Effect of virotherapy treatment on the dynamics of tumor growth through fractional calculus

Abstract

It has been reported that the quick progression and abnormal division of cells give rise to tumors. New cells are produced to fulfill new duties and replace older cells in the body of the host. It is obvious that tumors are deadly and affect practically every aspect of human life. Businesses, families, patients, and society as a whole lose financial resources and opportunities as a result of cancer and its treatment. Therefore, it is crucial to look at how tumors behave dynamically as a result of therapy and to highlight for policymakers and health officials the key elements of the system. In this research work, we formulate the transmission phenomena of tumor growth with the effect of virotherapy treatment in the fractional framework. Besides that, we focussed in our work on qualitative analysis and dynamic behavior of tumor dynamics. Using Banach’s and Schaefer’s theorems and the fixed-point theorem, the uniqueness and existence of the solution to the suggested model of tumor are examined. The necessary circumstances are established for our tumor system’s Ulam-Hyers stability. Laplace Adomian decomposition approach is used to examine the solution pathways and show the contribution of input factors on the dynamics of tumor growth. More specifically, we focused on the time series and chaotic behavior of the suggested system with change in input factors. The system’s most crucial components are recommended for regulating the dynamics of the tumor system.

Keywords

Virotherapy treatment tumor growth fractional calculus dynamical system numerical scheme dynamical behavior

Introduction

Cancer, the world’s second-largest health concern, results from the uncontrolled development of cancer cells and is caused by a variety of factors including tobacco, chemicals, radiation, acquired changes, and hormones. The stage of cancer is determined by the size of the tumor and the development of abnormal cells in the body. There are more than 200 different kinds of cancer. According to the American Cancer Society, prostate cancer is the most frequent kind of cancer in men and breast cancer is the most common type of cancer in women (Siegel et al., 2020). Lung cancer fatalities are frequent in the United States, according to the data. The development of an appropriate cancer therapy is a large study topic in medical science. Surgery has a better prognosis than other therapies in the early stages of solid cancer. Oncolytic virotherapy is a cancer treatment that includes infecting and breaking down malignant tumor cells with a virus without harming healthy cells. This therapy also stimulates the immune system, which helps to jump-start the body’s defenses (Russell et al., 2012). Oncolytic virotherapy works by attacking cancer as a virus would usually assault the body, without the need of chemotherapy or radiation. As a result, it is not susceptible to the same medication and radiation resistance that conventional therapies encounter. Because of how virotherapy works, it may be used in conjunction with other therapies; it can be given before or after surgery, or in between radiation or chemotherapy visits (Harrington et al., 2019; Russell et al., 2012; Varghese and Rabkin, 2002). The typical duration of virotherapy treatment is 3 years, with regular monitoring, and oncolytic virotherapy avoids the negative side effects that other cancer therapies, such as chemotherapy and radiation, are known to have (Zamarin et al., 2014). A virus must be capable of reproduction and selective infection in order to be suitable for oncolytic virotherapy.

In recent years, researchers have looked at the prospect of a single-shot cure, a vascular delivery threshold limit, and a new approach for viruses to target cancer cells. The first research of herpes simplex virus-1 (HSV-1) being utilized in children and young adults for its oncolytic characteristics was published in 2017. The first phase 1 study of a mutant HSV-1 virus, HSV1716, was completed by researchers from Nationwide Children’s Hospital and Cincinnati Children’s Hospital Medical Center. They discovered that the HSV1716 was both endurable and nontoxic once phase 1 is completed (Streby et al., 2017). Viruses from nine families have advanced to clinical trials for oncolysis as of 2018. Although these viruses have yielded promising outcomes, their effectiveness as a single agent treatment is restricted (Tadesse and Bekuma, 2018). Oncolytic viruses are now being studied to see if they can help with immunotherapy, particularly in malignancies that are sensitive to checkpoint inhibitors (Reale et al., 2019).

In the development of treatment methods, mathematical modeling and simulation of cancer at various biological scales are becoming increasingly essential. Mathematicians have used experimental data and analytical approaches to construct mathematical models that may be analyzed to identify crucial model parameters, as well as the short- and long-term dynamics of such a treatment strategy, from the beginning of oncolytic virotherapy. Previous mathematical studies have utilized a constant source rate to integrate virotherapy treatment into their models; these same articles have concentrated on the dynamics of infected and uninfected cell populations in their primary equations, without a free virus equation (Agarwal and Bhadauria, 2011; Malinzi et al., 2015). Some research (Malinzi et al., 2015; Piotrowska, 2016) incorporated an immune reaction in their system of differential equations, whereas others (Friedman and Lai, 2018; Jenner et al., 2018a, 2018b; Wodarz, 2016) did not include an immune response to malignant cells in their system of differential equations. Several scholars developed and tested mathematical models of cancer based on various assumptions (Bunimovich-Mendrazitsky et al., 2007; Mukhopadhyay and Bhattacharyya, 2009; Sarkar and Banerjee, 2005; Sherratt and Chaplain, 2001; Szymanska, 2003); however, further analysis is required to inspect the effect of treatment on the dynamics of tumor growth. The motivation of this research is to visualize the long-term behavior of the fractional dynamics of tumor that reflects the function of virotherapy on tumors as well as the impact of a cancer-specific immune response.

