Stochastic dynamics of a nonlinear tumor-immune competitive system

Abstract

The paper uses nine coupled ordinary differential equations (ODEs) to describe a tumor-immune competitive system. The model is then reduced to four nonlinear coupled ODEs, encompassing tumor cells, cytotoxic T-lymphocytes, macrophages, and dendritic cells by utilizing quasi-steady-state approximations. We explore the dynamics of biologically feasible steady states and their local stability analysis. We introduce stochastic fluctuation terms into the deterministic system to account for uncertainty and variability in the tumor-immune interaction system. The uniqueness and existence of our stochastic system is established by applying It$\hat{o}$’s lemma. Additionally, it is demonstrated that the solution of the stochastic system is both stochastically ultimately bounded and permanent. We established the criteria for determining the extinction of tumor cell population, and conditions under which our stochastic system exhibits asymptotic stability in a mean square sense are derived. Numerical illustrations are performed to validate both deterministic and stochastic models under different intensities of population fluctuation. Our model explores that in the absence of intensity fluctuations, the deterministic model remains stable around a high tumor-presence steady state. Additionally, the tumor cell population quickly approaches zero for sufficiently small values of intensity fluctuation parameters. If the intensity value of population fluctuations is increased, then the cell populations are more fluctuated. Moreover, the stochastic mean solution confirms the influence of stochastic noise on the cell population.

1 Introduction

Over the past few decades, malignant tumors or cancers have emerged as one of the deadliest global health challenges. The tumors consist of uncontrolled growth of abnormal tissues and cells, leading to the invasion of nearby body parts and the spread to surrounding organs. Various risk factors, such as obesity, unhealthy lifestyle, alcohol consumption, and smoking have been linked to the increased incidence of cancer [1]. In the field of medical science, a pivotal question in oncology revolves around understanding how our immune system can effectively combat tumor proliferation [2, 3]. Immune responses play a crucial role in regulating tumor growth and progression, posing a significant challenge in the quest for effective cancer treatment strategies.

Our immune system is an intricate network comprising various types of cells, proteins, and cytokines that work collaboratively to safeguard our body against external invaders. The primary components of our immune system consist of cytotoxic T-lymphocytes (CD8+ T cells), B cells, NK cells or natural killer cells, Helper T cells, macrophages, and specific immune-stimulatory cytokines such as IL-10 or interleukin-10, IL-12 or interleukin-12 and IFN-$\gamma$ or interferon-gamma, etc. Macrophages, NK cells, and CD8+ T cells are commonly referred to as effector cells [4]. Their main function is to eliminate or destroy tumor cells, thereby playing pivotal roles in the immune response against cancer cells [5]. On the other hand, immuno-stimulatory cytokines, while unable to directly eliminate tumor cells, play an important role in enhancing the functions of effector cells. They stimulate these cells to inhibit the proliferation of tumor cells. A vital link between the innate and adaptive immune system is formed by antigen-presenting dendritic cells. These specialized cells act as messengers and are responsible for producing IL-12, a cytokine that plays a crucial role in promoting CD8+ T cells. Regulatory T-cells or Tregs and transforming growth factor-$\beta$ (TGF-$\beta$) play critical role in the tumor-immune interaction system, primarily contributing to the immuno-suppressive component. However, in the tumor-immune interaction system, the dynamics are more complex. This interplay between the tumor cells and immune system can be likened to a prey-predator relationship, where tumor cells act as prey and effector cells take on the role of predators [6, 7].

The balance between effector cells (like cytotoxic T cells, macrophages and natural killer cells) and immune-suppressive components such as regulatory T cells (Tregs) and TGF-$\beta$ plays a crucial role in shaping the outcome of the ‘prey-predator’ relationship between tumor cells and effector cells. When there are more effector cells than immune-suppressive components, they can effectively identify and remove tumor cells, resulting in tumor control or regression. In this situation, the immune systems like a predator, effectively managing the population of tumor cells. On the other hand, when immune-suppressive components dominate the immune cells, they can suppress the activity of effector cells. This allows tumor cells to evade immune surveillance. In this scenario, the imbalance favors tumor cells, resembling a ‘prey-predator’ dynamic where the tumor cells act as predators, taking advantage of the weakened immune response. An excessive presence of immune-suppressive components can also promote tumor progression. Tregs and TGF-$\beta$ have the capability to suppress effector cell function, which diminishes the immune response against the tumor cells. In summary, the balance between effector cells and immune-suppressive components significantly impacts the outcome of the interaction between the immune system and tumor cells.

Biologically, our deterministic system faces certain limitations. The parameters governing the system such as activation rate, growth rate, death rate, and carrying capacity are subject to fluctuations induced by various environmental factors including body weight, air pollution, temperature, etc. Environmental fluctuations are a crucial phenomena in the tumor-immune interaction model. In most of the cases, equilibrium points of the deterministic system oscillate randomly and deterministic equilibrium points are no longer in a fixed state. As a result, the deterministic system may not accurately predict the future of the biological system. To address these challenges and achieve a more realistic representation of natural phenomena, stochastic systems offer a promising alternative. Unlike deterministic systems, stochastic models consider randomness and uncertainty observed in biological systems. Two approaches exist for developing stochastic models that correspond to deterministic systems. The first method involves introducing stochastic perturbation terms of Gaussian white noise type into important parameters, thereby formulating the system as an It$\hat{o}$ stochastic differential equation. In the second approach, random fluctuating terms are directly added into the deterministic system without changing any parameters. In this manuscript, we explain the impact of randomly fluctuating terms on our system and investigate its effects around the interior fixed point of the deterministic system. As a result, the stochastic system offers a valuable framework for studying and understanding the dynamic behavior of biological populations in response to environmental fluctuations and uncertainties.

The introduction of stochastic perturbation terms can enhance the accuracy of predictions about the future of a biological system by accounting for environmental fluctuations and uncertainties. These perturbation terms simulate the inherent randomness and variability in biological processes, allowing models to capture the unpredictable nature of environmental influences. By incorporating stochasticity, predictions can better reflect real-world dynamics, where environmental factors can vary widely and unpredictably over time. In the absence of stochastic fluctuations, the system may still exhibit stability under certain conditions. While stability may be observed in the absence of fluctuations, incorporating stochasticity provides a more accurate representation of the system’s dynamics and enhances the predictive capability of the model in the face of environmental uncertainties. In many cases, random environmental fluctuations can indeed contribute to the stability of competitive systems. The introduction of stochastic perturbation terms may not necessarily lead to enhanced stability. Random fluctuations in our system can perturb the balance by disturbing the interactions between tumor cells, immune cells, and other components of the microenvironment. In simply, although random fluctuations can sometimes help stabilize competitive systems, their effects on biological systems such as tumor-immune interactions are more complex. They can lead to unpredictable outcomes, unlike the stabilizing effects seen in other competitive systems. Therefore, we need to thoroughly study and understand the impact of stochastic perturbations in this context, as they may not always produce the same stabilizing effects.

Introducing stochastic fluctuations into a mathematical model of tumor-immune interactions can help understand the role of lifestyle factors such as obesity and smoking in tumor proliferation by accounting for the inherent randomness and variability in biological systems. Lifestyle factors like obesity and smoking can influence the immune response and tumor growth through various mechanisms, including inflammation, oxidative stress, and immune suppression. By incorporating stochastic fluctuations into the model, researchers can simulate the unpredictable nature of these lifestyle factors’ effects on immune cell dynamics, tumor growth rates, and treatment responses. This stochastic approach allows for a more realistic representation of the complex interplay between lifestyle factors, immune function, and tumor development, helping to identify critical points where interventions may be most effective in preventing or treating malignant tumors in individuals with specific lifestyle risk factors.

Many research articles on the tumor-immune competitive system can be found in [4,5,6, 8,9,10,11,12,13,14,15,16,17,18,19], but there are few articles published on stochastic cancer models. Kloeden & Platen [20] introduced a stochastic system to study the limitations of appropriate overlap in a randomly fluctuating environment. Li et al. [21] studied a cancer growth model using a stochastic system and observed extinction in the presence of the immune system. Lefever et al. [22] explored that when the variance of population fluctuations is increased around a mean value, a transition exists in the bifurcation diagram, which eventually disappears. Li et al. [23] observed stochastic fluctuation-induced tumor extinction and recurrence in a tumor growth model based on a catalytic Michaelis-Menten reaction. The authors found that at a certain level, environmental fluctuations facilitate tumor extinction. Oroji et al. [24] introduced a stochastic cancer model and this model has been categorized into three subpopulations for radiotherapy. They calculated the entire lifespan of tumor cells and observed the behavior of tumor cells under the application of two different treatment strategies. A tumor-immune interaction system was considered as a deterministic prey-predator system in [25]. In this paper, the authors tried to control the tumor cells by applying stochastic fluctuations in their deterministic system. Sardar et al. [26] introduced a deterministic model of tumor cell growth with the Allee effect. The authors extended the deterministic system into a stochastic system by applying the parameter perturbation method. Additionally, they conducted numerical simulations of the stochastic system corresponding to the Allee effect, weak Allee effect, and strong Allee effect. Kim et al. [27] developed a stochastic tumor model with virotherapy. To determine the probability of tumor extinction, the authors altered the parameter values. Additionally, they revealed that high infection rate viruses and optimal cytotoxicity are more effective for cancer treatment. Caravanga et al. [28] presented a hybrid-stochastic version of the tumor-immune interaction model in presence of cytokine IL-2, which is introduced by Kirschner and Panetta [4]. In their study, the researchers demonstrated that tumor growth can be reduced through random environmental fluctuations. Tumor-immune competitive systems with immunotherapeutic drug has played an important role to eradicate the tumor cell population [29, 30].

