Mathematical modeling of tumor-immune dynamics: stability, control, and synchronization via fractional calculus and numerical optimization

Abstract

This research introduces two distinct mathematical models to investigate the interactions between the tumor-immune system, both formulated within a random (stochastic) framework. The first model employs fractal-fractional derivatives, specifically the Atangana-Baleanu operator, to analyze tumor-immune dynamics from both qualitative and quantitative perspectives. We establish the well-posedness of this model by demonstrating the existence and uniqueness of solutions through fixed point theorems and examine stability via nonlinear analysis. Numerical simulations are performed using Lagrangian-piecewise interpolation across various fractional and fractal parameters, providing visual insights into the complex interplay between immune cells and cancer cells under different conditions. The second model consists of coupled nonlinear difference equations based on the Caputo fractional operator. Its solutions’ existence is guaranteed through classical fixed point theorems, and further properties such as stability, controllability, and synchronization are thoroughly explored to deepen understanding of the system’s behavior. Both models are thoroughly analyzed within a stochastic setting, which considers randomness inherent in biological systems, offering a more realistic depiction of tumor-immune interactions. Numerical simulations for specific scenarios reveal the dynamic characteristics and practical implications of the models, enhancing our insights into tumor-immune processes from a probabilistic perspective.

Introduction

Cancer is one of the most dangerous diseases that the world is still struggling to cure. It is easier to find the causes of it to prevent it from happening than it is to go through the long and painful process of trying to cure it. Cancer is caused by many different reasons, most of them are related to self-neglect and the lack of exercise which lead to bad eating habits that would cause the dead cells to regrow abnormally¹.

To break down the idea of cancer and to better understand it we need to know the process that turns a regular functioning cell into a cancerous cell. Our bodies have millions upon millions of cells that are needed for our day-to-day activities and to function correctly, what is great about these cells is that they are renewable. Our cells use the nutrients in our bodies to grow, they also multiply based on the needs of our bodies. The scientific name of this multiplication in the number of cells is called Cell division. Our cells also get damaged or die due to a number of reasons, some of these dead or damaged cells might regrow although they are not supposed to. The uncontrollable growth of these damaged/dead cells is how cancer forms itself. Once a cancerous cell forms, it will start to invade and spread pretty quickly through the body. Invading other nearby cells would cause tumors, some of these tumors are easy to detect and cure at their early stages, and some of them are not. The easy to cure tumors are called benign tumors, they don’t spread, nor do they grow back once removed unlike cancerous tumors².

There is also a type of cancer that doesn’t cause tumors, nor could it show the same visible symptoms as the other types of cancer, it is called leukemia also known as blood cancer. The tumors in most types of cancer are caused by the abnormal growth of the cancer infected cells that form lumps of tissues. The cancer infected cells grow at a faster than normal rate, which gives them the ability to invade and attack the other normal-functioning cells. The rate of which cancerous cells spread out makes them able to shut down the organs that they cover which results in the necessity of the removal of the organ to slow down or stop the spread of cancer in that specific area of the body and to begin with the appropriate steps of curing it³.

The relationship between the immune system and cancer is intricate and dynamic; while the immune system–comprising various organs and cells that defend health by distinguishing self from harmful pathogens like bacteria, viruses, and fungi–can occasionally detect and eliminate cancerous cells, malignancies often develop mechanisms to evade immune detection or create flaws in immune responses that lead to tumor formation. Most immune cells, known as white blood cells, originate from hematopoietic stem cells in the bone marrow and circulate through the bloodstream, migrating to specific tissues where they perform their protective roles, yet the interaction between immune surveillance and cancer development remains a complex balance of immune activation and evasion strategies⁴.

Using fractional calculus^5,6 in modeling the tumor-immune system helps in better understanding how tumors grow and how the immune system fights back. Unlike traditional models that assume changes happen instantaneously, fractional models consider the past behavior of the system, which is important because immune responses and tumor growth are affected by their history over time. This makes the models more realistic because they can capture delays and lingering effects that happen in the body^7,8.

When studying the stability of tumor-immune systems, fractional calculus provides tools that help determine whether the system will settle down to a steady state or continue to fluctuate. These tools consider the system’s memory, giving a clearer picture of what conditions lead to the tumor being controlled or eliminated. This understanding is useful for predicting how tumors might behave over time and how interventions can help. In terms of controlling tumor growth, fractional calculus allows for the design of better treatment strategies. Controllers based on fractional models can adapt more naturally to the system’s delays and lingering effects, making treatments like immune therapy more effective. They can also help in adjusting doses or timing to ensure the best possible outcome, even when there are uncertainties or unexpected changes. Finally, fractional calculus also helps in analyzing data and tuning models to match real-world tumor growth^9,10. This means the models can be improved to give more accurate predictions for individual patients, leading to personalized treatment plans. Overall, using fractional derivatives offers a more complete and practical way to understand and control tumor-immune interactions^11,12.