Literature predicts that fractional theory has extensive uses in science, engineering, and other fields of research (Jan et al., 2019; Munoz-Vazquez et al., 2019a, 2019b, 2020a, 2020b; Qureshi and Jan, 2021). Recent research suggests that fractional differential equations (FDE) can more accurately capture and represent biological processes in nature (Fatmawati et al., 2020; Jan et al., 2020; Shah et al., 2021; Srivastava et al., 2021); moreover, hereditary and crossover behavior can be effectively addressed through FDE. To gain a better understanding, we decide to look at the dynamics of the tumor in relation to the effects of the virotherapy treatment. Further research work is structured as follows: In the “Results of fractional theory” section, the basic concept and outcomes of fractional theory are listed. To depict a more realistic perspective, we organized the dynamics of tumor development with the influence of therapy in the “Fractional dynamics of tumor” section. In the “Existence theory” section, we investigate the suggested model, and the “Ulam-Hyers stability” section deduces the necessary conditions for Ulam-Hyers stability. In the “Proposed fractional system’s solution” section, we developed a numerical method to illustrate the time series of tumor dynamics with the influence of various factors. The article’s conclusion and final statement have been delivered in the “Concluding remarks” section.

Results of fractional theory

We will outline the basic ideas and concepts of fractional calculus in this part, which will be used to examine our system. It also has a wide range of useful applications in several scientific disciplines. For the study of recommended system, the fundamental concepts of the Caputo fractional are given below:

Definition 1. ( Kilbas et al., 2006) Assume a function $k (t)$ in a manner that $k (t) \in L^{1} ([a, b], R)$ , then the integral of fractional derivative is given by

I_{a^{+}}^{℘} k (t) = \frac{1}{Γ (℘)} \int_{0}^{t} (t - l)^{℘ - 1} k (l) dl

(1)

where $Γ$ indicates gamma function which generalizes the factorial function used in multiple differentiation and repeated integrations and the fractional order is ℘ with $0 < ℘ \leq 1$ .

Definition 2. ( Kilbas et al., 2006) Let us take a function $k (t)$ , then the Caputo fractional derivative is given by

^{c} D_{0^{+}}^{℘} k (t) = \frac{1}{Γ (n - ℘)} \int_{0}^{t} (t - l)^{n - ℘ - 1} k^{n} (l) dl, n - 1 < ℘ \leq n

(2)

Lemma 1. (Kilbas et al., 2006) Assume the below system

{\begin{matrix} ^{c} D_{0^{+}}^{℘} k (t) = v (t), t \in [0, τ], \\ k (0) = v_{0}, n - 1 < ℘ < n \end{matrix}

(3)

in which $v (t) \in C ([0, τ])$ . The solution of equation (3) is given by

k (t) = \sum_{i = 0}^{n - 1} d_{i} t^{i}, for i = 0, 1, \dots, n - 1

(4)

where $d_{i}$ in the above belongs to $R .$

Definition 3. (Podlubny, 1998) The Laplace transform for the Caputo Fractional operator is given as

{£ [}^{c} D_{0^{+}}^{℘} k (t)] = l^{℘} k (l) - \sum_{h = 0}^{n - 1} l^{℘ - h - 1} k^{h} (0), n - 1 < ℘ < n

(5)

Also, assume that

| | k | | = max_{t \in [0, τ]} {| k |, for all k \in Y}

(6)

be the norm on a Banach space $Y = C ([0, τ])$ .

Theorem 1. ( Granas and Dugundji, 2003) Assume a continuous mapping $H : Y \to Y$ on a Banach space $Y$ . Then, there exists a fixed point of $H$ , if

E = {k \in Y : k = λ Hk, λ \in (0, 1)}

(7)

is bounded.

Fractional dynamics of tumor

Here, we structure the dynamics of tumor to inspect the long-term behavior of the tumor cell population. In our formulation, we considered $(C_{U} (t))$ uninfected cells of tumor, $(C_{I} (t))$ infected cells of tumor, $(C_{E} (t))$ effector T cells and the last one is $(C_{V} (t))$ virion. In this formulation, we indicated the carrying capacity of tumor cells by $k$ while the growth rate of $(C_{U} (t))$ is denoted by $r$ . We assumed the decay rate of infected cells, effector T cells and virions to be $δ_{I}, δ_{E}$ , and $δ_{V}$ , respectively. The growth rate of T cells through infected tumor cells is indicated by $c$ while the rate of decay of infected and uninfected cells through T cells is denoted by $γ_{I}$ and $γ_{U}$ , respectively. Then, the dynamics of tumor is given by

{\begin{matrix} \frac{d C_{U}}{dt} = r C_{U} (1 - \frac{C_{I} + C_{U}}{k}) - β C_{U} C_{V} - γ_{U} C_{U} C_{E}, \\ \frac{d C_{I}}{dt} = β C_{U} C_{V} - γ_{I} C_{I} C_{E} - δ_{I} C_{I}, \\ \frac{d C_{E}}{dt} = c C_{I} - δ_{E} C_{E}, \\ \frac{d C_{V}}{dt} = N (t) + α δ_{I} C_{I} - δ_{V} C_{V} \end{matrix}

(8)

where the rate of infection is $β$ from $C_{U}$ . Furthermore, the conditions is given by

C_{U} (0) \geq 0, C_{I} (0) \geq 0, C_{E} (0) \geq 0 and C_{V}

Recent research recommended that the findings of FDE are more authentic and accurately represent biological phenomena. Motivated by the precise description of fractional operators, we organized our tumor model (8) as