Parameter perturbation methods involve systematically varying model parameters to observe their impact on tumor growth and response to treatment. This approach allows researchers to identify critical parameters that strongly influence tumor dynamics and treatment outcomes. By perturbing parameters related to factors such as tumor growth rate, immune cell activity, and treatment efficacy, researchers can gain insights into which factors are most important in determining tumor behavior and response to therapy. Additionally, studying phenomena like the Allee effect, which describes how populations may struggle to grow or survive at low densities, can provide valuable insights into tumor growth dynamics. In the context of cancer, the Allee effect suggests that tumors may face challenges in proliferating and spreading when their population density is low. Understanding how the Allee effect influences tumor growth dynamics can help predict the likelihood of tumor eradication and inform the design of more effective treatment strategies. Overall, the combined use of parameter perturbation methods and the examination of phenomena like the Allee effect can enhance our comprehension of tumor growth dynamics and improve our ability to predict and achieve tumor eradication in cancer patients. These approaches provide valuable tools for refining mathematical models of cancer progression and guiding the development of more targeted and personalized treatment approaches.

The rest of this research paper is delineated in the following manner. In Sect. 2, we introduce a deterministic system of four nonlinear coupled ordinary differential equations (ODEs) and conduct a stability analysis around singular points. In Sect. 3, we convert our deterministic system into a stochastic system by introducing a random fluctuating term. Some basic properties, that is, existence, uniqueness, stochastically ultimate bounded, stochastically permanence, and extinction of our stochastic model for the initial density are studied in Sect. 4. Section 5 investigates the asymptotically mean square stable of our stochastic system. In Sect. 6, we have estimated the values of some parameters. Sect. 7 deals with numerical analysis of both the deterministic and stochastic system. Finally, our research paper concludes with a summary in Sect. 8.

2 Deterministic model

In this section, we introduced a mathematical model using a system of nine coupled ordinary differential equations (ODEs), namely tumor cells (T), cytotoxic T-lymphocytes or CD8+ T cells $(T_{8})$, macrophages (M), dendritic cells (D), Tregs or regulatory T-cells ($T_{g})$, interleukin-10 or IL-10 $(I_{10})$, transforming growth factor-$\beta$ or TGF-$\beta$ $(T_{\beta })$, interleukin-12 or IL-12 $(I_{12})$ and interferon-$\gamma$ or IFN-$\gamma$ $(I_{\gamma })$. Then, our deterministic model is stated as follows

$$\begin{aligned} \frac{dT}{dt}= & {} r_{T}T (1-b_{T}T) - \frac{(\alpha _{T}^{'}M+\gamma _{T}^{'}T_{8})T}{g_{T}^{'}+I_{10}}, \nonumber \\ \frac{dT_{8}}{dt}= & {} \frac{\alpha _{8}^{'}I_{12}}{g_{8}^{'}+T_{g}} - \delta _{8}T_{8}, \nonumber \\ \frac{dM}{dt}= & {} s_{m} + \frac{\alpha _{m}^{'}I_{\gamma }}{(g_{m}^{'}+I_{\gamma })}\cdot \frac{1}{(g_{m1}^{'}+T_{\beta })}\nonumber \\{} & {} - \gamma _{m}MT- \delta _{m}M, \nonumber \\ \frac{dD}{dt}= & {} s_{d} + \frac{\alpha _{d}T}{g_{d}+T} - \delta _{d}D, \nonumber \\ \frac{dT_{g}}{dt}= & {} \alpha _{g}T_{8} - \delta _{g}T_{g}, \nonumber \\ \frac{dI_{10}}{dt}= & {} \alpha _{10}M - \delta _{10}I_{10}, \nonumber \\ \frac{dT_{\beta }}{dt}= & {} s_{\beta } + \alpha _{\beta }T - \delta _{\beta }T_{\beta }, \nonumber \\ \frac{dI_{12}}{dt}= & {} \alpha _{12}D - \delta _{12}I_{12}, \nonumber \\ \frac{dI_{\gamma }}{dt}= & {} \alpha _{\gamma }T_{8} - \delta _{\gamma }I_{\gamma }. \end{aligned}$$

(1)

• The first equation in (1) represents the tumor cell density at any given time t. Initial term $r_{T}T (1-b_{T}T)$ describes the logistic growth of tumor cells in the absence of any immune response [12]. Here, $r_{T}$ stands for the intrinsic growth rate, and $\frac{1}{b_{T}}$ denotes the maximum carrying capacity of tumor cells. The next component describes the elimination of tumor cells through interactions with macrophages [2] and CD8+ T cells [5], each with elimination rates denoted by $\alpha _{T}^{'}$ and $\gamma _{T}^{'}$, respectively. The term $g_{T}^{'}$ represents the half-saturation constant, where $\frac{1}{g_{T}^{'}+I_{10}}$ serves as a immunosuppressive factor affecting both macrophages and CD8+ T cells.

• The second equation in (1) refers the density of CD8+ T cells. The first term represents the activation of CD8+ T cells at a rate $\alpha _{8}^{'}$. This activation relies on the presence of CD4+T cells, which are boosted by the cytokine IL-12 [31]. However, the activation of CD8+ T cells is countered by regulatory T-cells [32] at a suppressive rate $g_{8}^{'}$. The decay rate of CD8+ T cells is denoted by $\delta _{8}$.

• The third equation in (1) describes to the density of macrophages, where $s_{m}$ denotes the constant influx rate of macrophages [33]. The recruitment of macrophages ($\alpha _{m}^{'}$) is directly affected by the presence of IFN-$\gamma$ [2, 33], with $g_{m}^{'}$ indicating the half-saturation constant. Additionally, the term $\frac{1}{g_{m1}^{'}+T_{\beta }}$ serves as an immunosuppressive factor for macrophages, where $g_{m1}^{'}$ represents the half-saturation constant. The third component highlights the deactivation of macrophages resulting from their interactions with tumor cells at a rate denoted by $\gamma _{m}$ [2]. Additionally, $\delta _{m}$ represents the decay rate of macrophages.

• The fourth equation in (1) elaborates on the density of antigen-presenting dendritic cells. The term $s_{d}$ denotes the constant source rate of dendritic cells [34]. The coefficient $\alpha _{d}$ signifies the activation rate of dendritic cells induced by the direct presence of tumor cells, with $g_{d}$ acting as the half-saturation constant following Michaelis-Menten kinetics [35]. The death rate of dendritic cells is represented by $\delta _{d}$.

• The fifth equation in (1) clarifies the behavior of regulatory T-cells (Tregs). Tregs are produced from activated CD8+ T cells [36] with an activation term $\alpha _{g}$, while their natural degradation rate is denoted by $\delta _{g}$.

• The sixth equation in the model (1) illustrates the density of the anti-inflammatory cytokine IL-10. IL-10 is activated by macrophages [33] with a rate denoted by $\alpha _{10}$, while decay rate is represented by $\delta _{10}$.

• The seventh equation in the model (1) depicts the level of TGF-$\beta$. The term $s_{\beta }$ signifies the constant production rate of TGF-$\beta$ [2]. The second term represents the contribution from tumor cells, which is directly proportional to their size, with $\alpha _{\beta }$ denoting the release rate per tumor cell [37]. The final term represents the degradation of the immunosuppressive cytokine TGF-$\beta$ at a constant rate of $\delta _{\beta }$.

• The eighth equation in the model (1) describes the concentration of IL-12. IL-12 is produced by dendritic cells, with $\alpha _{12}$ indicating the release rate per antigen-specific dendritic cell [31]. The degradation rate of IL-12 is denoted by $\delta _{12}$.

• The ninth equation in the system (1) outlines the behavior of IFN-$\gamma$. We hypothesize that IFN-$\gamma$ is produced by CD8+ T cells [2, 37] with a production rate $\alpha _{\gamma }$. The final term represents the degradation of IFN-$\gamma$ at a constant rate of $\delta _{\gamma }$.

