A New Fractional Model for Cancer Therapy with M1 Oncolytic Virus

)e aim of this work is to propose and analyze a new mathematical model formulated by fractional differential equations (FDEs) that describes the dynamics of oncolytic M1 virotherapy. )e well-posedness of the proposed model is proved through existence, uniqueness, nonnegativity, and boundedness of solutions. Furthermore, we study all equilibrium points and conditions needed for their existence. We also analyze the global stability of these equilibrium points and investigate their instability conditions. Finally, we state some numerical simulations in order to exemplify our theoretical results.


Introduction
Cancer is a collection of related diseases where some of the body's cells divide continuously and spread into surrounding tissues. Cancer is caused by certain changes to genes. It can start almost anywhere in the human body. Old or damaged cells survive when they should die; new cells form when the body does not need them. ese extra cells can divide continuously and may form tumors. A tumor becomes dangerous when it begins to form extensions to neighboring areas (metastasis) [1]. is is why it is important to detect cancer as early as possible in order to avoid this migration. Cancer treatment is adapted according to each situation. ere are different cancer treatments used alone or in combination, such as surgery, radiotherapy, chemotherapy, hormone therapy, immunotherapy, and virotherapy. Virotherapy is one of the new therapies; it consists in using a virus after having reprogrammed it.
is virus is called oncolytic virus. Oncolytic viruses infect and destroy cancer cells; they use the cell's genetic machinery to make copies of themselves and subsequently spread to surrounding uninfected cells [2].
According to a medical experiment, in vitro, in vivo, and ex vivo studies showed potent oncolytic efficacy and high tumor tropism of alphavirus M1, which is a naturally occurring and a selective oncolytic virus targeting zinc-finger antiviral protein (ZAP) deficient cancer cells [3]. To model the role of the M1 virus in oncolytic virotherapy, Wang et al. [4] proposed a nonlinear system governed by ordinary differential equations (ODEs) that describe the growth of normal cells, tumor cells, and the M1 virus with limited nutrients. Elaiw et al. [5] extended the model presented in [4] by including spatial effects and anti-tumor immune response mediated by cytotoxic T lymphocyte (CTL) cells. e results in [5] indicated that the immune response has a negative impact on oncolytic M1 virotherapy, and it reduced its efficiency.
On the other hand, all the above mathematical models neglected the memory effect by considering only integerorder derivatives. However, fractional-order derivative provides an excellent tool for describing memory and hereditary properties which exist in most biological systems. For instance, Cole [6] proved that the membranes of cells of the biological organism have fractional-order electrical conductance since the memory means that the system's response is dependent not only on the current state but on its complete history. erefore, the classical integer-order derivative does not reflect this memory effect because it is a local operator, unlike the fractional derivative. e main purpose of this study is to develop a mathematical model governed by fractional-order differential equations (FDEs) to study the effect of memory on the dynamics of oncolytic M1 virotherapy. So, the rest of the paper is outlined as follows: the next section is devoted to the formulation of the model, including the well-posedness and the existence of equilibria. Section 3 focuses on stability analysis. Section 4 deals with numerical simulations in order to illustrate our main analytical results. Finally, a brief conclusion is given in Section 5.

Model Formulation and Preliminaries
In this section, we propose the following FDE model: where S(t), N(t), T(t), and V(t) are the concentrations of nutrient, normal cells, tumor cells, and M 1 virus at time t, respectively. e parameters A and B are the recruitment rates of nutrient and M 1 virus, respectively. Also, B represents the minimum effective dosage of medication. e normal and tumor cells consume the nutrient at rates β 1 SN and β 2 ST, respectively. e growth rate of normal cells as a result of consuming the nutrient is given by r 1 β 1 SN, while the growth rate of tumor cells is given by r 2 β 2 ST. e virus infects and kills tumor cells at rate β 3 TV, and it replicates at rate r 3 β 3 TV. e parameter d is the washout constant rate of nutrient and bacteria. e parameters ε 1 , ε 2 , and ε 3 are the natural death rates of normal cells, tumor cells, and M 1 virus, respectively. e operator D α denotes the Caputo fractional derivative with α ∈ (0, 1] that describes the memory effect. It is important to note that the ODE mathematical model verifying potent oncolytic efficacy of M1 virus [4] is a special case of our model presented by system (1), and it suffices to take α � 1. Furthermore, to prove that our model is biologically well-posed, we assume that the initial conditions of (1) satisfy: Proof. It is not hard to show that the vector function of system (1) satisfies the first condition of Lemma 4 in [7]. It remains to prove the second condition. Let where us, is implies that the second condition of Lemma 4 in [7] is satisfied.
On the other hand and according to (1), we have By Lemmas 5 and 6 in [7], we deduce that the solution of (1) is nonnegative.
It remains to prove the boundedness of solutions. en we consider the following function: Complexity which implies that S, N, T, and V are bounded. is completes the proof. Now, we establish the equilibrium points of our model. It is obvious that any equilibrium point of system (1) satisfies the following algebraic equations: From (13), we have N � 0 or S � (d + ε 1 )/r 1 β 1 . Similarly, equation (14) leads to T � 0 or r 2 β 2 S � d + ε 2 + β 3 V: is number reflects the ability of absorbing nutrients by normal cells. It is called absorbing number [4]. When A 1 > 1, system (1) has another equilib- , and where Since a 1 > 0 and a 3 < 0, we have δ � a 2 2 − 4a 1 a 3 ≥ 0. us equation (17) has two roots given by is number reflects the ability of absorbing nutrients by tumor cells. It can be called the absorbing number of nutrients by tumor cells. Hence, system (1) has another equilibrium point when Substituting S and T in (12), we obtain Furthermore, N > 0 implies that us, system (1) has another equilibrium point when A 2 > A 1 : All the above cases are summerized in the following result. □ Theorem 2. Let A 1 and A 2 be defined by (16) and (20). en