{\begin{matrix} _{0}^{L C} D_{t}^{ϑ} C_{U} = r C_{U} (1 - \frac{C_{I} + C_{U}}{k}) - β C_{U} C_{V} - γ_{U} C_{U} C_{E}, \\ _{0}^{L C} D_{t}^{ϑ} C_{I} = β C_{U} C_{V} - γ_{I} C_{I} C_{E} - δ_{I} C_{I}, \\ _{0}^{L C} D_{t}^{ϑ} C_{E} = c C_{I} - δ_{E} C_{E}, \\ _{0}^{L C} D_{t}^{ϑ} C_{V} = N (t) + α δ_{I} C_{I} - δ_{V} C_{V} \end{matrix}

(9)

where $_{0}^{LC} D_{t}^{ϑ}$ indicates Liouville-Caputo’s derivative and fractional order is given by $ϑ$ . The tumor-free equilibrium of recommended system (9) of tumor cells infection is denoted by $E_{0}$ and is given by $E_{0} (C_{U}, C_{I}, C_{E}, C_{V})$ .

Existence theory

Here, the recommended fractional model’s qualitative theory (9) will be provided. In order to do this, we go about it as follows

{\begin{matrix} S_{1} (t, C_{U}, C_{I}, C_{E}, C_{V}) = r C_{U} (1 - \frac{C_{I} + C_{U}}{k}) - β C_{U} C_{V} - γ_{U} C_{U} C_{E}, \\ S_{2} (t, C_{U}, C_{I}, C_{E}, C_{V}) = β C_{U} C_{V} - γ_{I} C_{I} C_{E} - δ_{I} C_{I}, \\ S_{3} (t, C_{U}, C_{I}, C_{E}, C_{V}) = c C_{I} - δ_{E} C_{E}, \\ S_{4} (t, C_{U}, C_{I}, C_{E}, C_{V}) = N (t) + α δ_{I} C_{I} - δ_{V} C_{V} \end{matrix}

(10)

System (10) of tumor growth can be represented as

{\begin{matrix} ^{c} D_{0^{+}}^{ϑ} S (t) = P (t, S (t)), t \in [0, τ], \\ S (0) = S_{0}, 0 < ϑ \leq 1 \end{matrix}

(11)

where

{\begin{matrix} S (t) = C_{U} (t), \\ C_{I} (t), \\ C_{E} (t), \\ C_{V} (t) . \end{matrix} {\begin{matrix} , S_{0} (t) = C_{U 0}, \\ C_{I 0}, \\ C_{E 0}, \\ C_{V 0} . \end{matrix} {\begin{matrix} P (t, S (t)) = S_{1} (t, C_{U}, C_{I}, C_{E}, C_{V}) \\ S_{2} (t, C_{U}, C_{I}, C_{E}, C_{V}) \\ S_{3} (t, C_{U}, C_{I}, C_{E}, C_{V}) \\ S_{4} (t, C_{U}, C_{I}, C_{E}, C_{V}) \end{matrix}

(12)

Through Lemma (1), the integral of equation (11) is given by the following

S (t) = S_{0} (t) + \frac{1}{Γ (ϑ)} \int_{0}^{t} (t - l)^{ϑ - 1} P (l, S (l)) dl

(13)

Furthermore, we have the below for the recommended system:

(Z1) There exists a constants $H_{P}, K_{P}$ , and $s \in [0, 1)$ in a way that

| P (t, S (t)) | \leq H_{P} | S |^{s} + K_{P}

(14)

(Z2) There exists $M P > 0$ , and all $S$ , $\bar{S} \in Y$ in a way that

| P (t, S) - P (t, \bar{S}) | \leq M_{P} [| S - \bar{S} |]

(15)

In the next step, we introduce a map $U$ on $Y$ as

U S (t) = S_{0} (t) + \frac{1}{Γ (ϑ)} \int_{0}^{t} (t - l)^{ϑ - 1} P (l, S (l)) dl

(16)

One can find at least one solution of system (11) in the case of assumption $Z 1$ and $Z 2$ . This idea will be utilized for the solution of the recommended tumor growth system.

Theorem 2. If the condition $Z 1$ and $Z 2$ holds true, then there is at least one solution of the model (9) of tumor growth.

Proof. For the demanded result, we will utilize the well-known Schaefer’s fixed-point theorem. This result will be proved in four steps in the following way:

A1: Here, the continuity of the operator $U$ will be proved. For which, we assume that $S_{i}$ is continuous for $i = 1, 2, \dots, 9$ which provides that $P (t, S (t))$ is continuous. In the next step, assume $S_{j}$ , $S \in Y$ in a way that $S_{j} \to S$ , and we must have $U S j \to U S$ . Furthermore, take

\begin{matrix} | | U S_{j} - U S | | = max_{t \in [0, τ]} | \frac{1}{Γ (ϑ)} \int_{0}^{t} (t - l)^{ϑ - 1} P_{j} (l, S_{j} (l)) dl \\ - \frac{1}{Γ (ϑ)} \int_{0}^{t} (t - l)^{ϑ - 1} P (l, S (l)) dl | \\ \leq max_{t \in [0, τ]} \int_{0}^{t} | \frac{{(t - l)}^{ϑ - 1}}{Γ (ϑ)} | | P_{j} (l, S_{j} (l)) - P (l, S (l)) | dl \\ \leq \frac{τ^{ϑ} L_{P}}{Γ (ϑ + 1)} | | S_{j} - S | | \to 0 as j \to \infty \end{matrix}

(17)

Thus, the continuity of $P$ ensure the continuity of the operator $U$ .