Tumor growth encompasses multiple time scales. The expansion of the tumor cell population spanning over periods ranging from months to years, as well as weeks to months depends on their characteristics. According to the clinical observations, the activation of CD8+ T cells, macrophages, and dendritic cells (antigen-presenting) occurs in shorter time frames, like days to weeks. On the other hand, the degradation and secretion of both immuno-suppressive and immuno-stimulatory cytokines occur on shorter time scales (for example, seconds to hours). To enhance our comprehension of the intricate dynamics involved in the interaction between tumors and the immune system, we apply the quasi-steady-state approximations [38] in our mathematical model for the cytokines concentration. Then, from the cytokine equations of (1), we have

$$\begin{aligned} T_{g}= & {} \frac{\alpha _{g}}{\delta _{g}}T_{8},~I_{10}=\frac{\alpha _{10}}{\delta _{10}}M,\\ T_{\beta }= & {} \frac{s_{\beta }}{\delta _{\beta }}+\frac{\alpha _{\beta }}{\delta _{\beta }}T, \\ I_{12}= & {} \frac{\alpha _{12}}{\delta _{12}}D,~I_{\gamma }=\frac{\alpha _{\gamma }}{\delta _{\gamma }}T_{8}. \end{aligned}$$

After substituting these cytokine expressions into the first to fourth equations of the deterministic system (1), we get the following four nonlinear ODEs of tumor-immune interaction system

$$\begin{aligned} \frac{dT}{dt}= & {} r_{T}T (1-b_{T}T) - \frac{(\alpha _{T}M+\gamma _{T}T_{8})T}{g_{T}+M}, \nonumber \\ \frac{dT_{8}}{dt}= & {} \frac{\alpha _{8}D}{g_{8}+T_{8}} - \delta _{8}T_{8}, \nonumber \\ \frac{dM}{dt}= & {} s_{m} + \frac{\alpha _{m}T_{8}}{(g_{m}+T_{8})}\cdot \frac{1}{(g_{m1}+T)} - \gamma _{m}MT- \delta _{m}M, \nonumber \\ \frac{dD}{dt}= & {} s_{d} + \frac{\alpha _{d}T}{g_{d}+T} - \delta _{d}D, \end{aligned}$$

(2)

with the given following initial densities:

$$\begin{aligned} T(0)= & {} T_{0}> 0,~~~ T_{8}(0) =T_{80}> 0,\\ M(0)= & {} M_{0}> 0,~~~ D(0) = D_{0} > 0, \end{aligned}$$

where

$$\begin{aligned} \alpha _{T}= & {} \frac{\delta _{10}}{\alpha _{10}}\alpha _{T}^{'},~~~ \gamma _{T}=\frac{\delta _{10}}{\alpha _{10}}\gamma _{T}^{'},\\ g_{T}= & {} \frac{\delta _{10}}{\alpha _{10}}g_{T}^{'},~~~\alpha _{8} = \frac{\delta _{g}}{\alpha _{g}}\frac{\alpha _{12}}{\delta _{12}}\alpha _{8}^{'}, \\ g_{8}= & {} \frac{\delta _{g}}{\alpha _{g}}g_{8}^{'}, ~~~ \alpha _{m}= \frac{\delta _{\beta }}{\alpha _{\beta }}\alpha _{m}^{'},\\ g_{m}= & {} \frac{\delta _{\gamma }}{\alpha _{\gamma }}g_{m}^{'}, ~~~g_{m1}=\frac{\delta _{\beta }}{\alpha _{\beta }}g_{m1}^{'}+\frac{s_{\beta }}{\alpha _{\beta }}. \end{aligned}$$

Here,

• The first term of the first equation of (2) represents the logistic growth of tumor cell population with no immune response, where $r_{T}$ is the intrinsic growth rate of tumor cells and $\frac{1}{b_{T}}$ is the maximum population size of tumor cells.

• The function $\frac{(\alpha _{T}M+\gamma _{T}T_{8})T}{g_{T}+M}$ represents the deactivation term of tumor cells due to interaction with macrophages and cytotoxic T-lymphocytes with the deactivation rate $\alpha _{T}$ and $\gamma _{T}$, respectively. $\frac{1}{g_{T}+M}$ represents the major immuno-suppressive factor for both macrophages and CD8+ T cells, where $g_{T}$ is the half-saturation constant.

• $\alpha _{8}$ designates the activation rate of CD8+ T cells due to the direct presence of dendritic cells and the half-saturation constant of CD8+ T cells is $g_{8}$.

• The natural death rate of CD8+ T cells is $\delta _{8}$.

• $s_{m}$ signifies the constant influx rate of macrophages.

• The function $\frac{\alpha _{m}T_{8}}{(g_{m}+T_{8})}\cdot \frac{1}{(g_{m1}+T)}$ describes the activation term of macrophages. $\alpha _{m}$ is the activation rate of macrophages due to the direct presence of CD8+ T cells, where $g_{m}$ is the half-saturation constant. $\frac{1}{(g_{m1}+T)}$ is the major immuno-suppressive factor of macrophages and $g_{m1}$ is the suppressive parameter.

• $\gamma _{m}$ is the decay rate of macrophages due to the interaction with tumor cells.

• $\delta _{m}$ is the natural death rate of macrophages.

• The constant source rate of dendritic cells is $s_{d}$.

• The activation rate of dendritic cells due to the direct presence of tumor cells is represented by the coefficient $\alpha _{d}$, with $g_{d}$ being the half-saturation constant.

• $\delta _{d}$ is natural death rate of dendritic cells.

2.1 Equilibria

The deterministic model (2) has two feasible singular points, namely

(i) tumor-free singular point $E^{1}{\equiv }(T^{1}, T^{1}_{8},M^{1}, D^{1})=\Big ( 0, T_{8}^{1}, \frac{s_{m}g_{m1}(g_{m}+T_{8}^{1})+\alpha _{m}T_{8}^{1}}{\delta _{m}g_{m1}(g_{m}+T_{8}^{1})}, \frac{s_{d}}{\delta _{d}}\Big )$, where $T^{1}_{8}$ is the unique positive root of the given equation

$$\begin{aligned} (T_{8}^{1})^{2}+g_{8}T_{8}^{1}-\frac{\alpha _{8}s_{d}}{\delta _{8}\delta _{d}}= & {} 0. \end{aligned}$$

(3)

From the above equation, we have the following positive root

$$\begin{aligned} T_{8}^{1}= & {} \frac{-g_{8}+\sqrt{\Big (g_{8}^{2}+4\frac{\alpha _{8}s_{d}}{\delta _{8}\delta _{d}}\Big )}}{2}. \end{aligned}$$

To study the stability analysis of (2), we evaluate the variational matrix around the singular point $E(T, T_{8}, M, D)$ is

$$\begin{aligned} J(E)= & {} \left[ \begin{array}{cccc} r_{T}-2r_{T}b_{T}T-\frac{(\alpha _{T}M+\gamma _{T}T_{8})}{g_{T}+M} &{} -\frac{\gamma _{T}T}{g_{T}+M} &{} -\frac{(g_{T}\alpha _{T}-\gamma _{T}T_{8})T}{(g_{T}+M)^{2}} &{} 0 \\ 0 &{} -\frac{\alpha _{8}D}{(g_{8}+T_{8})^{2}}-\delta _{8} &{} 0 &{} \frac{\alpha _{8}}{g_{8}+T_{8}} \\ -\frac{\alpha _{m}T_{8}}{(g_{m}+T_{8})(g_{m1}+T)^{2}}-\gamma _{m}M &{} \frac{\alpha _{m}g_{m}}{(g_{m}+T_{8})^{2}(g_{m1}+T)} &{} -\gamma _{m}T-\delta _{m} &{} 0 \\ \frac{g_{d}\alpha _{d}}{(g_{d}+T)^{2}} &{} 0 &{} 0 &{} -\delta _{d} \\ \end{array} \right] . \end{aligned}$$

At the tumor-free singular point $E^{1}$, the variational matrix $J(E^{1})$ of (2) has the following eigenvalues:

$$\begin{aligned} \lambda ^{1}_{1}= & {} r_{T}-\frac{\alpha _{T}M^{1}+\gamma _{T}T_{8}^{1}}{g_{T}+M^{1}}, \\ \lambda ^{1}_{2}= & {} -\frac{\alpha _{8} D^{1}}{(g_{8}+T_{8}^{1})^{2}} - \delta _{8}, \\ \lambda ^{1}_{3}= & {} -\delta _{m}, \\ \lambda ^{1}_{4}= & {} -\delta _{d}. \\ \end{aligned}$$

It is observing that when $\lambda ^{1}_{1} < 0$, all eigenvalues of J($E^{1}$) are negative.. Therefore, the tumor-free singular point is locally asymptotically stable if $r_{T} < \frac{\alpha _{T}M^{1}+\gamma _{T}T_{8}^{1}}{g_{T}+M^{1}}$, otherwise unstable. Then, we have the following theorem.