In reference¹³, the authors examined dendritic cell transfection immunotherapy and provided an extensive discussion on modeling approaches for immunotherapy treatments, building upon earlier works^14,15, which predominantly treat the therapy as a continuous process employing techniques characteristic of optimal control theory in continuous time; however, in¹⁶, discrete injection times were introduced, and¹⁷ developed a mathematical model based on discrete fractional equations with initial conditions to explore tumor-immune interactions. Additionally, Fatmawati and M. A. Khan analyzed dengue infection dynamics using fractal-fractional operators applied to real-world statistical data¹⁸, while Srivastava and Saad modeled Ebola virus spread with fractal-fractional operators, comparing their numerical outcomes to finite-difference methods for integer orders and observing close correspondence^19–22. Lastly²³, focused on a mathematical tumor-immune model formulated through fractal-fractional derivatives, highlighting the diverse applications of fractional calculus in capturing complex biological phenomena.

In²⁴, the authors formulated the tumor-immune interaction within the framework of fractional derivatives. They conducted qualitative and dynamical analyses of the system, establishing existence and uniqueness of solutions via Banach’s and Schaefer’s fixed point theorems. The study also derived conditions for Ulam–Hyers stability of the proposed fractional model. A novel numerical scheme was implemented to examine the influence of parameters on tumor-immune dynamics. Furthermore, they explored the system’s chaotic behavior and the effects of varying fractional orders. Key aspects of tumor-immune cell interactions were identified and recommended to policymakers, emphasizing their significance in immunotherapy success and clinical outcomes. In²⁵, the authors undertook several key efforts: they developed a fractional-order tumor-immune interaction model, partitioning the total population into three subgroups–macrophages, activated macrophages, and tumor cells–to capture system complexity. They analyzed the influence of the fractional derivative on the stability and dynamical behavior of solutions using the Caputo fractional operator, which facilitated the handling of initial conditions. The existence and uniqueness of solutions were rigorously established, supported by numerical simulations that validated the analytical results. Furthermore, the model was employed to describe the growth and regression kinetics of B-lymphoma BCL1 in the spleen of mice. Numerical experiments conducted across various fractional orders, particularly around =0.80, demonstrated a close alignment with experimental data, with the fractional model providing a superior fit compared to integer-order models. These findings underscore the enhanced accuracy of fractional calculus in depicting biological dynamics.

In this paper, we consider two different fractional tumor growth model in both discrete time and continuous time, and then, we investigate the stability, controllability, synchronicity and numerical results for the mentioned models. The numerical results and graphical representations are computed and generated using Maple, ensuring precise and reliable visualization of the system’s behavior.

Preliminaries

In this part, we gather all the necessary prerequisites and tools required for the analysis of our mathematical models.

Fractional calculations

Here, we propose some basic concepts of fractional calculus.

Fractal-fractional derivatives and integrals

Here, suppose is a continuous and fractal differentiable on . Here, we introduce Atangana-Baleanu fractional-order derivative, which focuses on exponential decay law, power law, and the generalized Mittag-Leffler function, respectively. We denote the fractal-fractional with exponential decay kernel, power law kernel, and Mittag-Leffler kernel via FFE,FFP, and FFM, respectively. Hence, the fractal-fractional operators are given as follows²⁶:

Consider the fractional order and fractal dimension on unit interval [0, 1]. According to the categories below for kernels, the fractal-fractional derivative of with order and dimension in the Riemann-liouville sense is defined by

• power law type kernel:

  
• exponentially decaying type kernel:

  
• generalized Mittag-Leffler type kernel:

  

  where and is the Mittag-Leffler function defined in²⁷.

As above, the fractal-fractional integral of with order based on the type of kernel has the following forms:

• power law type kernel:

  
• exponentially decaying type kernel:

  
• generalized Mittag-Leffler type kernel:

  

Fractional sums

Here, we first present the concepts of fractional sum and fractional difference, then, we provide some auxiliary lemmas needed for the rest of the paper. In this subsection, we refer the reader to the references^28–32.

Consider and The th fractional-order sum of is given by

where and

Similarly, the Caputo-like delta fractional difference is defined by

where and

Lemma 1

Consider the delta discrete fractional equation (DFE) below,

1

Then, the delta DFE (1) has the following equivalent integral difference equation

2

where and

Lemma 2

For and with we have that

Lemma 3

For we have

Generalized special functions

In this part, we present some classical and well-known special functions.