A2: Here, the boundedness of the operator $U$ will be proved. Assume any $S \in Y$ , then the below will be fulfill through the operator $U$

\begin{matrix} | | U S | | = max_{t \in [0, τ]} | S_{o} (t) + \frac{1}{Γ (ϑ)} \int_{0}^{t} (t - l)^{ϑ - 1} P (l, S (l)) dl | \\ \leq | S_{0} | max_{t \in [0, τ]} \frac{1}{Γ (ϑ)} \int_{0}^{t} | (t - l)^{ϑ - 1} | | P (l, S (l)) | dl \\ \leq S_{0} | + \frac{τ^{ϑ}}{Γ (ϑ + 1)} [H_{P} | | S | |^{s} + K_{P}] \end{matrix}

(18)

For the required result, we take any $S \in G$ , as the set $G$ is bounded; therefore, we can find an $H \geq 0$ in a way that

| | S | | \leq H, \forall S \in G

(19)

Hence, the application of the above combine with $S \in G$ implies that

\begin{matrix} | | UV | | \leq | S_{0} | + \frac{τ^{ϑ}}{Γ (ϑ + 1)} [H_{P} | | S | |^{s} + K_{P}] \leq | S_{0} | \\ + \frac{τ^{ϑ}}{Γ (ϑ + 1)} [H_{P} H^{s} + K_{P}] \end{matrix}

(20)

This implies the boundedness of $U (G)$ .

A3: For equi-continuity, we take $t_{1}, t_{2} \in [0, τ]$ in a way that $t_{1} \geq t_{2}$ , then

\begin{matrix} | U S (t_{1}) - U S (t_{1}) | = | \frac{1}{Γ (ϑ)} \int_{0}^{t_{1}} | (t_{1} - l)^{ϑ - 1} | | P (l, S (l)) | dl \\ - \frac{1}{Γ (ϑ)} \int_{0}^{t_{2}} | (t_{2} - l)^{ϑ - 1} | | P (l, S (l)) | dl | + \\ \leq | \frac{1}{Γ (ϑ)} \int_{0}^{t_{1}} | (t_{1} - l)^{ϑ - 1} | - \frac{1}{Γ (ϑ)} \int_{0}^{t_{2}} \\ | (t_{2} - l)^{ϑ - 1} | | | P (l, S (l)) | dl \\ \leq \frac{τ^{ϑ}}{Γ (ϑ + 1)} [H_{P} | | S | |^{s} + K_{P}] [t_{1}^{ϑ} - t_{2}^{ϑ}] \\ \to 0 as t_{1} \to t_{2} \end{matrix}

(21)

Consequently, the relative compactness of $U (G)$ is obtained through the theorem of Arzela-Ascoli.

A4: Finally, the below set is taken

E = {S \in Y : S = λ U S, λ \in (0, 1)}

(22)

Assume $S \in E$ for the boundedness of the set $E$ , then the below is satisfied for $t \in [0, τ]$

| | S | | = λ | | U S | | \leq λ [| S_{0} | \frac{τ^{ϑ}}{Γ (ϑ + 1)} H_{P} | | S | |^{s} + K_{P}]

(23)

Thus, $E$ is bounded. Schaefer’s theorem shows that the operator $U$ has a fixed point as a consequence, which implies that the recommended system (11) of tumor has at least one solution.

Remark 1. If condition $Z 1$ holds true for $s = 1$ , the Theorem (2) fulfills for $\frac{τ^{α} K_{Q}}{Γ (α + 1)} < 1$ .

Theorem 3. There exists a solution of the recommended fractional system (11) if $\frac{τ^{α} K_{Q}}{Γ (α + 1)} < 1$ satisfied.

Proof. We take $S, \bar{S} \in Y$ for the required result and apply the Banach’s contraction theorem as follows

\begin{matrix} | | U S - U \bar{S} | | \leq max_{t \in [0, τ]} \frac{1}{Γ (ϑ)} \int_{0}^{t} | (t - l)^{ϑ - 1} | | P (l, S (l)) - P (l, \bar{S} (l)) | dl \\ \leq \frac{τ^{ϑ} H_{P}}{Γ (ϑ + 1)} | | S - \bar{S} | | \end{matrix}

(24)

This implies that the operator $U$ has a fixed point, and as a result, there exists a solution of the recommended system (11) of fractional derivative.

Ulam-Hyers stability

Ulam-Hyers stability for our suggested model of tumor progression will be established in this part. Ulam first introduced the concept of this stability in 1940; Hyers expanded it (Hyers, 1941; Ullam, 1940). In several fields of study, many scientists applied the Ulam-Hyers stability theory (Ali et al., 2019; Benkerrouche et al., 2021; Rassias, 1978). The main idea is given as:

Take an operator $T : Y \to Y$ in a way that

T S = S for S \in Y

(25)

Definition 4. If for any solution $S \in Y$ and $ϵ > 0$ , the following holds true

| | S - T S | | \leq ϵ for t \in [0, τ]

(26)

Then, equation (25) is Ulam-Hyers type stable (UHS).

There exists a solution $\bar{S}$ of equation (25) in a way that $C_{s} > 0$ and the following mentioned fulfills

| | \bar{S} - S | | \leq C_{s} ϵ, t \in [0, τ]

(27)

Definition 5. If for any solution $S$ of equation (27) and any other solution $\bar{S}$ of equation (25), we have

| | \bar{S} - S | | \leq X (ϵ)

(28)

in which $X \in C (R, R)$ and zero is the image of zero, then the above equation (25) is generalized UHS.