Theorem 1

The tumor-free steady state $E^{1}$ of the deterministic system (2) will be locally asymptotically stable if $r_{T} - \frac{\alpha _{T}M^{1}+\gamma _{T}T_{8}^{1}}{g_{T}+M^{1}} < 0$, otherwise unstable.

(ii) The interior singular point is $E^{*}(T^{*}, T_{8}^{*}, M^{*}, D^{*})$, where $(T^{*}, T_{8}^{*}, M^{*}, D^{*})$ is the positive root of the equations $\frac{dT}{dt}=\frac{dT_{8}}{dt}=\frac{dM}{dt}=\frac{dD}{dt}=0$. From an analytical perspective, determining the explicit form of the interior singular point $E^{*}$ proves to be highly challenging. Hence, we investigate its existence and stability by numerical simulation.

3 Stochastic model

So far we have discussed the suggested tumor-immune interaction model in a deterministic setting. We aim to explore the dynamics of the changed system influenced by white noise, offering a more realistic representation compared to its deterministic counterpart. Although the deterministic tumor-immune interaction model provides significant insight into the dynamics of tumor growth, it does not describe the disease’s likelihood of extinction. Due to the presence of some environmental fluctuations, our parameters such as the growth rate, death rate, and activation rate of our cell population also fluctuated. In this case, we introduce stochastic perturbations into our deterministic system to study the effect of environmental fluctuation. To study environmental fluctuation, there are mainly two processes to develop a stochastic perturbation of a deterministic model. In the first type, some important parameters are perturbed by the Gaussian white-noise type in the stochastic differential equation [26]. On the other hand, we introduce a randomly fluctuating term directly into the deterministic system without changing any parameter [25]. In the present paper, we consider the randomly fluctuating effects in our system and aim to investigate their impact around the interior fixed point of the deterministic system (2).

In real-life scenarios, biological systems exhibit inherent variability due to factors such as genetic diversity, environmental fluctuations. This variability can show up as stochastic fluctuations in cell populations, cytokine concentrations, and immune responses within tumor microenvironments. Environmental factors can impact tumor growth and immune system function. The environmental perturbations introduce stochasticity into the system, leading to unpredictable fluctuations in tumor progression and immune response dynamics. Clinical studies have highlighted the stochastic nature of cancer progression and treatment response. Despite similar initial conditions and treatments, patients with the same type and stage of cancer may exhibit varying outcomes, including differences in tumor growth rates, immune responses. This variability underscores the importance of considering stochastic fluctuations in cancer modeling. Experimental studies using in the models of cancer have demonstrated the presence of stochastic fluctuations including proliferation, activation, death. These stochastic fluctuations can arise from inherent biological noise, random molecular interactions, and environmental variability. By providing a comprehensive overview of these biological and medical considerations, we justify the application of white noise for stochastic perturbations in the tumor-immune interaction model. This background information helps establish the biological relevance of incorporating stochasticity into the model and underscores the importance of capturing real-life variability in cancer biology.

The motivation behind introducing stochastic equations in the tumor-immune interaction model lies in capturing the inherent randomness and uncertainty observed in biological systems. In real-world scenarios, biological processes are influenced by environmental factors such as fluctuations in temperature, oxygen levels, and nutrient availability can impact both tumor growth and immune cell function. These environmental fluctuations can introduce randomness into the system, affecting the dynamics of tumor-immune interactions. By incorporating stochasticity into the model, we aim to create a more realistic representation that accounts for the unpredictable nature of these processes. Overall, the motivation for using stochastic equations in the tumor-immune interaction model is to develop a more comprehensive understanding of the complex, inherently stochastic nature of biological systems and their responses to environmental fluctuations. This enables us to generate more realistic predictions and insights that can inform the development of effective cancer treatment strategies.

In the system (2), we consider that the stochastic random perturbations of the state variables around interior singular point $E^{*}(T^{*},~T^{*}_{8},~M^{*},~D^{*})$ are white noise type, which are directly proportional to the distances of singular point, that is, $T^{*},~T^{*}_{8},~M^{*},~D^{*}$ from $T(t), T_{8}(t), M(t), D(t)$. Therefore, our model has stochastic nature with following form

$$\begin{aligned} \begin{array}{llll} dT &{}=&{} \bigg [r_{T}T (1-b_{T}T) - \frac{(\alpha _{T}M+\gamma _{T}T_{8})T}{g_{T}+M}\bigg ]dt \\ &{}&{}+ \rho _{1}(T-T^{*})dB_{1}(t), \\ dT_{8} &{}=&{} \bigg [\frac{\alpha _{8}D}{g_{8}+T_{8}} - \delta _{8}T_{8}\bigg ]dt + \rho _{2}(T_{8}-T_{8}^{*})dB_{2}(t), \\ dM &{}=&{} \bigg [s_{m} + \frac{\alpha _{m}T_{8}}{(g_{m}+T_{8})}\cdot \frac{1}{(g_{m1}+T)} - \gamma _{m}MT- \delta _{m}M\bigg ]dt \\ &{}&{}+ \rho _{3}(M-M^{*})dB_{3}(t), \\ dD &{}=&{} \bigg [s_{d} + \frac{\alpha _{d}T}{g_{d}+T} - \delta _{d}D\bigg ]dt + \rho _{4}(D-D^{*})dB_{4}(t). \end{array}\nonumber \\ \end{aligned}$$

(4)

$\rho _{i} ~(i=1,~2,~3,~4)$ are positive real constants, which are known as intensity of white noise. $B_{i}(t) ~(i=1,~2,~3,~4)$ represents a 4-dimensional independent standard Wiener process [20]. We investigate the asymptotic stochastic stability behavior around the singular point $E^{*}$ for the stochastic system (4) and comparing the results with the deterministic system (2). In our stochastic model (4), we used the Ito stochastic differential process in the following form

$$\begin{aligned} dX_{t}= & {} h(t, X_{t})dt + n(t, X_{t})dB(t),\nonumber \\ X_{t_{0}}= & {} X_{0}, t \in [t_{0}, t_{f}], \end{aligned}$$

(5)

where the solution $\{X_{t}, t\in [t_{0}, t_{f}](t > 0)\}$ is an Ito process, h is the slowly varying continuous component or drift coefficient, n represents the rapidly varying continuous random component or diffusion matrix [20]. Also, B(t) denotes a 4-dimensional random process which have scalar Wiener process with increments $\triangle B_{i}(t) = B_{i}(t+\triangle t)-B_{i}(t), (i=1,~2,~3,~4)$ are independent of Gaussian random variables $N(0,~\triangle t)$. After stochastic integration, the equation (5) can be written in the following form

$$\begin{aligned} X_{t}= & {} X(t) = X_{0} + \int _{t_{0}}^{t}h(s, X_{s})ds \nonumber \\{} & {} +\int _{t_{0}}^{t}n(s, X_{s})dB(s), \end{aligned}$$

(6)

where second term of the right hand side of (6) is known as Riemann-Stieltjes integral and last term is known as an It$\hat{o}$ integral. Comparing (4) and (5), we have

$$\begin{aligned} X_{t}= & {} (T,~T_{8},~M,~D)^{T},\nonumber \\ B(t)= & {} (B_{1}(t),~B_{2}(t),~B_{3}(t),~B_{4}(t))^{T}, \end{aligned}$$

(7)

$$\begin{aligned} h= & {} \left[ \begin{array}{cccc} r_{T}T (1-b_{T}T) - \frac{(\alpha _{T}M+\gamma _{T}T_{8})T}{g_{T}+M} \\ \frac{\alpha _{8}D}{g_{8}+T_{8}} - \delta _{8}T_{8} \\ s_{m} + \frac{\alpha _{m}T_{8}}{(g_{m}+T_{8})}\cdot \frac{1}{(g_{m1}+T)} - \gamma _{m}MT- \delta _{m}M \\ s_{d} + \frac{\alpha _{d}T}{g_{d}+T} - \delta _{d}D \end{array} \right] , \end{aligned}$$

(8)

and

$$\begin{aligned}{} & {} n= \nonumber \\{} & {} \left[ \begin{array}{cccc} \rho _{1}(T-T^{*}) &{} 0 &{} 0 &{} 0 \\ 0 &{} \rho _{2}(T_{8}-T_{8}^{*}) &{} 0 &{} 0 \\ 0 &{} 0 &{} \rho _{3}(M-M^{*}) &{} 0 \\ 0 &{} 0 &{} 0 &{} \rho _{4}(D-D^{*}) \\ \end{array} \right] .\nonumber \\ \end{aligned}$$

(9)

The system characterized by the stochastic equation (4) is described as having multiplicative noise, given that the diffusion matrix (9) varies based on the solution $X_{t}=(T,T_{8},M,D)^{T}$. Also, the stochastic system (4) is said to have a diagonal noise, since there is a diagonal form in the diffusion matrix n.