Fox type functions

Let be a proper contour of the Mellin–Barnes type in the complex –plane. The function (sometimes called Fox’s –function) is defined by

in which and

We now introduce some special cases of the function including the exponential function, one parameter Mittag–Leffler function, Wright function, Gauss Hypergeometric function, –function, Fox–Wright function, Meijer –function, respectively, as follows:

• 

• 

• 

• 

• 

• 

• 

Notice that for and we define and

Mittag–Leffler type functions

Suppose that and The m–parameter Mittag–Leffler function is defined by

in which with for all and Note that for every is given by

Here, we introduce some special cases of (), as follows:

• 

• 

• 

• 

• 

Aggregate window maps

Let and be a diagonal matrix in which for every An n–ary aggregate window map is a mapping s.t., and

Also, for all if then,

Some examples of aggregation maps are given as follows:

• geometric mean function:

• arithmetric mean function:

• maximum function:

• minimum function:

• sum function:

• product function:

Fractal fractional model of tumor–immune interaction

The immune system protects the body against infection and illness that viruses, bacteria, parasites or fungi can cause. It is a collection of responses and reactions that the body makes to infection or damaged cells. Cancer can weaken the immune system through spreading into lungs, bones, lymphoma, liver or leukaemia. Some cells of the immune system can recognise cancer cells as abnormal. Cancer cells are different from normal cells due to their shape, size, differentiation, number, ability and function to travel to organ systems and distant tissues. T cells and dendritic cells are two central types of the immune system and play significant roles in the adaptive immune response. A dendritic cells’s main function is to process antigen material and propose it on the cell surface to the T cells of the immune system (see Fig. 1). The basic mechanism of cancer immunity is to kill tumor cells via the help of regulatory T cells ( cell), CD4 + T and CD8 + T cells. One of the tasks of CD4 + T cells is to produce Interleukin-2 (IL-2) as a result of antigen stimulation response. IL-2 is a type of cytokine signaling molecule in the immune system that regulates the activities of white blood cells³³.

Fig. 1.

Dendritic cells activate T cells and trigger immune responses (see https://www.sciencedirect.com/science/article/pii/B9780081027233105372).

As shown in Fig. 2, in the left picture, IL-2 is mostly produced through CD4 + T cells and then consumed at the same site by cells, CD4+ and CD8+ T cells. In the right picture, activated DCs home to the draining lymph nodes, where activated CD4+ and CD8+ T cells produce large amounts of IL-2 and then the produced IL-2 consumed via cells.

Fig. 2.

Steady state (left) and immune response (right) in a lymph node (see https://www.sciencedirect.com/science/article/pii/B9780081027233105372).

This section employs innovative operators to examine the interaction between the immune response and cancer cells. The tumor-immune dynamics are modeled both qualitatively and quantitatively using the Atangana-Baleanu fractal-fractional derivative. We establish the well-posedness of the model by proving existence and uniqueness via fixed point theorems, and demonstrate stability results through nonlinear analysis. Numerical simulations are conducted using Lagrangian-piecewise interpolation across various fractal-fractional operators, with results visualized by modeling the interplay between immune and cancer cells for different fractional and fractal parameters.

A fractal-fractional tumor-immune model for is formulated as follows^34,35:

3

along with the following initial conditions:

4

where for respectively, denote CD4 + T cells, CD8 + T cells, cancer cells, dendritic cells, IL-2, and other symbols are given in Table 1.

Table 1.

Parameter descriptions of the tumor-immune model (3) (see³⁵).

Parameters	Description	Values
	CD4 T birth rate	0.0001
	CD4 T Proliferation rate	0.1000
	CD4 T death rate	1.0000
	Carrying capacity of CD4 T	0.0050
	CD8 T birth rate	0.0001
	CD8 T proliferation rate	0.0100
	Carrying capacity of CD8 T	1.0000
	CD8 T death rate	0.0050
	1/2 satur const of tumor	0.0200
	Carrying capacity of tumor	1.0000
	Killing by CD8 of tumor	0.1000
	CD8 T killing of DC	0.1000
	IL-2 production by CD4 T	0.0100
	IL-2 degradation rate	0.0100
	IL-2 uptake by CD8 T

The stabilization results

Here, we will establish the conditions for the existence and uniqueness of solutions, as well as the stability results of the proposed model, within the framework of fractal-fractional derivatives based on the Atangana-Baleanu concept.