Remark 2. The solution $\bar{S} \in Y$ satisfies equation (27), if the following holds true

(1) $| ℜ (t) | \leq ϵ, for all t \in [0, τ]$ with $ℜ \in C ([0, τ]; R)$

(2) $T \bar{S} (U) = \bar{S} + ℜ (U), for all t \in [0, τ]$

After small perturbation, the system (11) can be represented as

{\begin{matrix} ^{c} D_{0^{+}}^{ϑ} S (t) = P (t, S (t)) + ℜ (t), \\ S (0) = S_{0} \end{matrix}

(29)

Lemma 2. The below condition satisfied by the system (29):

| S (t) - U S (t) | \leq a ϵ, with a = \frac{τ^{ϑ}}{Γ (ϑ + 1)}

(30)

Proof. We can prove this result with the help of Remark (2) and Lemma (1).

Theorem 4. If $\frac{τ^{ϑ} M_{P}}{Γ (ϑ + 1)} < 1$ satisfies on Lemma (2), then the solution of system (11) is UHS and generalized UHS.

Proof. Let us assume the solution $S \in Y$ of the system (29) and $\bar{S} \in Y$ be a unique solution of the system (11) to prove the required result, then we have

\begin{matrix} | S (t) - \bar{S} (t) | = | S (t) - \bar{S} (t) | \\ \leq | S (t) - U \bar{S} (t) | \\ \leq | S (t) - U \bar{S} (t) | \\ \leq a ϵ + \frac{τ^{ϑ} M_{P}}{Γ (ϑ + 1)} | S (t) - \bar{S} (t) | \\ \leq \frac{a ϵ}{1 - \frac{τ^{ϑ} M_{P}}{Γ (ϑ + 1)}} \end{matrix}

(31)

Thus, the solution of system (11) is UHS and generalized UHS.

Definition 6. Let us consider that $Ω \in C [[0, τ], R]$ , the solution of equation (25) is Ulam-Hyers-Rassias stable (UHRS) if for any solution $S \in Y$ of the below

| | S - T S | | \leq Ω (t) ϵ, for t \in [0, τ] and ϵ > 0

(32)

There exists a solution $\bar{S}$ of equation (25) having $C_{s} > 0$ satisfying

| | \bar{S} - S | | \leq C_{s} Ω (t) ϵ, \forall t \in [0, τ]

(33)

Definition 7. Take the solution $S$ of equation (32) and $\bar{S}$ is the unique solution of equation (25) in a way that

| | \bar{S} - S | | \leq C_{s, Ω} Ω (t) ϵ, \forall t \in [0, τ]

(34)

With $ϵ > 0$ and $Ω \in C [[0, τ], R]$ in a way that $C_{s, Ω}$ . Then, the solution of equation (25) is generalized UHRS.

Remark 3. The solution $\bar{S} \in Y$ verifies equation (27) if the below satisfies

(a) $| ℜ (t) | \leq ϵ Ω (t), \forall t \in [0, τ]$ , where $ℜ (t) \in C ([0, τ]; R)$

(b) $T \bar{S} (t) = \bar{S} + ℜ (t), \forall t \in [0, τ]$

Lemma 3. After a small perturbation in Lemma (2), the below inequality satisfies

| S (t) - U S (U) | \leq a Ω (t) ϵ, where a = \frac{τ^{ϑ}}{Γ (ϑ + 1)}

(35)

Proof. Applying Remark (3) and Lemma (1), we can easily prove the result.

Theorem 5. If $\frac{τ^{ϑ} M_{P}}{Γ (ϑ + 1)} < 1$ , then on Lemma (3), the solution to the system (11) is UHRS and generalized UHRS.

Proof. Here, we take any solution $S \in Y$ in for the required result and assume a unique solution $\bar{S} \in Y$ of the system (11), then we have the below

\begin{matrix} | S (t) - \bar{S} (t) | = | S (t) - \bar{S} (t) | \\ \leq | S (t) - U \bar{S} (t) | \\ \leq | S (t) - U \bar{S} (t) | \\ \leq a Ω (t) ϵ + \frac{τ^{ϑ} M_{P}}{Γ (ϑ + 1)} | S (t) - \bar{S} (t) | \\ \leq \frac{a Ω (t) ϵ}{1 - \frac{τ^{ϑ} M_{P}}{Γ (ϑ + 1)}} \end{matrix}

(36)

Consequently, the solution of equation (11) is UHRS and generalized UHRS.

Proposed fractional system’s solution

The Laplace transform will be used in this instance to provide a generic method for the recommend fractional system (9) of tumor. In this numerical method, we go through the following steps

{\begin{matrix} £ [C_{U} (t)] = \frac{C_{U 0}}{s} + \frac{1}{s^{ϑ}} £ [r C_{U} (1 - \frac{C_{I} + C_{U}}{k}) - β C_{U} C_{V} - γ_{U} C_{U} C_{E}], \\ £ [C_{I} (t)] = \frac{C_{I 0}}{s} + \frac{1}{s^{ϑ}} £ [β C_{U} C_{V} - γ_{I} C_{I} C_{E} - δ_{I} C_{I}], \\ £ [C_{E} (t)] = \frac{C_{E 0}}{s} + \frac{1}{s^{ϑ}} £ [c C_{I} - δ_{E} C_{E}], \\ £ [C_{V} (t)] = \frac{C_{V 0}}{s} + \frac{1}{s^{ϑ}} £ [N (t) + α δ_{I} C_{I} - δ_{V} C_{V}] \end{matrix}