3.1 Preliminaries

We consider a complete probability space $(\Phi ,~\{F_{t}\}_{t\ge 0},~\mathbb {P})$ with filtration $\{F_{t}\}_{t\ge 0}$ satisfies $X(t)=(T(t),~T_{8}(t),~M(t),~D(t)),~|X(t)|=(T^{2}(t)+T^{2}_{8}(t)+M^{2}(t)+D^{2}(t))^{\frac{1}{2}}$ and $\mathbb {R}^{c}_{+}=\{x \in \mathbb {R}^{c}: \varphi _{j} > 0,~j=1,2,3,...,c \}$. To study our stochastic system (4), we need some definitions which are very important in our case.

Definition 3.1

[39] Let us assume that (T(0),  $T_{8}(0), M(0), D(0)) \in \mathbb {R}^{4}_{+}$ be the initial densities of the cell populations, then the solution $X(t)=(T(t),~T_{8}(t),$ M(t), D(t)) of the stochastic system (4) is said to be stochastically ultimate bounded (SUB), if for every $\kappa \in (0,1)$, there is a constant $\mu =\mu (\kappa )>0$, such that

$$\begin{aligned} \limsup \limits _{t\rightarrow + \infty }\mathbb {P}\{|X(t)| > \mu \}< & {} \kappa . \end{aligned}$$

Definition 3.2

[39] Let us assume that $(T(0),~T_{8}(0),~M(0),~D(0)) \in \mathbb {R}^{4}_{+}$ be the initial densities of cell populations, then the solution $X(t)=(T(t),~T_{8}(t),~M(t),~D(t))$ of the stochastic system (4) is said to be stochastically permanence (SP), if for every $\kappa \in (0,1)$, there is a constant $\mu =\mu (\kappa )>0$, such that

$$\begin{aligned} \liminf \limits _{t\rightarrow + \infty }\mathbb {P}\{|X(t)| \le \mu \}> & {} 1-\kappa ,\\ \liminf \limits _{t\rightarrow + \infty }\mathbb {P}\{|X(t)| \ge \varphi \}> & {} 1-\kappa , \end{aligned}$$

where $\varphi = \varphi (\kappa )$.

4 Basic properties of the stochastic model (4)

In this section, we perform some fundamental properties of our stochastic model (4) including existence & uniqueness, stochastically ultimate bounded (SUB) and stochastically permanence (SP) with respect to the initial densities. To show these properties, we have constructed some suitable $\mathbb {C}^{2}$-function [39].

Theorem 2

The solution $X_{t} = X(t)=(T(t),~T_{8}(t),$  M(t),  D(t)) of stochastic model (4) is unique on $t\ge 0$ for any given initial densities $X_{0}=(T(0),~T_{8}(0),$$M(0),~D(0)) \in \mathbb {R}_{+}^{4}$ and the solution will remain in $\mathbb {R}_{+}$ with $\mathbb {P}\{T(t),~T_{8}(t),~M(t),~D(t) \in \mathbb {R}_{+},~\forall ~t\ge 0 \} = 1$, that is, probability 1 almost surely.

Proof

For given any initial densities $X_{0}=(T(0),$$~T_{8}(0),~M(0),~D(0))$, coefficients which are to be used in the stochastic system (4) are continuous and satisfy the locally Lipschtiz condition. If $t_{e}$ be the explosion time [39], then there must exists a unique solution $X_{t} = X(t) =(T(t),~T_{8}(t),~M(t),~D(t))\in \mathbb {R}_{+}^{4}$ of the stochastic system (4) for $t \in [0, t_{e})$. To prove the nature of the solution is globally, we will must show that $t_{e} = \infty$ almost surely. We assume that all of the initial densities $(T(0),~T_{8}(0),~M(0),~D(0))$ lies between the interval $\bigg [\frac{1}{p_{0}}, p_{0}\bigg ]$, where $p_{0} > 0$ be a sufficiently large constant. We consider final time $t_{p}$ as

$$\begin{aligned} t_{p}= & {} \left\{ \begin{array}{cccc} t \in [0, t_{e}) &{}:&{} \min \{T(t),~T_{8}(t),~M(t),~D(t)\} \le \frac{1}{p} \\ &{}or&{} \max \{T(t),~T_{8}(t),~M(t),~D(t)\} \ge p \end{array} \right\} , \end{aligned}$$

and $\inf \phi = \infty$, where $\phi$ is the void set. It is clear that, if p tends to $\infty$, then the function $t_{p}$ is increased. We have $t_{\infty } = \lim \limits _{p\rightarrow + \infty } t_{p}$ implies that $t_{\infty } \le t_{e}$ almost surely.

Therefore, in order to show that $t_{e} = \infty$, we have must to prove that $t_{\infty } = \infty$ almost surely. If possible, let us assume that $t_{\infty } < \infty$ almost surely. Then, there exists two constants $t_{T} > 0$ and $\epsilon \in (0,1)$ in such a way that $\mathbb {P}\{t_{\infty } \le t_{T} \} > \epsilon$. Hence, there exists an integer $p_{1} \ge p_{0}$ such that

$$\begin{aligned} \mathbb {P} \{t_{p} \le t_{T}\} \ge \epsilon ~\forall ~ p \ge p_{1}. \end{aligned}$$

(10)

We define a $\mathbb {C}^{2}$-function $V: \mathbb {R}^{4}_{+} \rightarrow \mathbb {R}_{+}$ such that $V(T,~T_{8},~M,~D) = (T + 1 - \ln T)+(T_{8} + 1 - \ln T_{8})+(M + 1 - \ln M)+(D + 1 - \ln D)$. Using It$\hat{o}$’s formula, we get

$$\begin{aligned}{} & {} dV(T,~T_{8},~M,~D) \\{} & {} = LV(T,~T_{8},~M,~D)dt + \rho _{1}(T-T^{*})\Big (1-\frac{1}{T}\Big )dB_{1}\\{} & {} \quad +\rho _{2}(T_{8}-T_{8}^{*})\Big (1-\frac{1}{T_{8}}\Big )dB_{2} \\{} & {} \quad + \rho _{3}(M-M^{*})\Big (1-\frac{1}{M}\Big )dB_{3} \\{} & {} \qquad + \rho _{4}(D-D^{*})\Big (1-\frac{1}{D}\Big )dB_{4}, \end{aligned}$$

where

$$\begin{aligned}{} & {} LV(T,~T_{8},~M,~D) \\{} & {} = \bigg (1-\frac{1}{T}\bigg )\bigg [r_{T}T (1-b_{T}T) - \frac{(\alpha _{T}M+\gamma _{T}T_{8})T}{g_{T}+M}\bigg ] \\{} & {} \quad + \frac{\rho _{1}^{2}}{2}\bigg (1-\frac{T^{*}}{T}\bigg )^{2}\\{} & {} \quad + \bigg (1-\frac{1}{T_{8}}\bigg )\bigg [\frac{\alpha _{8}D}{g_{8}+T_{8}} - \delta _{8}T_{8}\bigg ] + \frac{\rho _{2}^{2}}{2}\bigg (1-\frac{T_{8}^{*}}{T_{8}}\bigg )^{2} \\{} & {} \quad + \bigg (1-\frac{1}{M}\bigg )\bigg [s_{m} + \frac{\alpha _{m}T_{8}}{(g_{m}+T_{8})} \cdot \frac{1}{(g_{m1}+T)} \\{} & {} \quad - \gamma _{m}MT- \delta _{m}M \bigg ] + \frac{\rho _{3}^{2}}{2}\bigg (1-\frac{M^{*}}{M}\bigg )^{2} \\{} & {} \quad + \bigg (1-\frac{1}{D}\bigg )\bigg [s_{d} + \frac{\alpha _{d}T}{g_{d}+T} - \delta _{d}D \bigg ] + \frac{\rho _{4}^{2}}{2}\bigg (1-\frac{D^{*}}{D}\bigg )^{2}. \end{aligned}$$

Positiveness of $T,~T_{8},~M,~D$ implies that

$$\begin{aligned}{} & {} LV(T,~T_{8},~M,~D) \\{} & {} \le \delta _{8}+s_{m}+\delta _{m}+s_{d}+\delta _{d}\\{} & {} \quad +\frac{\rho _{1}^{2}}{2}\bigg (1+\bigg (\frac{T^{*}}{T}\bigg )^{2}\bigg )+ \frac{\rho _{2}^{2}}{2}\bigg (1+\bigg (\frac{T_{8}^{*}}{T_{8}}\bigg )^{2}\bigg )\\{} & {} \quad + \frac{\rho _{3}^{2}}{2}\bigg (1+\bigg (\frac{M^{*}}{M}\bigg )^{2}\bigg ) + \frac{\rho _{4}^{2}}{2}\bigg (1+\bigg (\frac{D^{*}}{D}\bigg )^{2}\bigg )\\{} & {} \quad + T(r_{T}+r_{T}b_{T}+\gamma _{m}) \\{} & {} \quad + \frac{(\alpha _{T}M+\gamma _{T}T_{8})}{g_{T}+M} +\frac{\alpha _{8}D}{g_{8}+T_{8}} \\{} & {} \quad + \frac{\alpha _{m}T_{8}}{(g_{m}+T_{8})}\cdot \frac{1}{(g_{m1}+T)} + \frac{\alpha _{d}T}{g_{d}+T}. \end{aligned}$$