Here, we can rewrite the fractal-fractional system (3) in the sense of Atangana-Baleanu-Riemann-Liouville (ABRL) derivative, as follows^35–37:

5

where and

We now rephrase the above system, as follows:

Notice that the Riemann-Liouville derivative has some disadvantages when trying to model real-world phenomena with fractional-order differential equations, for instance, the Riemann-Liouville derivative of a constant is not zero. Theses disadvantages reduce the field of application of the Riemann-Liouville fractional derivative. With the Caputo definition,the fractional derivative of a constant is equal to zero and more importantly it allows traditional initial and boundary conditions to be included in the formulation of the problem³⁸.

Considering the above note, we replace the Atangana-Baleanu-Riemann-Liouville derivative by the Atangana-Baleanu-Caputo derivative ^35,37 and then using the definition of fractional integral, we get

6

where

and

Here, define an operator by

7

In what follows, we first introduce random normed spaces and then, we discuss some premises before demonstrating and expressing the main findings in this subsection.

A minimum continuous triangular norms (in short, MCTN)²⁷ on [0, 1] is an operation which for every is defined by and satisfied the following conditions:

• boundary condition: ;

• commutativity: ;

• associativity: ;

• monotonicity: ;

• continuity: .

Consider , the set of matrix distribution functions, including left continuous and increasing functions s.t. and . Now are all (proper) mappings for which . Note that proper matrix distribution function’s are the matrix distribution function’s of real random variables that . In , we say that iff for every In addition, which is defined by 0 and 1, for and respectively, belongs to and , for every matrix distribution function For example, for and for is a distribution function on [0, 1].

Let be a linear space, be an MCTN and be a distribution function. The triple is a random normed space³⁹ if for every and

• iff ;

• ;

• 

For example, the distribution function for every and for every , defines a random norm and is a random normed space.

A complete matrix random normed space is called a random Banach space.

We now consider the following Lipschitz and growth conditions for the nonlinear function :

for every and there is a positive constant s.t.,

8

for every and there are positive constants and s.t.,

9

Theorem 1

Consider the conditions and and let be a continuous function, and where Then, the considered tumor model has

1. at least one solution,

2. a unique solution.

Proof

1. Since is continuous, then, the operator given in (7) is also continuous. Consider Now, for every and we have that

   

   where Thus, is uniformly bounded, where denotes the beta function⁴⁰. Also, one can easily observe that the operator is equicontinuous and thus is completely continuous via the Arzela-Ascoli theorem. Hence, making use of Schauder’s fixed point result, the considered tumor model has at least one solution.
2. For we get

   

   where Thus, is a contraction. Hence, the proposed model has a unique solution via the Banach contraction principle.

Now, we are going to prove the stability result of our tumor model.

Definition 1

(⁴⁰) The considered tumor model is stable if there exists s.t., for every and for every satisfying the inequality

there exists a unique solution of the tumor model s.t.,

We now consider a small perturbation s.t., Suppose the following assumptions for :

• 

• 

Theorem 2

1. The solution of perturbed model with satisfies the following relation

   10

   in which and

2. Consider the condition and part (1) of Theorem 2. The tumor model is stable, if

Proof

1. The proof is straightforward. Making use of (7) and doing some mainpulations on the LHS of (10), we can obtain the desired result.

2. Suppose is a unique solution and is any solution of the tumor model. Thus, for every

   

   Where Thus, we conclude that where and

Numerical results

In this part, we consider the numerical method used in²³ through Lagrangian piecewise interpolation to describe the obtained numerical solutions.

The considered method is based on the Adams-Bashforth numerical method²³ presented in the following steps:

Step 1. Considering (5) in the Caputo sense and using the Atangana-Baleanu integral, we get the following

Step 2. Applying numerical scheme at we get

Step 3. Making use of the approximation of the integrals, we have

Step 4. Approximate the kernel inside the integrals via Lagranian piecewise interpolation within the interval as follows:

Step 5. Applying Lagrangian polynomial piecewise interpolation, for every we have that

This technique uses Lagrangian piecewise interpolation to approximate the solutions that were obtained computationally. Essentially, the method involves breaking the problem into smaller segments and applying polynomial interpolation within each segment to ensure smooth and accurate estimates of the solution at different points. The underlying framework of the approach relies on the Adams-Bashforth method, a well-known explicit multistep technique that advances the solution forward in time based on previous data points. This combination allows for efficient and reliable simulation of the system’s dynamics over the considered timeframe.

Considering the initial conditions and and also, using the parametric values given in Table 1, the numerical results are presented via Maple at different values of fractal and fractional orders in Figs. 3–5. Below, using the diagrams obtained and references^8,11, we provide an interpretation of the functions of some key immune cells in combating tumors over time.

Fig. 3.

The contour plots of the dynamics of helper cells during 35 to 195 days (), for the fixed fractional order and fixed fractal order .

Fig. 5.

