rescaling the system to ease mathematical treatment, we use same parameter

¼ ð Þ 1 � *u rx*ð Þ� 1 � *x* � *y axv*

(16)

¼ *axv* � ð Þ 1 � *u y*

with fixed initial conditions *x*ð Þ 0 , *y*ð Þ 0 , *v*ð Þ 0 and a fixed final time T.

¼ *b*ð Þ 1 � *u y* � *axv* � *cv:*

According to The Pontryagin's Maximum Principle, if *u*ð Þ*:* ∈ Ω is optimal for the problem under consideration, the minimizer with the initial conditions and fixed final time *T*, then there exists a nontrivial absolutely continuous mapping

System (16) along with the initial conditions and the final time *T* has a unique optimal solution *<sup>x</sup>*<sup>∗</sup> ð Þ� , *<sup>y</sup>* <sup>∗</sup> ð Þ� , *<sup>v</sup>* <sup>∗</sup> ð Þ ð Þ� associated to an optimal control *<sup>u</sup>*<sup>∗</sup> ð Þ on 0, ½ � *<sup>T</sup> :*

For the simulations and numerical results of the basic model and those of the

summarized in **Table 1**. We also combing those results in the **Figures 1**–**4** below. Note that *α* ¼ 1 represents the simulation of the basic model (2.2), whereas the other values of *α* describe the memory of the derivatives of the basic model. The parameter values are *r* ¼ 0*:*36, *a* ¼ 0*:*11, and *c* ¼ 0*:*2. By considering *b* ¼ 9, the following equilibrium points can be obtained *E*<sup>0</sup> ¼ ð Þ 0, 0, 0, 0 , *E*<sup>1</sup> ¼ ð Þ 1, 0, 0 , *E*<sup>2</sup> ¼ ð Þ 0*:*6, 0*:*0730, 2*:*5729 . Here the bifurcation parameter values are *μ*<sup>1</sup> ¼ 5 and *μ*<sup>2</sup> ¼ 27*:*766. When 5<*b*<27*:*766, *E*<sup>þ</sup> is locally asymptotically stable while *E*<sup>1</sup> is unstable. The equilibrium point *E*<sup>0</sup> is always unstable. **Figure 1** shows the treatment will eventually reach the equilibrium point *E*<sup>1</sup> that is locally asymptotically stable. **Figures 2** and **3** show periodic solutions rising from Hopf bifurcation, and **Figure 4**

*Optimal state variables for the controlled and the uncontrolled systems subject to the initial values x* ¼ 0*:*5, *y* ¼ 0*:*5*, and v* ¼ 1*:*5*, b* ¼ 4*, and the admissible control set versus trajectories without control measures.*

fractional approach, we use the same parameter values used in [10] and

<sup>1</sup> , Ψ<sup>∗</sup>

*rx*ð Þ 1 � *x* � *y* Ψ<sup>1</sup> � *y*ð Þ Ψ<sup>2</sup> þ *b*Ψ<sup>3</sup>

*<sup>i</sup>* ð Þ¼ T 0, *i* ¼ 1, 2, 3*:* Furthermore,

<sup>1</sup> , and Ψ<sup>∗</sup>

*<sup>B</sup> :*

<sup>3</sup> , such that with

rescaling that was used for the previous models and we get

d*x* d*t*

*Mathematical Modeling and Dynamics of Oncolytic Virotherapy*

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

d*y* d*t*

d*v* d*t*

Now we come to the main result in this section, [19]

<sup>Ψ</sup> : ½ �! 0, 1 <sup>3</sup>

**Figure 5.**

**149**

.

transversality conditions Ψ<sup>∗</sup>

The proof is given in [19].

Moreover, there exists adjoint functions Ψ<sup>∗</sup>

*<sup>u</sup>*<sup>∗</sup> ðÞ¼ *<sup>t</sup>*

**6. Numerical simulation and discussion**

The general form of the control function *u t*ð Þ is given by

$$\mathbf{J}(\mathbf{u}(\mathbf{t})) = \Psi(\mathbf{x}(\ \mathbf{T})) + \int\_0^1 \mathbf{L}(\mathbf{x}(\mathbf{t}), \mathbf{u}(\mathbf{t}), \mathbf{T}(\mathbf{t})) \mathbf{dt} \,\mathbf{u}$$

where *x t*ð Þ is the system state which evolves according to the state equation

$$\dot{\boldsymbol{x}} = \mathbf{f}(\boldsymbol{\varkappa}(t), \boldsymbol{\mu}(t), t) \quad \boldsymbol{\varkappa}(\mathbf{0}) = \boldsymbol{\varkappa}\_0 \quad t \in [\mathbf{0}, T].$$