We define positive constants are as follows

$$\begin{aligned} k_{1}= & {} \delta _{8}+s_{m}+\delta _{m}+s_{d}+\delta _{d}\\{} & {} +\frac{\rho _{1}^{2}}{2}\bigg (1+\bigg (\frac{T^{*}}{T}\bigg )^{2}\bigg )+ \frac{\rho _{2}^{2}}{2}\bigg (1+\bigg (\frac{T_{8}^{*}}{T_{8}}\bigg )^{2}\bigg )\\{} & {} +\frac{\rho _{3}^{2}}{2}\bigg (1+\bigg (\frac{M^{*}}{M}\bigg )^{2}\bigg )\\{} & {} + \frac{\rho _{4}^{2}}{2}\bigg (1+\bigg (\frac{D^{*}}{D}\bigg )^{2}\bigg ), \\ k_{2}= & {} r_{T}+r_{T}b_{T}+\gamma _{m}. \end{aligned}$$

After some algebraic manipulation, we have

$$\begin{aligned} LV(T,~T_{8},~M,~D)\le & {} k_{1}+k_{2}T+\frac{(\alpha _{T}M+\gamma _{T}T_{8})}{g_{T}+M}\\{} & {} +\frac{\alpha _{8}D}{g_{8}+T_{8}}+ \frac{\alpha _{m}T_{8}}{(g_{m}+T_{8})}\\{} & {} \cdot \frac{1}{(g_{m1}+T)}+ \frac{\alpha _{d}T}{g_{d}+T}. \end{aligned}$$

If we take superior of the coefficients of the above inequality, then there exists a positive constant G such that

$$\begin{aligned} LV\le & {} G. \end{aligned}$$

LV is a diffusion operator of It$\hat{o}$’s process with $\mathbb {C}^{2}$-function V and if x be It$\hat{o}$’s process with

$$\begin{aligned} dx= & {} f dt + \rho dB, \end{aligned}$$

then we get

$$\begin{aligned}{} & {} dV(T,~T_{8},~M,~D) \\{} & {} \le G dt + \rho _{1}(T-T^{*})\Big (1-\frac{1}{T}\Big )dB_{1}\\{} & {} \quad + \rho _{2}(T_{8}-T_{8}^{*})\Big (1-\frac{1}{T_{8}}\Big )dB_{2} \\{} & {} \quad + \rho _{3}(M-M^{*})\Big (1-\frac{1}{M}\Big )dB_{3}\\{} & {} \quad + \rho _{4}(D-D^{*})\Big (1-\frac{1}{D}\Big )dB_{4}. \end{aligned}$$

By taking $t_{1} \le t_{T}$, we obtain

$$\begin{aligned}{} & {} \int _{0}^{t_{p} \wedge t_{1}} dV(T,~T_{8},~M,~D) \\{} & {} \le \int _{0}^{t_{p} \wedge t_{1}} G dt + \int _{0}^{t_{p} \wedge t_{1}}\rho _{1}(T-T^{*}) \Big (1-\frac{1}{T}\Big )dB_{1} \\{} & {} \quad + \int _{0}^{t_{p} \wedge t_{1}}\rho _{2}(T_{8}-T_{8}^{*})\Big (1-\frac{1}{T_{8}}\Big )dB_{2} \\{} & {} \quad + \int _{0}^{t_{p} \wedge t_{1}}\rho _{3}(M-M^{*})\Big (1-\frac{1}{M}\Big )dB_{3} \\{} & {} \quad + \int _{0}^{t_{p} \wedge t_{1}}\rho _{4}(D-D^{*})\Big (1-\frac{1}{D}\Big )dB_{4}, \end{aligned}$$

where $t_{p} \wedge t_{1} = \min (t_{p}, t_{1})$. This leads to

$$\begin{aligned}{} & {} V(T(t_{p}\wedge t_{1}),~T_{8}(t_{p}\wedge t_{1}),~M(t_{p}\wedge t_{1}),~D(t_{p}\wedge t_{1})) \\{} & {} \le V(T_{0},~T_{8_{0}},~M_{0},~D_{0}) + \int _{0}^{t_{p} \wedge t_{1}} G dt \\{} & {} \quad + \int _{0}^{t_{p} \wedge t_{1}}\rho _{1}(T-T^{*})\Big (1-\frac{1}{T}\Big )dB_{1} \\{} & {} \quad + \int _{0}^{t_{p} \wedge t_{1}}\rho _{2}(T_{8}-T_{8}^{*})\Big (1-\frac{1}{T_{8}}\Big )dB_{2} \\{} & {} \quad + \int _{0}^{t_{p} \wedge t_{1}}\rho _{3}(M-M^{*})\Big (1-\frac{1}{M}\Big )dB_{3} \\{} & {} \quad + \int _{0}^{t_{p} \wedge t_{1}}\rho _{4}(D-D^{*})\Big (1-\frac{1}{D}\Big )dB_{4}. \end{aligned}$$

At first, we take the expectation on both sides, followed by the application of Fubini’s theorem [40] and the properties of It$\hat{o}$’s integral, we have

$$\begin{aligned}{} & {} \mathbb {E}V(T(t_{p}\wedge t_{1}),~T_{8}(t_{p}\wedge t_{1}),~M(t_{p}\wedge t_{1}),~D(t_{p}\wedge t_{1})) \\{} & {} \quad \le V(T_{0},~T_{8_{0}},~M_{0},~D_{0}) + G \mathbb {E}(t_{p}\wedge t_{1}). \end{aligned}$$

Let us assume that $\Phi _{p} = \{t_{p} \le t_{T} \}$, for $p\ge p_{1}$. Then, from equation (10), we have $\mathbb {P}(\Phi _{p}) \ge \epsilon$.

Now, for every $\omega \in \Phi _{p}$, there is at least one of ($T(t_{p},\omega ),~T_{8}(t_{p}, \omega ),~M(t_{p}, \omega ),~D(t_{p}, \omega )$) is equal to either p or $\frac{1}{p}$ and $V(T(t_{p}),~T_{8}(t_{p}),~M(t_{p}),~D(t_{p}))$ is also not less than the smallest of $\min \{(p+1-\log p), \Big (\frac{1}{p}+1+\log p\Big ) \}$. Consequently,

$$\begin{aligned}{} & {} V(T_{0},~T_{8_{0}},~M_{0},~D_{0}) \\{} & {} \quad \ge \epsilon \max [(p+1-\log p) \wedge \Big (\frac{1}{p}+1+\log p\Big )\max ]. \end{aligned}$$

Indicator function of $\Phi _{p}$ is $1_{\Phi _{p}}(\omega )$ defined as

$$\begin{aligned} 1_{\Phi _{p}}(\omega )= & {} \left\{ \begin{array}{cccc} &{}1&{}, if~ \omega \in \Phi _{p}, \\ &{}0&{}, if~ \omega \notin \Phi _{p}. \end{array} \right. \end{aligned}$$

If $p\rightarrow \infty$, we have $\infty > V(T_{0},~T_{8_{0}},~M_{0},~D_{0})+G t_{T} = \infty$, which resulting in a contradiction. So, we have $t_{\infty } = \infty$. Hence, the proof of the theorem. $\square$

Theorem 3

The solution of the stochastic system (4) is stochastically ultimate bounded for initial densities $(T(0),~T_{8}(0),~M(0),~D(0)) \in \mathbb {R}^{4}_{+}$.