The contour plots of the dynamics of IL-2 during 200 to 500 days (), for the fixed fractional order and different fractal order .

The contour plots illustrating the dynamics of helper cells over time are presented in Fig. 3. These plots provide a visual representation of how the population or activity levels of helper cells evolve throughout the specified time period, highlighting regions of high and low activity. The contours effectively depict the spatial and temporal variations, revealing patterns such as zones of rapid proliferation or decay, which are crucial for understanding the overall immune response dynamics. This visual tool allows for a clearer interpretation of the complex interactions and fluctuations occurring within the helper cell population during different stages of the immune process.

Over time, the dynamics of helper cells exhibit significant fluctuations characterized by initial proliferation followed by gradual decline, reflecting their role in modulating immune responses. These changes are governed by complex nonlinear interactions and feedback mechanisms within the immune system, leading to dynamic patterns that evolve as the system adapts to ongoing stimuli. Such temporal variations are essential for maintaining immune homeostasis and responding effectively to pathological challenges.

During an immune response, helper cells exhibit significant changes over time, displaying a pattern of rapid growth followed by a gradual decrease in their population. At the start, these cells multiply quickly to help activate other components of the immune system. As the response progresses, their numbers steadily decline, which reflects the body’s mechanism to control and resolve inflammation. These fluctuations are driven by complex nonlinear processes involving various immune signaling pathways, including cytokines and feedback loops, as well as communication between cells. Such mechanisms help the immune system maintain a balance, ensuring it responds adequately to threats without going overboard and causing damage to healthy tissue. The ability of helper cells to dynamically rise and fall over time is essential for immune homeostasis, allowing the body to ramp up defenses when needed and ramp down afterward. This dynamic behavior also enables the immune system to switch seamlessly between activating and suppressing immune functions, thus providing effective pathogen protection while preserving the stability and integrity of the organism as a whole.

The specific values chosen for the fractional order and the fractal order were determined based on a comprehensive integration of theoretical insights and empirical data^41,42.

Fractional Order : This parameter captures the memory-dependent behavior characteristic of tumor-immune interactions. Fractional derivatives inherently account for historical influences within the system, and selecting a fractional order below one enables the model to simulate sub-diffusive and anomalous transport phenomena commonly observed in biological tissues. The value of 0.30 was established through detailed sensitivity analysis and optimization procedures, aiming to produce solutions that align closely with observed biological patterns and experimental measurements.

Fractal Order : This parameter reflects the complexity and irregularity of the tumor microenvironment, particularly its fractal-like vasculature and cellular organization. Fractal analysis of imaging data provided evidence supporting this approximate value, indicating that it effectively models the spatial heterogeneity within tumor tissues. Furthermore, numerical experiments confirmed that this fractal order maintains the stability of the model and reproduces realistic tissue architecture, encapsulating the chaotic and irregular nature of tumor growth.

In essence, these parameter selections stem from an amalgamation of biological data interpretation, mathematical calibration, and stability testing, all aimed at constructing a robust and biologically meaningful model of the tumor-immune system dynamics.

The subsequent figures in the study further illustrate the behavior of the system for various selected parameter values. These variations are grounded in comprehensive numerical experiments and reflect the exploration of different scenarios consistent with empirical data. The parameter choices for these plots were informed by biological evidence, literature review, and the need to evaluate the robustness and sensitivity of the model under diverse conditions. By systematically investigating a range of parameter sets, we aim to demonstrate the model’s capacity to capture the complex and heterogeneous nature of tumor-immune interactions across different biological contexts. This approach enhances the credibility of the model and provides insights into how parameter variations influence tumor progression and immune response dynamics^41–43.

In tumor-immune interactions, the dynamics of dendritic cells change significantly over time. Initially, dendritic cells are highly active and capable of recognizing and presenting tumor antigens, which stimulates an effective anti-tumor immune response. As time progresses, the tumor environment evolves, often becoming immunosuppressive, leading to a decline in dendritic cell function. These cells may become less mature, less capable of antigen presentation, or even adopt suppressive roles due to signals from the tumor. This shift impairs the immune system’s ability to detect and attack the tumor cells. Consequently, the weakening of dendritic cell activity allows the tumor to evade immune surveillance, facilitating its growth and potential metastasis. In simple terms, early on, dendritic cells act as powerful defenders, but over time, the tumor tricks the immune system into weakening, enabling it to expand unchecked. In Fig. 4, you can see the activity of the dynamics of dendritic cells over time.

Fig. 4.

The contour plots of the dynamics of dendritic cells during 40 to 440 days (), for the fixed fractional order and fixed fractal order .