(37)

with

{\begin{matrix} C_{U} (t) = \sum_{j = 0}^{\infty} C_{Uj} (t), \\ C_{I} (t) = \sum_{j = 0}^{\infty} C_{Ij} (t), \\ C_{E} (t) = \sum_{j = 0}^{\infty} C_{Ej} (t), \\ C_{V} (t) = \sum_{j = 0}^{\infty} C_{Vj} (t) \end{matrix}

(38)

Here, Adomian polynomials are used to break down our system’s nonlinear terms, and we go through the following steps

\begin{matrix} C_{U} (t) C_{I} (t) = \sum_{j = 0}^{\infty} A_{j} (t), with A_{j} (t) = \frac{1}{j!} \frac{d^{j}}{d z^{j}} \\ [\sum_{k = 0}^{j} z^{k} C_{Uk} (t) \sum_{k = 0}^{j} z^{k} C_{Ik} (t)] z = 0, \end{matrix}

\begin{matrix} C_{U} (t) C_{U} (t) = \sum_{j = 0}^{\infty} B_{j} (t), with B_{j} (t) = \frac{1}{j!} \frac{d^{j}}{{dz}^{j}} \\ [\sum_{k = 0}^{j} z^{k} C_{Uk} (t) \sum_{k = 0}^{j} z^{k} C_{Uk} (t)] z = 0, \end{matrix}

\begin{matrix} C_{U} (t) C_{V} (t) = \sum_{j = 0}^{\infty} C_{j} (t), with C_{j} (t) = \frac{1}{j!} \frac{d^{j}}{d z^{j}} \\ [\sum_{k = 0}^{j} z^{k} C_{Uk} (t) \sum_{k = 0}^{j} z^{k} C_{Vk} (t)] z = 0, \end{matrix}

\begin{matrix} C_{U} (t) C_{E} (t) = \sum_{j = 0}^{\infty} D_{j} (t), with D_{j} (t) = \frac{1}{j!} \frac{d^{j}}{d z^{j}} \\ [\sum_{k = 0}^{j} z^{k} C_{Uk} (t) \sum_{k = 0}^{j} z^{k} C_{Ek} (t)] z = 0, \end{matrix}

and

\begin{matrix} C_{I} (t) C_{E} (t) = \sum_{j = 0}^{\infty} E_{j} (t), with E_{j} (t) = \frac{1}{j!} \frac{d^{j}}{d z^{j}} \\ [\sum_{k = 0}^{j} z^{k} C_{Ik} (t) \sum_{k = 0}^{j} z^{k} C_{Ek} (t)] z = 0 \end{matrix}

Then, we have

{\begin{matrix} £ [\sum_{j = 0}^{\infty} C_{Uj} (t)] = \frac{C_{U 0}}{s} + \frac{1}{s^{α}} \\ £ [r (C_{Uj} - \frac{1}{k} \sum_{j = 0}^{\infty} A_{j} (t) + \sum_{j = 0}^{\infty} B_{j} (t)) - β \sum_{j = 0}^{\infty} C_{j} (t) - γ_{U} \sum_{j = 0}^{\infty} D_{j} (t)], \\ £ [\sum_{j = 0}^{\infty} C_{Ij} (t)] = \frac{C_{I 0}}{s} + \frac{1}{s^{ϑ}} £ [β \sum_{j = 0}^{\infty} C_{j} (t) - γ_{I} \sum_{j = 0}^{\infty} E_{j} (t) - δ_{I} C_{Ij}], \\ £ [\sum_{j = 0}^{\infty} C_{Ej} (t)] = \frac{C_{E 0}}{s} + \frac{1}{s^{ϑ}} £ [c C_{Ij} - δ_{E} C_{Ej}], \\ £ [\sum_{j = 0}^{\infty} C_{Vj} (t))] = \frac{C_{V 0}}{s} + \frac{1}{s^{ϑ}} £ [N (t) + α δ_{I} C_{Ij} - δ_{V} C_{Vj}] \end{matrix}

(39)

{\begin{matrix} £ [C_{U 0} (t)] = \frac{C_{U 0}}{s}, \\ £ [C_{I 0} (t)] = \frac{C_{I 0}}{s}, \\ £ [C_{E 0} (t)] = \frac{C_{E 0}}{s}, \\ £ [C_{V 0} (t)] = \frac{C_{V 0}}{s} \end{matrix}

(40)

{\begin{matrix} £ [C_{U 1} (t)] = \frac{C_{U 0}}{s} + \frac{1}{s^{α}} \\ £ [r (C_{U 0} - \frac{1}{k} A_{0} (t) + B_{0} (t)) - β C_{0} (t) - γ_{U} D_{0} (t)], \\ £ [C_{I 1} (t)] = \frac{C_{I 0}}{s} + \frac{1}{s^{ϑ}} £ [β C_{0} (t) - γ_{I} E_{0} (t) - δ_{I} C_{I 0}], \\ £ [C_{E 1} (t)] = \frac{C_{E 0}}{s} + \frac{1}{s^{ϑ}} £ [c C_{I 0} - δ_{E} C_{E 0}], \\ £ [C_{V 1} (t))] = \frac{C_{V 0}}{s} + \frac{1}{s^{ϑ}} £ [N (t) + α δ_{I} C_{I 0} - δ_{V} C_{V 0}] \end{matrix}