The Hamiltonian is defined as

$$\mathbf{H}(\mathbf{x}, \Psi, \boldsymbol{\mu}, \mathbf{t}) = \Psi^{\mathrm{T}}(\mathbf{t})\mathbf{f}(\boldsymbol{\kappa}, \boldsymbol{\mu}, \mathbf{t}) + \mathrm{L}(\boldsymbol{\kappa}, \boldsymbol{\mu}, \mathbf{t}),$$

where Ψð Þt is a vector of costate variables of the same dimension as the state variable *x*ð Þt such that, [18]

$$
\dot{\Psi}(t) = -\frac{\partial H}{\partial \mathbf{x}}\,.
$$

Applying the control theory approach, we reformulate the basic model by introducing a control function *u t*ð Þ which represents efforts on damaging the tumor cells AND 1ð Þ � *u t*ð Þ represents the growth rate of the infected cells. After incorporating the control *u* into the basic model, we obtain the following model with control

$$\begin{aligned} \frac{d\mathbf{x}}{dt} &= (\mathbf{1} - \mathbf{u}(\mathbf{t})) \lambda \mathbf{x} \left(\mathbf{1} - \frac{\mathbf{x} + \mathbf{y}}{\mathbf{K}}\right) - \beta \mathbf{x} v \\\\ \frac{d\mathbf{y}}{dt} &= \beta \mathbf{x} v - (\mathbf{1} - \mathbf{u}(\mathbf{t})) \delta \mathbf{y} \\\\ \frac{d\mathbf{v}}{dt} &= \mathbf{b} \delta(\mathbf{1} - \mathbf{u}(\mathbf{t})) \mathbf{y} - \beta \mathbf{x} v - \gamma v. \end{aligned} \tag{15}$$

The control is usually assumed to be bounded by maximim value less than 1 and greater than 0. For our current model, we assume the maximum value is 0*:*9, a choice that make our proposed model more realistic from a medical view point.

The objective function will be the function that will host our optimal value *u*<sup>∗</sup> and it is given by

$$\mathbf{J}(\boldsymbol{\mu}(\mathbf{t})) = \int\_0^\mathrm{T} \mathbf{y}(\mathbf{t}) + \frac{\mathbf{1}}{2} \mathbf{B} \mathbf{u}^2 \,\mathrm{d}\mathbf{t} \,\mathrm{d}\mathbf{t}.$$

Where *B* is a measure of the relative cost of interventions associated to the control *u t*ð Þ. Our goal is to minimize the number of the infected tumor cells by choosing an appropriate strategy that can lower the number of free viruses as well. As a result of that, the cost of treatment will be lowered.

The admissible set of control functions is defined as

$$\mathfrak{Q} = \left\{ u(\cdot) \in \mathcal{L}^{\infty}(\mathbf{0}, t\_f) \, : \, 0 \lessdot u(t) \lessdot u\_{\text{max}}, \forall t \in [\mathbf{0}, T] \right\}.$$

*Mathematical Modeling and Dynamics of Oncolytic Virotherapy DOI: http://dx.doi.org/10.5772/intechopen.96963*

Rescaling the system to ease mathematical treatment, we use same parameter rescaling that was used for the previous models and we get

$$\begin{aligned} \frac{d\mathbf{x}}{dt} &= (\mathbf{1} - u)x\mathbf{x}(\mathbf{1} - \mathbf{x} - \mathbf{y}) - a\mathbf{x}v\\ \frac{d\mathbf{y}}{dt} &= a\mathbf{x}v - (\mathbf{1} - u)\mathbf{y} \\ \frac{d\mathbf{v}}{dt} &= b(\mathbf{1} - u)\mathbf{y} - a\mathbf{x}v - cv. \end{aligned} \tag{16}$$

with fixed initial conditions *x*ð Þ 0 , *y*ð Þ 0 , *v*ð Þ 0 and a fixed final time T.

According to The Pontryagin's Maximum Principle, if *u*ð Þ*:* ∈ Ω is optimal for the problem under consideration, the minimizer with the initial conditions and fixed final time *T*, then there exists a nontrivial absolutely continuous mapping <sup>Ψ</sup> : ½ �! 0, 1 <sup>3</sup> .

Now we come to the main result in this section, [19]

System (16) along with the initial conditions and the final time *T* has a unique optimal solution *<sup>x</sup>*<sup>∗</sup> ð Þ� , *<sup>y</sup>* <sup>∗</sup> ð Þ� , *<sup>v</sup>* <sup>∗</sup> ð Þ ð Þ� associated to an optimal control *<sup>u</sup>*<sup>∗</sup> ð Þ on 0, ½ � *<sup>T</sup> :* Moreover, there exists adjoint functions Ψ<sup>∗</sup> <sup>1</sup> , Ψ<sup>∗</sup> <sup>1</sup> , and Ψ<sup>∗</sup> <sup>3</sup> , such that with transversality conditions Ψ<sup>∗</sup> *<sup>i</sup>* ð Þ¼ T 0, *i* ¼ 1, 2, 3*:* Furthermore,

$$
\mu^\*\left(t\right) = \frac{r\varkappa(1-\varkappa-\jmath)\Psi\_1 - \jmath(\Psi\_2+b\Psi\_3)}{B}.
$$

The proof is given in [19].