In tumor-immune response, the dynamics of IL-2 play a crucial role in shaping the immune response over time. Initially, IL-2 levels increase as activated T cells produce this cytokine to promote their proliferation and enhance immune activity against the tumor. As the immune response progresses, IL-2 concentration may fluctuate due to changes in T cell populations and regulatory mechanisms aimed at preventing excessive inflammation. Over time, the sustained presence or decline of IL-2 impacts the effectiveness of immune cells in targeting tumor cells. In our study, we have presented the contour plots of the dynamics of IL-2 throughout this process in Fig. 5, illustrating how its levels evolve over time.

In Table 2, we present the obtained error for diverse values of As you can observe the value of the obtained optimal error increases, when the values of and increase.

Table 2.

The obtained error for diverse values of .

0.004	0.001	0.0075		0.001	0.0032
	0.002	0.0084		0.002	0.0049
	0.003	0.0097		0.003	0.0057
	0.004	0.0108	0.002	0.004	0.0071
0.008	0.005	0.0121		0.005	0.0098
	0.006	0.0135		0.006	0.0104
	0.007	0.0154		0.007	0.0115
	0.008	0.0166	0.006	0.008	0.0123

Discrete fractional model of tumor–immune interaction

In this study, we develop a mathematical framework based on discrete fractional equations with initial conditions to examine tumor-immune system interactions. This model consists of a coupled set of nonlinear difference equations formulated using the Caputo fractional operator. To establish the existence of solutions, we employ fixed point theorems, specifically Banach’s and Leray–Schauder’s, providing rigorous analytical results. Furthermore, we explore various properties of the model, including stability, controllability, and synchronization, to understand the system’s behavior more deeply. The dynamic characteristics of the tumor-immune fractional map are also examined through numerical simulations for specific scenarios, offering insights into the model’s practical implications.

In⁴⁴, the authors describe the following tumor–immune interaction model

11

where is the constant source rate of effector cells, denotes the accumulation of effector cells in the tumor site in which and are positive constants, is positive constant, is the natural death rate of effector cells, is the coefficient of the maximal growth of tumor, is the environment capacity, and is the positive constant.

In⁴⁵, is supposed to be which causes model (11) to adapt the following form:

12

where and describes the immune response to the tumor cells.

Hence, (12) is the dimensionless counterpart can also be given by

13

where for display the density of effector cells and tumor cells at time , respectively, and also, and are positive.

The tumor model (13) can be written in the delta type first order equation, as follows¹⁷:

14

Then, the discrete fractional tumor–immune model in the sense of Caputo can be formulated as¹⁷,

15

where and is the initial point of

Rewrite (15), as

where

with the initial values

According to (15), the initial value problem for every is given by

16

in which and for

Stabilization results

Initially, we analyze a nonlinear problem and establish a solution framework that can be applied to determine the existence of solutions. Then, we provide the essential criteria needed for the model to meet the assumptions required to achieve stability and controllability.

Theorem 3

Let be defined on and Then, the solution of the initial value problem

17

is given by

18

where

Proof

Suppose is a solution of (17). Making use of Lemma 2, we have that

Applying the fractional sum operator of order and letting we get

19

where

Let and define the operator as

20

where We now consider the following conditions:

There is a positive s.t., for every and

There are positive and s.t., for every and

Let be a non decreasing function, s.t., for every and

Theorem 4

Consider the conditions and and set

Plus, let

21

Then, the considered tumor model on has

1. at least one solution,

2. a unique solution.

Proof

1. Set where Consider and s.t., According to and Lemma 3, we get

   

   where Now, we have that

   

   which implies that for every and We now prove that is completely continuous in Consider and s.t., Then, via Lemma 3, and we get

   

   for every Thus, we can conclude that is an equi-continuous. Based on the Mazur Lemma⁴⁶ and the condition is relatively compact. Suppose Thus, where Then, is relatively compact. Applying the Ascoli-Arzela theorem⁴⁷, we can conclude that for every contains a uniformly convergent subsequence on which implies that the set is relatively compact and continuous. We now derive that has a fixed point that is a solution of our tumor model by the Leray-Schauder Theorem⁴⁸.
2. Making use of and Lemma 3, for every and we get

   

   Thus, is a contraction and then as a result, the considered model has a unique solution via fixed point Theorem.

Definition 2

(²⁷) The tumor model (16) is stable, if and for every solution of the inequality

22

there exist a solution of (16), with

23

where and for By setting in (23), where and in which for then, we say (16) is generalized stable.

Definition 3

(²⁷) The tumor model (16) is controllable, if and for every solution of the inequality

24

there exist a solution of (16), with

25

where and for By setting in (24) and (25), in which for then, we say (16) is generalized controllable.