Stability Analysis
In this section, we focus on the stability analysis of the equilibria E 0 , E 1 , E 2 , and E 3 .
Proof. In order to show the first part of this theorem, we consider the following Lyapunov functional: Based on the property of fractional derivatives given in [8], we get 4 Complexity [9], we deduce that E 0 is globally asymptotically stable for A 1 ≤ 1 and It remains to investigate the dynamical property of E 0 in case when A 1 > 1 or A 2 > 1 + (Bβ 3 /(d + ε 2 )(d + ε 3 )). For this purpose, we compute the characteristic equation at E 0 that is given by where We have

en the tumor-free equilibrium E 1 is globally asymptotically stable if
and it is unstable if Proof. Consider the following Lyapunov functional: en D α L 1 ≤ 0 when Obviously, D α L 1 � 0 if and only if S � S 1 , N � N 1 , T � 0, and V � V 1 .
en the largest invariant set contained in (S, N, T, V)|D α L 1 (t) � 0 is the singleton E 1 . By LaSalle's invariance principle, we deduce that E 1 is globally asymptotically stable for On the contrary, the characteristic equation at E 1 is given by where f(λ) � (d + λ + β 1 N 1 )(d + ε 1 + λ − r 1 β 1 S 1 ) + r 1 β 2 1 N 1 S 1 . One of the eigenvalues of (39) is 6 Complexity We observe that λ 1 > 0 if us, E 1 is unstable when and (A 2 /A 1 ) > 1. en the treatment failure equilibrium E 2 is globally asymptotically stable if and becomes unstable if Proof. Consider the following Lyapunov functional: By computation, we find us, S 2 − S 3 ≤ 0 implies that Consequently, D α L 2 ≤ 0 when en the largest invariant set contained in (S, N, T, V)|D α L 2 (t) � 0 is the singleton E 2 . By LaSalle's invariance principle, we deduce that E 2 is globally asymptotically stable for On the other side, the characteristic equation at E 2 is given by where Complexity (52) One of the eigenvalues of (51) is We can observe from the proof of part (a) that λ 2 > 0 if us, E 2 is unstable when

Theorem 6. e partial success equilibrium E 3 is globally asymptotically stable if
Proof. Consider the following Lyapunov functional erefore D α L 3 ≤ 0, with equality if and only if S � S 3 and V � V 3 . By a simple computation, we show that D α L 3 � 0 if and only if S � S 3 , N � N 3 , T � T 3 and V � V 3 . It follows from LaSalle's invariance principale that E 3 is globally asymptotically stable under the conditions that this point exists. ■ □

Numerical Simulations
In this section, we give some numerical simulations to illustrate and validate our theoretical results, and we present some biological interpretations. We choose the time interval from t � 0 to t � 400 with a step size Δt � 0.1. We take A � 0.02, d � 0.02, B � 0.01, r 1 � 0.8, and r 3 � 0.5. e parameters β 1 , β 2 , β 3 , ε 1 , ε 2 , ε 3 , and r 2 of model (1) are taken as free parameters.

Conclusion
In this paper, we have studied the dynamics of an oncolytic M1 virotherapy model, considering the memory effect denoted by the Caputo fractional derivative. e well-posedness of the proposed model was proved through nonnegativity and boundedness of solutions. We found that the model has four possible equilibrium points, namely, the competition-free equilibrium E 0 , the tumor-free equilibrium E 1 , the treatment failure equilibrium E 2 , and the partial success equilibrium E 3 . By constructing suitable Lyapunov functionals, the global stability of E 0 is determined by two threshold parameters that are the absorbing number of nutrients by normal cells A 1 and the absorbing number of nutrients by tumor cells A 2 , when A 1 ≤ 1 and A 2 ≤ 1 + (Bβ 3 /(d + ε 2 )(d + ε 3 )), E 0 is globally asymptotically stable, and these conditions determine when normal and tumor cells are lost, which may not be useful to test the viability of treatment. e tumor-free equilibrium E 1 exists and is globally asymptotically stable if A 1 > 1 and and these conditions show that the M1 virus succeeds to eliminate the tumor, which is helpful in improving virotherapy. e treatment failure equilibrium E 2 exists and is globally asymptotically stable if Complexity A 2 > A 1 , and these conditions refer to the failure of the treatment, as indicated by his name. e partial success equilibrium E 3 exists and is globally asymptotically stable if ese results indicate the partial success of M1 virus in decreasing tumor cells and increasing normal cells, which can reduce the tumor's size and stabilize the disease progression.
From the above analytical results, we remark that the Caputo fractional derivative's memory does not affect the stability analysis of equilibria. Based on the numerical simulations, we observe that the fractional order affects the speed of convergence and the time for arriving to equilibria (Figures 1-4).
e results obtained in this study are based on the fractional derivative in sense of Caputo with singular kernel. It will be more interesting to model the dynamics of oncolytic M1 virotherapy by using the new generalized fractional derivative with nonsingular kernel [10]. Moreover, we will extend our model presented in (1) by taking into account other biological factors such as diffusion [11,12] and immunity [13,14].

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.