Proof

In the previous theorem, we prove that there is a unique solution in $\mathbb {R}^{4}_{+}$ for all $t \ge 0$ almost surely. Now, we define a $\mathbb {C}^{2}$-function $V: \mathbb {R}^{4}_{+} \rightarrow \mathbb {R}_{+}$ such that

$$\begin{aligned} V(T,~T_{8},~M,~D)= & {} e^{t}(T^{\theta }+T_{8}^{\theta }+M^{\theta }+D^{\theta }), \end{aligned}$$

(11)

where $(T,~T_{8},~M,~D) \in \mathbb {R}^{4}_{+}$ and $\theta \in (0, 1)$. Differentiating equation (11) by It$\hat{o}$’s formula, we get

$$\begin{aligned}{} & {} dV(T,~T_{8},~M,~D) = e^{t}\bigg [(T^{\theta }+T_{8}^{\theta }+M^{\theta }+D^{\theta })\\{} & {} \quad +\theta T^{\theta -1}\bigg (r_{T}T (1-b_{T}T) - \frac{(\alpha _{T}M+\gamma _{T}T_{8})T}{g_{T}+M}\bigg ) \\{} & {} \quad + \theta T_{8}^{\theta -1}\bigg (\frac{\alpha _{8}D}{g_{8}+T_{8}} - \delta _{8}T_{8}\bigg )\\{} & {} \quad + \theta M^{\theta -1}\\{} & {} \quad \bigg (s_{m} + \frac{\alpha _{m}T_{8}}{(g_{m}+T_{8})}\cdot \frac{1}{(g_{m1}+T)} - \gamma _{m}MT- \delta _{m}M \bigg ) \\{} & {} \quad + \theta D^{\theta -1}\bigg (s_{d} + \frac{\alpha _{d}T}{g_{d}+T} - \delta _{d}D\bigg ) \\{} & {} \quad + \frac{\theta (\theta -1)}{2}\bigg \{\rho ^{2}_{1}T^{\theta }\bigg (1-\frac{T^{*}}{T} \bigg )^{2}+\rho ^{2}_{2}T_{8}^{\theta }\bigg (1-\frac{T_{8}^{*}}{T_{8}} \bigg )^{2} \\{} & {} \quad + \rho ^{2}_{3}M^{\theta }\bigg (1-\frac{M^{*}}{M} \bigg )^{2} + \rho ^{2}_{4}D^{\theta } \bigg (1-\frac{D^{*}}{D} \bigg )^{2} \bigg \}\bigg ]dt \\{} & {} \quad + e^{t}\theta \bigg [\rho _{1}T^{\theta }\bigg (1-\frac{T^{*}}{T} \bigg )dB_{1} \\{} & {} \quad + \rho _{2}T_{8}^{\theta }\bigg (1-\frac{T_{8}^{*}}{T_{8}} \bigg )dB_{2} \\{} & {} \quad + \rho _{3}M^{\theta }\bigg (1-\frac{M^{*}}{M} \bigg )dB_{3} \\{} & {} \quad + \rho _{4}D^{\theta }\bigg (1-\frac{D^{*}}{D} \bigg )dB_{4} \bigg ] \\{} & {} \le e^{t}\bigg [G + \frac{\theta (\theta -1)}{2}\bigg \{\rho ^{2}_{1}T^{\theta }\bigg (1-\frac{T^{*}}{T} \bigg )^{2}\\{} & {} \quad +\rho ^{2}_{2}T_{8}^{\theta }\bigg (1-\frac{T_{8}^{*}}{T_{8}} \bigg )^{2}\\{} & {} \quad +\rho ^{2}_{3}M^{\theta }\bigg (1-\frac{M^{*}}{M} \bigg )^{2} \\{} & {} \quad + \rho ^{2}_{4}D^{\theta }\bigg (1-\frac{D^{*}}{D} \bigg )^{2} \bigg \}\bigg ]dt \\{} & {} \quad + e^{t}\theta \bigg [\rho _{1}T^{\theta }\bigg (1-\frac{T^{*}}{T} \bigg )dB_{1} + \rho _{2}T_{8}^{\theta }\bigg (1-\frac{T_{8}^{*}}{T_{8}} \bigg )dB_{2} \\{} & {} \quad + \rho _{3}M^{\theta }\bigg (1-\frac{M^{*}}{M} \bigg )dB_{3} + \rho _{4}D^{\theta }\bigg (1-\frac{D^{*}}{D} \bigg )dB_{4} \bigg ], \\{} & {} \quad (\text {taking superior of the coefficient of the inequality})\\{} & {} \le G e^{t} dt + e^{t}\theta \bigg [\rho _{1}T^{\theta }\bigg (1-\frac{T^{*}}{T} \bigg )dB_{1} \\{} & {} \quad + \rho _{2}T_{8}^{\theta }\bigg (1-\frac{T_{8}^{*}}{T_{8}} \bigg )dB_{2} \\{} & {} \quad + \rho _{3}M^{\theta }\bigg (1-\frac{M^{*}}{M} \bigg )dB_{3} \\{} & {} \quad + \rho _{4}D^{\theta }\bigg (1-\frac{D^{*}}{D} \bigg )dB_{4} \bigg ], ~~~~~~ {[\text {since,}\,(\theta -1)<0]}. \end{aligned}$$

Integrating both sides of the above inequality from 0 to $t_{p}\wedge t$ and taking expectation leads to the following inequality

$$\begin{aligned}{} & {} \mathbb {E}V(T(t_{p}\wedge t),~T_{8}(t_{p}\wedge t),~M(t_{p}\wedge t),~D(t_{p}\wedge t)) \\{} & {} \quad \le V(T(0),~T_{8}(0),~M(0),~D(0))\\{} & {} \qquad + G \mathbb {E}\int _{0}^{t_{p} \wedge t} e^{p}dp. \end{aligned}$$

If we take $p \rightarrow +\infty$, then we have

$$\begin{aligned}{} & {} \mathbb {E}V(T(t),~T_{8}(t),~M(t),~D(t)) \\{} & {} \quad \le V(T(0),~T_{8}(0),~M(0),~D(0)) + G (e^{t}-1), \end{aligned}$$

which implies that

$$\begin{aligned}{} & {} e^{-t}\mathbb {E}V(T(t),~T_{8}(t),~M(t),~D(t)) \\{} & {} \quad \le e^{-t}V(T(0),~T_{8}(0),~M(0),~D(0)) + G. \end{aligned}$$

Now, we can express that

$$\begin{aligned} |X(t)|^{\theta }= & {} (T^{2}(t)+T_{8}^{2}(t)+M^{2}(t)+D^{2}(t))^{\frac{\theta }{2}} \\\le & {} {4^{\frac{\theta }{2}} \max \{T^{\theta }(t),~T_{8}^{\theta }(t),~M^{\theta }(t),~D^{\theta }(t) \}} \\\le & {} 2^{\theta } \{T^{\theta }(t)+T_{8}^{\theta }(t)+M^{\theta }(t)+D^{\theta }(t) \}. \end{aligned}$$

Now, the equation (11) leads to the following form

$$\begin{aligned}{} & {} T^{\theta }+T_{8}^{\theta }+M^{\theta }+D^{\theta } = e^{-t}V(T,~T_{8},~M,~D) \\{} & {} \quad {\Rightarrow |X(t)|^{\theta }} \le {2^{\theta }( e^{-t}V(T,~T_{8},~M,~D))} \\{} & {} \quad {\Rightarrow \mathbb {E}|X(t)|^{\theta }} \\{} & {} \quad \le {\mathbb {E}[2^{\theta }( e^{-t}V(T,~T_{8},~M,~D))]} \\{} & {} \quad \Rightarrow \mathbb {E}|X(t)|^{\theta } \\{} & {} \quad \le 2^{\theta } (e^{-t}V(T(0),~T_{8}(0),~M(0),~D(0))+ G), \end{aligned}$$

which gives

$$\begin{aligned} \lim \limits _{t\rightarrow + \infty } \mathbb {E}|X(t)|^{\theta }\le & {} 2^{\theta } G < +\infty . \end{aligned}$$

We choose $\theta =\frac{1}{2}$, then there exists a constant $\mu _{1}> 0$, such that

$$\begin{aligned} \limsup \limits _{t\rightarrow + \infty } \mathbb {E}|X(t)|^{\frac{1}{2}}\le & {} \mu _{1}. \end{aligned}$$

Using Chebyshev’s inequality and taking $\mu =\frac{\mu _{1}^{2}}{\kappa ^{2}}$, we get

$$\begin{aligned}{} & {} \mathbb {P}\{|X(t)|> \mu \} \le \frac{ \mathbb {E}|X(t)|^{\frac{1}{2}}}{\mu ^{\frac{1}{2}}} \\{} & {} \Rightarrow \limsup \limits _{t\rightarrow + \infty }\mathbb {P}\{|X(t)| > \mu \} \\{} & {} \quad \le \frac{\mu _{1}}{\mu ^{\frac{1}{2}}} = \kappa . \end{aligned}$$

Therefore, the solution of stochastic model (4) is stochastically ultimate bounded with the given initial densities. $\square$

Theorem 4

For any initial densities $(T(0),~T_{8}(0),$$M(0),~D(0)) \in \mathbb {R}^{4}_{+}$, the solution of the stochastic model (4) is stochastically permanence.