Remark 1

A function is a solution of (22) (or (24)), if there exists a function s.t., for every and we get

1. (or

2. 

Theorem 5

Suppose and Then, we have the following:

1. If be a solution of (22), then,

   
2. If be a solution of (24), then,

   

Proof

1. Making use of Remark 1 and Lemma 2, we get

   

   This implies that

   

   for every

2. This follows from part (1) of Theorem 5.

Theorem 6

Consider the inequality (21), and Then, the considered tumor model is

1. (generalized) stable,

2. (generalized) controllable.

Proof

1. Based on the solution (18), Theorem 5, Lemma 3, for every and we can conclude that

   

   Then, we get where By setting with we get Thus, our tumor model is generalized stable.
2. Based on the solution (18), Theorem 5, and Lemma 3, for every and we can conclude that

   

Thus, we can conclude that where By letting we get Thus, the tumor model is generalized controllable.

Numerical results

Making use of Lemma 1, the solution of (15) is given by

26

Now, consider the discrete kernel function and let and Thus, we have that

27

where and are initial values. According to the biological research literature^49,50, we consider the parameters value given in Table 3 for numerical simulation.

Table 3.

Parameter values obtained through biological research.

Dimensional parameters	Description of parameters	Dimensional values	Dimensionless parameters	Dimensionless values
	Source rate of constant	cell		0.1181
	Rate of natural death	0.0412		0.3743
	Innate tumor groth rate	0.18		1.636
	Ability of to carry			0.002
	Immune reaction to	, (daycells)		0.04

Dynamics of changes and for the fixed fractional order are shown in Figs. 6 and 7. Below, based on the generated diagrams and references^8,11, we offer an explanation of how certain crucial immune cells function in the fight against tumors throughout different stages.

Fig. 6.

Dynamic changes of effector cells during 50 to 200 days (), for the fixed fractional order .

Fig. 7.

Dynamic changes of tumor cells during 100 to 350 days (), for the fixed fractional order .

In our tumor-immune model, we examine how effector cells–these are the immune system’s fighter cells, such as cytotoxic T cells, that attack and destroy tumor cells–change over time. Figure 6 illustrates how the population of these effector cells varies throughout the progression of the disease. At the beginning, the number of effector cells is usually quite low because the immune system has not yet fully detected the tumor. As the tumor grows and begins to produce signals that alert the immune system, effector cells start to recognize the threat. In response, their numbers gradually increase, reflecting the immune system’s efforts to fight the tumor. This increase continues until the effector cell population reaches a maximum point, often called the peak response. This peak shows that the immune system is actively working to eliminate the tumor, with many effector cells attacking the cancer cells simultaneously. After this peak, several scenarios can occur. If the immune system successfully controls the tumor, the number of effector cells may stay high or gradually decline as the tumor shrinks. However, if the tumor continues to evade immune detection or suppress immune activity, the effector cell population might decline, allowing the tumor to grow again. By observing this dynamic pattern, we can understand how the immune response fluctuates in real-time during tumor development and treatment. These insights are crucial for designing therapies that can enhance immune activity and improve cancer control.

In our system, Fig. 7 illustrates how tumor cells evolve over time as the disease progresses. At the initial stage, the number of tumor cells is typically low, reflecting the early development of the tumor. As time advances, these cells begin to multiply and grow more rapidly, causing a noticeable increase in their population. This rapid expansion often occurs during the middle phase, indicating the aggressive growth phase of the tumor. However, depending on the immune response and other factors in the system, the growth of tumor cells may slow down or sometimes even decline after reaching a certain point. This can happen when immune cells start becoming more effective at identifying and destroying tumor cells, leading to a reduction in tumor size. Alternatively, if the tumor manages to evade the immune system, its growth may continue unchecked, resulting in a steady or exponential increase. Overall, the graph reflects a dynamic process with tumor cells initially increasing, potentially reaching a peak, and then either stabilizing, shrinking, or continuing to grow based on the balance between tumor development and immune response. Understanding these changing patterns helps in designing better strategies to control or eliminate tumors.

Consider the following random control functions and defined by

and

where for .

In Table 4, we calculate for and and various parameters and As you can observe and present the relative maximal thresholds and also, and present the relative minimal thresholds, respectively. As a result, relative optimal threshold can present the best approximation error estimate via Fox–type controllers for our problem.

Table 4.

for and and various parameters and .