#### **6. Numerical simulation and discussion**

For the simulations and numerical results of the basic model and those of the fractional approach, we use the same parameter values used in [10] and summarized in **Table 1**. We also combing those results in the **Figures 1**–**4** below. Note that *α* ¼ 1 represents the simulation of the basic model (2.2), whereas the other values of *α* describe the memory of the derivatives of the basic model. The parameter values are *r* ¼ 0*:*36, *a* ¼ 0*:*11, and *c* ¼ 0*:*2. By considering *b* ¼ 9, the following equilibrium points can be obtained *E*<sup>0</sup> ¼ ð Þ 0, 0, 0, 0 , *E*<sup>1</sup> ¼ ð Þ 1, 0, 0 , *E*<sup>2</sup> ¼ ð Þ 0*:*6, 0*:*0730, 2*:*5729 . Here the bifurcation parameter values are *μ*<sup>1</sup> ¼ 5 and *μ*<sup>2</sup> ¼ 27*:*766. When 5<*b*<27*:*766, *E*<sup>þ</sup> is locally asymptotically stable while *E*<sup>1</sup> is unstable. The equilibrium point *E*<sup>0</sup> is always unstable. **Figure 1** shows the treatment will eventually reach the equilibrium point *E*<sup>1</sup> that is locally asymptotically stable. **Figures 2** and **3** show periodic solutions rising from Hopf bifurcation, and **Figure 4**

**Figure 5.**

*Optimal state variables for the controlled and the uncontrolled systems subject to the initial values x* ¼ 0*:*5, *y* ¼ 0*:*5*, and v* ¼ 1*:*5*, b* ¼ 4*, and the admissible control set versus trajectories without control measures.*

shows the dynamics of tumor cells vs infected tumor cells when *α* ¼ 0*:*98 which is almost the same as the result of the usual derivative *α* ¼ 1. *E*<sup>þ</sup> is locally asymptotically stable when 5< *b*< 27*:*766.

200 days for burst size *b* ¼ 9, and 1000 days for the oscillation to capture that

Numerical results show that the existence of the control can improve the growth of the normal cells until approximately 60 days of the therapy and will be stabilized after then. Whereas the number of the infected cells will be dropped significantly after the fifth day of the treatment until they are completely terminated in the day 50. The dynamics is hugely determined by the burst size in addition to the other control parameter values. The numerical results clearly show that the virotherapy can reduce the tumor load within days of the therapy and reduces number of the free viruses that are needed in the therapy. As a result, the cost of the therapy is

behavior as shown in the **Figures 5**–**8**.

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

*Mathematical Modeling and Dynamics of Oncolytic Virotherapy*

minimized. See **Figures 5**–**8**.

**Author details**

**151**

Abdullah Abu-Rqayiq

Texas A&M University-Corpus Christi, United States

provided the original work is properly cited.

\*Address all correspondence to: abdullah.aburqayiq@tamucc.edu

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

The numerical results of the optimal control system (5.2) can be obtained by implementing forward fourth-order Runge-Kutta method for state system and the backward one for the adjoint system. The method depends on the choice of an initial guess for the value of the control *u*. The optimal control system is estimated to predict the evolution of the tumors cells relative to specific choices of virus bust size. Simulation shows the results in time scale of 100 days for burst size *b* ¼ 4,

**Figure 6.**

*Optimal state variables for the controlled and the uncontrolled systems subject to the initial values x* ¼ 0*:*5, *y* ¼ 0*:*5*, and v* ¼ 1*:*5*, b* ¼ 9*, and the admissible control set versus trajectories without control measures.*

#### **Figure 7.**

*Damped oscillators appear for the controlled and the uncontrolled systems subject to the initial values x* ¼ 0*:*5, *y* ¼ 0*:*5*, and v* ¼ 1*:*5*, b* ¼ 26*, and the admissible control set versus trajectories without control measures.*

#### **Figure 8.**

*The optimal control u*<sup>∗</sup> *for the Oncolytic virotherapy model subject to the initial values x* <sup>¼</sup> <sup>0</sup>*:*5, *<sup>y</sup>* <sup>¼</sup> <sup>0</sup>*:*5*, and v* ¼ 1*:*5*, b* ¼ 9*, and the admissible control.*

#### *Mathematical Modeling and Dynamics of Oncolytic Virotherapy DOI: http://dx.doi.org/10.5772/intechopen.96963*

200 days for burst size *b* ¼ 9, and 1000 days for the oscillation to capture that behavior as shown in the **Figures 5**–**8**.

Numerical results show that the existence of the control can improve the growth of the normal cells until approximately 60 days of the therapy and will be stabilized after then. Whereas the number of the infected cells will be dropped significantly after the fifth day of the treatment until they are completely terminated in the day 50. The dynamics is hugely determined by the burst size in addition to the other control parameter values. The numerical results clearly show that the virotherapy can reduce the tumor load within days of the therapy and reduces number of the free viruses that are needed in the therapy. As a result, the cost of the therapy is minimized. See **Figures 5**–**8**.