(41)

and

{\begin{matrix} £ [C_{U 2} (t)] = \frac{C_{U 0}}{s} + \frac{1}{s^{α}} \\ £ [r (C_{U 1} - \frac{1}{k} A_{1} (t) + B_{1} (t)) - β C_{1} (t) - γ_{U} D_{1} (t)], \\ £ [C_{I 2} (t)] = \frac{C_{I 0}}{s} + \frac{1}{s^{ϑ}} £ [β C_{1} (t) - γ_{I} E_{1} (t) - δ_{I} C_{I 1}], \\ £ [C_{E 2} (t)] = \frac{C_{E 0}}{s} + \frac{1}{s^{ϑ}} £ [c C_{I 1} - δ_{E} C_{E 1}], \\ £ [C_{V 2} (t))] = \frac{C_{V 0}}{s} + \frac{1}{s^{ϑ}} £ [N (t) + α δ_{I} C_{I 1} - δ_{V} C_{V 1}] \end{matrix}

(42)

and in the same way, we have

{\begin{matrix} £ [C_{U (j + 1)} (t)] = \frac{C_{U 0}}{s} + \frac{1}{s^{α}} \\ £ [r (C_{Uj} - \frac{1}{k} A_{j} (t) + B_{j} (t)) - β C_{j} (t) - γ_{U} D_{j} (t)], \\ £ [C_{I (j + 1)} (t)] = \frac{C_{I 0}}{s} + \frac{1}{s^{ϑ}} £ [β C_{j} (t) - γ_{I} E_{j} (t) - δ_{I} C_{Ij}], \\ £ [C_{E (j + 1)} (t)] = \frac{C_{E 0}}{s} + \frac{1}{s^{ϑ}} £ [c C_{Ij} - δ_{E} C_{Ej}], \\ £ [C_{V (j + 1)} (t))] = \frac{C_{V 0}}{s} + \frac{1}{s^{ϑ}} £ [N (t) + α δ_{I} C_{Ij} - δ_{V} C_{Vj}] \end{matrix}

(43)

with the initial conditions

{\begin{matrix} C_{U 0} (t) = C_{U 0} \\ C_{I 0} (t) = C_{I 0} \\ C_{E 0} (t) = C_{E 0} \\ C_{V 0} (t) = C_{V 0} \end{matrix}

(44)

We take the following step for further simplification

{\begin{matrix} C_{U 1} (t) = £^{- 1} [\frac{1}{s^{ϑ}} £ [r (C_{U 0} - \frac{1}{k} A_{0} (t) + B_{0} (t)) - β C_{0} (t) - γ_{U} D_{0} (t)]], \\ C_{I 1} (t) = £^{- 1} [\frac{1}{s^{ϑ}} £ [β C_{0} (t) - γ_{I} E_{0} (t) - δ_{I} C_{I 0}]], \\ C_{E 1} (t) = £^{- 1} [\frac{1}{s^{ϑ}} £ [c C_{I 0} - δ_{E} C_{E 0}]], \\ C_{V 1} (t) = £^{- 1} [\frac{1}{s^{ϑ}} £ [N (t) + α δ_{I} C_{I 0} - δ_{V} C_{V 0}]] \end{matrix}

(45)

and

{\begin{matrix} C_{U 2} (t) = £^{- 1} [\frac{1}{s^{ϑ}} £ [r (C_{U 1} - \frac{1}{k} A_{1} (t) + B_{1} (t)) - β C_{1} (t) - γ_{U} D_{1} (t)]], \\ C_{I 1} (t) = £^{- 1} [\frac{1}{s^{ϑ}} £ [β C_{1} (t) - γ_{I} E_{1} (t) - δ_{I} C_{I 1}]], \\ C_{E 1} (t) = £^{- 1} [\frac{1}{s^{ϑ}} £ [c C_{I 1} - δ_{E} C_{E 1}]], \\ C_{V 1} (t) = £^{- 1} [\frac{1}{s^{ϑ}} £ [N (t) + α δ_{I} C_{I 1} - δ_{V} C_{V 1}]] \end{matrix}

(46)

Through the same steps, we get

{\begin{matrix} C_{U (j + 1)} (t) = £^{- 1} [\frac{1}{s^{ϑ}} £ [r (C_{Uj} - \frac{1}{k} A_{j} (t) + B_{j} (t)) - β C_{j} (t) - γ_{U} D_{j} (t)]], \\ C_{I (j + 1)} (t) = £^{- 1} [\frac{1}{s^{ϑ}} £ [β C_{j} (t) - γ_{I} E_{j} (t) - δ_{I} C_{Ij}]], \\ C_{E (j + 1)} (t) = £^{- 1} [\frac{1}{s^{ϑ}} £ [c C_{Ij} - δ_{E} C_{Ej}]], \\ C_{V (j + 1)} (t) = £^{- 1} [\frac{1}{s^{ϑ}} £ [N (t) + α δ_{I} C_{Ij} - δ_{V} C_{Vj}]] \end{matrix}

(47)

In the end, the below series solution is obtained

{\begin{matrix} C_{U} (t) = C_{U 0} (t) + C_{U 1} (t) + C_{U 2} (t) + C_{U 3} (t) + \dots, \\ C_{I} (t) = C_{I 0} (t) + C_{I 1} (t) + C_{I 2} (t) + C_{I 3} (t) + \dots, \\ C_{E} (t) = C_{E 0} (t) + C_{E 1} (t) + C_{E 2} (t) + C_{E 3} (t) + \dots, \\ C_{V} (t) = C_{V 0} (t) + C_{V 1} (t) + C_{V 2} (t) + C_{V 3} (t) + \dots . \end{matrix}

(48)

The suggested model (9) was simulated using the aforementioned numerical technique. For numerical reasons, the parameter values listed in Table 1 are utilized. Here, using numerical simulations, our major goal is to illustrate how the input component affects the dynamics of tumor growth. Through our experimental effort, we will emphasize the effects of therapy on the system and identify the most important components of the tumor development system. Numerical simulations with varying input parameters will be used to emphasize the system’s oscillatory and chaotic behavior.