	0.0038	0.0056	0.0077	0.0092
	0.0065	0.0081	0.0094	0.0118
	0.0084	0.0106	0.0118	0.0139
	0.0023	0.0041	0.0063	0.0084
	0.0071	0.0093	0.0104	0.0121
	0.0050	0.0067	0.0088	0.0110
	0.0028	0.0042	0.0071	0.0085
	0.0044	0.0065	0.0091	0.0101
	0.0079	0.0091	0.0117	0.0125
	0.0012	0.0037	0.0054	0.0069
	0.0060	0.0078	0.0105	0.0113
	0.0037	0.0051	0.0089	0.0092

The plots of dynamic for different values of are shown in Fig. 8. In our tumor-immune interaction model, the fractional order parameter plays a crucial role in determining the system’s dynamic behavior and control effectiveness. Figure 8 shows that as increases, the resulting optimal error also tends to increase. This trend indicates that higher values of , which lessen the influence of the system’s past states on its current behavior, make it more difficult to achieve stable and accurate control over tumor dynamics. From a stability perspective, lower fractional orders (smaller ) incorporate a greater dependence on the system’s previous states, enhancing responsiveness and controllability. This dependence helps maintain the system’s stability and reduces the control error. Conversely, increasing diminishes this influence, which can weaken stability and make tumor control more challenging, resulting in higher errors. In practical terms, adjusting affects how much the past states impact current system behavior. Smaller values improve stability and control accuracy, while larger values tend to decrease stability and increase the error. Understanding this relationship is essential for designing effective control strategies in tumor-immune systems by choosing an optimal fractional order that balances stability and error minimization.

Fig. 8.

The plots of the obtained optimal error for fixed .

Numerical synchronization results

Making use of (15), consider

and

where for represent the synchronization control parameters, and denote the states of master and slave, respectively. Now, we get

This implies that converge to zero when if we set

In Table 5, you can observe the numerical results of synchronization control parameters obtained through tumor-immune models (14) and (15). As you can see the obtained changes via the fractional derivative operator imply better estimation than the obtained results via the classical derivative. But it is important to mention that the speed of changes of the results obtained through the ordinary derivative are higher than those of the fractional-order derivative in order to converge to zero.

Table 5.

Numerical results of synchronization control parameters obtained through tumor-immume models (14) and (15).

Changes of synchronization controller		(0.001,0.002)	(0.002,0.003)	(0.003,0.004)	(0.004,0.005)
		0.0001026	0.0001945	0.0003747	0.0006402
Delta type first order tumor-immune model (14)		0.0000782	0.0001691	0.0002893	0.0005594
		0.0000923	0.0001197	0.0001274	0.0001430
Caputo discrete fractional tumor-immune model (15)		0.0000401	0.0000579	0.0000802	0.0001013

Since the fractional derivative accounts for system memory and hereditary properties, it provides a more accurate representation of the tumor-immune dynamics, especially in complex biological systems. Although the results obtained via this approach have slower convergence rates, their enhanced estimation accuracy can lead to more effective and reliable control strategies in practice. Conversely, the classical derivative, with faster convergence, may be suitable for applications that require rapid stabilization but might overlook subtle dynamical features captured better by fractional operators.

Conclusion

This study has introduced and thoroughly analyzed two distinct mathematical models to explore tumor-immune system interactions from different perspectives, both within a stochastic framework. The first model, based on fractal-fractional derivatives–specifically the Atangana-Baleanu operator–provides a detailed description of the system’s dynamics, capturing the memory-dependent behaviors inherent in biological processes. By establishing the existence and uniqueness of solutions through fixed point theorems, we confirmed the well-posedness of this model. Numerical simulations employing Lagrangian-piecewise interpolation revealed how variations in fractional and fractal parameters influence the interaction dynamics between immune cells and tumor cells, offering new insights into the stability and responsiveness of the system. In contrast, the second model utilizes coupled nonlinear difference equations formulated with the Caputo fractional operator. Through rigorous application of fixed point theorems, we demonstrated the existence of solutions and further examined key properties such as stability, controllability, and synchronization. This model serves as an independent approach to understanding tumor-immune interactions, with its analysis within a stochastic context adding a layer of realism to the complex biological phenomena under consideration. Numerical investigations of this model in specific scenarios shed light on its dynamic behavior and practical implications. In summary, despite their different formulations and underlying methodologies, both models contribute valuable insights into tumor-immune dynamics. The integration of fractional calculus and stochastic elements in each approach highlights their potential for enhancing our understanding of the complex biological processes involved. These findings lay a foundation for future research, including model refinement, incorporation of additional biological factors, and improved numerical techniques, ultimately aimed at advancing therapeutic strategies and personalized medicine.

Author contributions

S.R.A., methodology, writing–original draft preparation. R.S., supervision and project administration. F.R.A., editing–original draft preparation. O.T., supervision, project supervision and editing-original draft preparation. All authors have read and approved the final manuscript.

Funding

No funding.

Data availability

Data will be provided by corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.