Table 1.

Interpretation and values of input parameters of the system for numerical outcomes.

Parameters	Descriptions	Values	Reference
$N$	Dosage of the virotherapy	$8.672 \times 10^{7}$	Assumed
$α$	Virions released through infected cell lysis	3500	Kim et al. (2015)
$δ_{V}$	Virions decay rate in the system	2.3	Kim et al. (2015)
$δ_{I}$	Infected cells decay rate	1.00	Assume
$δ_{E}$	Effector T cells decay rate	0.35	Kim et al. (2015)
$γ_{I}$	Decay rate of infected cells through T cells	$1.5 \times 10^{- 7}$	Kim et al. (2015)
$γ_{U}$	Decay rate of uninfected cells through T cells	$1.5 \times 10^{- 7}$	Kim et al. (2015)
$r$	Infected tumor cells growth rate	0.31	Kim et al. (2015)
$β$	Rate of infection of uninfected cells	$8.9 \times 10^{- 13}$	Kim et al. (2015)
$c$	Growth rate of T cells	2.90	Kim et al. (2015)
$k$	Carrying capacity	2.9	Kim et al. (2015)
$ϑ$	Fractional order or index of memory	0.60	Assume

To emphasize the effect of fractional order on the system’s solution pathways, we have depicted the dynamical behavior of the proposed fractional system of tumor growth in the first scenario illustrated in Figures 1 –3 with various values of fractional order. In these simulations, the effect of a fractional parameter on the system’s dynamic behavior has been noted. The blue solid curves show the integer-order dynamics while the red doted curves illustrate fractional-order dynamics of tumor model. In these figures, the order of the fractional derivative is assumed to be 1, 0.85, and 0.65 in comparison with the integer-order derivative. It can be seen that the results of the fractional derivative are more flexible than integer-order and are recommended to the researchers for more accurate results. In the second scenario presented in Figures 4 –6, the chaotic graphs of the system of tumor dynamics has been presented with variation of different input parameters. To be more specific, we visualized the parameters responsible for the chaotic behavior of the system. We mainly varied the parameters $N, ϑ$ , and $r$ to observe the effect of these parameters on the chaos of the system to notice that whether these parameters can control the chaos of the system or not. In the third scenario illustrated in Figures 7–8, we visualize the concentrations of uninfected tumor cells, infected tumor cells, effector T cells, and the concentration of virion with the variation of different values of the system. In these simulations, we noticed the dynamical behavior of tumor growth system with the variation of $c$ and $K$ . Our findings demonstrated that the system’s nonlinearity causes substantial oscillation and chaos, both of which exhibit strong relationships with one another.

Figure 1.

Time series analysis of the proposed fractional model (9) of tumor growth with fractional order different $ϑ = 1$ .

Figure 2.

Time series analysis of the proposed fractional model (9) of tumor growth with fractional order different $ϑ = 0.85$ .

Figure 3.

Time series analysis of the proposed fractional model (9) of tumor growth with fractional order different $ϑ = 0.65$ .

Figure 4.

Illustration of chaotic plots of our fractional model (9) of tumor growth with $ϑ = 0.8, N = 8.672 \times 10^{7}$ , and $r = 0.31$ while other parameters are fixed.

Figure 5.

Illustration of chaotic plots of our fractional model (9) of tumor growth with $ϑ = 0.8, N = 5.672 \times 10^{7}$ , and $r = 0.31$ while other parameters are fixed.

Figure 6.

Illustration of chaotic plots of our fractional model (9) of tumor growth with $ϑ = 0.7, N = 8.672 \times 10^{7}$ , and $r = 0.61$ while other parameters are fixed.

Figure 7.

Dynamical behavior of the system (9) of tumor growth with the variation of input parameter $c$ , that is, $c = 3.6, 2.7, 1.8, 0.9$ .

Figure 8.

Dynamical behavior of the system (9) of tumor growth with the variation of input parameter $K$ , that is, $c = 3.6, 2.7, 1.8, 0.9$ .

Concluding remarks

Oncolytic viruses and viral immunotherapy are used in virotherapy to kill cancer cells directly or to activate the immune system to identify and kill cancer cells. Virotherapy for mesothelioma cancer is being studied in clinical studies. In this research work, the effect of virotherapy treatment on the dynamics of tumor growth through fractional calculus has been visualized. In addition, in our work, we concentrated on the qualitative analysis and dynamical behavior of tumor development dynamics. The fixed-point theorem is used to assess the uniqueness and existence of the solution of the suggested model of tumor growth within the context of Banach’s and Schaefer’s. We set up enough conditions for our tumor system’s Ulam-Hyers stability. To demonstrate the effect of the various factors on the dynamics of the tumor, the solution paths are examined using the Laplace Adomian decomposition method. To be more precise, we have demonstrated the chaotic and oscillatory behavior of the tumor development system as a function of varying input parameters. In our future work, we will extend our proposed model in the framework of impulsive differential equations and delay differential equation.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Rashid Jan

Salah Boulaaras

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