Proof

Let us assume that a $\mathbb {C}^{2}$-function $V: \mathbb {R}^{4}_{+} \rightarrow \mathbb {R}_{+}$ such that

$$\begin{aligned} V(T,~T_{8},~M,~D)= & {} \frac{1}{T+T_{8}+M+D}. \end{aligned}$$

(12)

Using by It$\hat{o}$’s formula, we have

$$\begin{aligned} dV= & {} LV dt - V^{2}[\rho _{1}(T-T^{*})dB_{1}+\rho _{2}(T_{8}-T_{8}^{*})dB_{2}\\{} & {} +\rho _{3}(M-M^{*})dB_{3}+\rho _{4}(D-D^{*})dB_{4}], \end{aligned}$$

where

$$\begin{aligned} LV= & {} -V^{2}\bigg [r_{T}T (1-b_{T}T) - \frac{(\alpha _{T}M+\gamma _{T}T_{8})T}{g_{T}+M} \\{} & {} + \frac{\alpha _{8}D}{g_{8}+T_{8}} - \delta _{8}T_{8} + s_{m} \\{} & {} + \frac{\alpha _{m}T_{8}}{(g_{m}+T_{8})}\cdot \frac{1}{(g_{m1}+T)} - \gamma _{m}MT \\{} & {} - \delta _{m}M + s_{d} + \frac{\alpha _{d}T}{g_{d}+T} - \delta _{d}D \bigg ] \\{} & {} + V^{3}[\rho _{1}^{2}(T-T^{*})^{2}+\rho _{2}^{2}(T_{8}-T_{8}^{*})^{2}\\{} & {} +\rho _{3}^{2}(M-M^{*})^{2}+\rho _{4}^{2}(D-D^{*})^{2}]. \end{aligned}$$

We take a positive constant $\psi$, such that $\psi < 1$ and define

$$\begin{aligned} U(T,~T_{8},~M,~D)= & {} (1+V(T,~T_{8},~M,~D))^{\psi }. \end{aligned}$$

Applying It$\hat{o}$’s formula, we get

$$\begin{aligned} LU= & {} \psi (1+V)^{\psi -1}\bigg [ -V^{2}\bigg \{r_{T}T (1-b_{T}T)\\{} & {} - \frac{(\alpha _{T}M+\gamma _{T}T_{8})T}{g_{T}+M} + \frac{\alpha _{8}D}{g_{8}+T_{8}} - \delta _{8}T_{8} \\{} & {} + s_{m} + \frac{\alpha _{m}T_{8}}{(g_{m}+T_{8})}\cdot \frac{1}{(g_{m1}+T)} \\{} & {} - \gamma _{m}MT - \delta _{m}M + s_{d} + \frac{\alpha _{d}T}{g_{d}+T} - \delta _{d}D \bigg \} \\{} & {} + V^{3}[\rho _{1}^{2}(T-T^{*})^{2}+\rho _{2}^{2}(T_{8} -T_{8}^{*})^{2}\\{} & {} +\rho _{3}^{2}(M-M^{*})^{2}+\rho _{4}^{2}(D-D^{*})^{2}]\bigg ]\\{} & {} + \frac{1}{2}\psi (\psi -1)(1+V)^{\psi -2}V^{4}[\rho _{1}^{2}(T-T^{*})^{2}\\{} & {} +\rho _{2}^{2}(T_{8}-T_{8}^{*})^{2} +\rho _{3}^{2}(M-M^{*})^{2} \\{} & {} + \rho _{4}^{2}(D-D^{*})^{2}] \\= & {} \psi (1+V)^{\psi -1}LV\\{} & {} +\frac{1}{2}\psi (\psi -1)(1+V)^{\psi -2}V^{4}[\rho _{1}^{2}(T-T^{*})^{2}\\{} & {} +\rho _{2}^{2}(T_{8}-T_{8}^{*})^{2} +\rho _{3}^{2}(M-M^{*})^{2} \\{} & {} + \rho _{4}^{2}(D-D^{*})^{2}]. \end{aligned}$$

Again, we choose very small positive constant, denoted as n, such that

$$\begin{aligned} W(T,~T_{8},~M,~D)= & {} e^{nt}(1+V(T,~T_{8},~M,~D))^{\psi }. \end{aligned}$$

Using It$\hat{o}$’s formula, we get

$$\begin{aligned}{} & {} LW = ne^{nt}(1+V)^{\psi } + e^{nt}L(1+V)^{\psi } \\{} & {} \quad = e^{nt}(1+V)^{\psi -2} \\{} & {} \qquad \bigg [n(1+V)^{2}-\psi (1+V) V^{2}\bigg \{r_{T}T (1-b_{T}T)\\{} & {} \qquad - \frac{(\alpha _{T}M+\gamma _{T}T_{8})T}{g_{T}+M} + \frac{\alpha _{8}D}{g_{8}+T_{8}} - \delta _{8}T_{8} \\{} & {} \qquad + s_{m} + \frac{\alpha _{m}T_{8}}{(g_{m}+T_{8})}\cdot \frac{1}{(g_{m1}+T)} - \gamma _{m}MT- \delta _{m}M \\{} & {} \qquad + s_{d} + \frac{\alpha _{d}T}{g_{d}+T} - \delta _{d}D \bigg \} + \psi (1+V) V^{3}[\rho _{1}^{2}(T-T^{*})^{2} \\{} & {} \qquad + \rho _{2}^{2}(T_{8}-T_{8}^{*})^{2}+\rho _{3}^{3}(M-M^{*})^{2}+\rho _{4}^{2}(D-D^{*})^{2}]\\{} & {} \qquad + \frac{1}{2}\psi (\psi -1)V^{4}[\rho _{1}^{2}(T-T^{*})^{2} + \rho _{2}^{2}(T_{8}-T_{8}^{*})^{2} \\{} & {} \qquad + \rho _{3}^{2}(M-M^{*})^{2}+\rho _{4}^{2}(D-D^{*})^{2}] \bigg ] \\{} & {} \quad \le e^{nt}(1+V)^{\psi -2} \bigg [n(1+V)^{2} + \psi (1+V) V^{2} \bigg \{ r_{T} b_{T} T^{2} \\{} & {} \qquad + \frac{(\alpha _{T} M + \gamma _{T} T{8})T}{g_{T}+M} +\delta _{8}T_{8}+\gamma _{m}MT \\{} & {} \qquad + \delta _{m}M + \delta _{d}D \bigg \} + \psi (1+V)V^{3} \bigg \{\rho _{1}^{2}(T-T^{*})^{2} \\{} & {} \qquad + \rho _{2}^{2}(T_{8}-T_{8}^{*})^{2}+ \rho _{3}^{2}(M-M^{*})^{2}+\rho _{4}^{2}(D-D^{*})^{2} \bigg \} \\{} & {} \qquad + \frac{1}{2}\psi (\psi -1)V^{4}[\rho _{1}^{2}(T-T^{*})^{2} + \rho _{2}^{2}(T_{8}-T_{8}^{*})^{2}\\{} & {} \qquad + \rho _{3}^{2}(M-M^{*})^{2}+\rho _{4}^{2}(D-D^{*})^{2}] \bigg ] \\{} & {} \quad \le e^{nt}(1+V)^{\psi -2} \bigg [n(1+V)^{2} + \psi (1+V) V^{2} \bigg \{ r_{T} b_{T} T^{2} \\{} & {} \qquad + \frac{(\alpha _{T} M + \gamma _{T} T{8})T}{g_{T}+M} +\delta _{8}T_{8}+\gamma _{m}MT \\{} & {} \qquad + \delta _{m}M + \delta _{d}D \bigg \} + \psi (1+V) V^{3} \bigg \{\rho _{1}^{2}(T-T^{*})^{2} \\{} & {} \qquad + \rho _{2}^{2}(T_{8}-T_{8}^{*})^{2}{+} \rho _{3}^{2}(M-M^{*})^{2}{+}\rho _{4}^{2}(D-D^{*})^{2} \bigg \} \bigg ],\\{} & {} \qquad (\text {since}\, (\psi -1) < 0). \end{aligned}$$

We take G is the superior of the coefficient and we have an expression as

$$\begin{aligned}{} & {} \psi (1+V) V^{3}[\rho _{1}^{2}(T-T^{*})^{2}+\rho _{2}^{2}(T_{8}-T_{8}^{*})^{2}\\{} & {} \quad +\rho _{3}^{2}(M-M^{*})^{2}+\rho _{4}^{2}(D-D^{*})^{2}] \\{} & {} \quad \le \psi (1+V) V (2 \max \limits _{i=1,2,3,4}\{\rho _{i}^{2}\}). \end{aligned}$$

Therefore,

$$\begin{aligned} LW \le me^{nt}, \end{aligned}$$

where

$$\begin{aligned} m= & {} (1+V)^{\psi -2}[n+V\{2n+\psi (2 \max \limits _{i=1,2,3,4}\{\rho _{i}^{2}\}) \} \\{} & {} + V^{2} (n+ G \psi + \psi (2 \max \limits _{i=1,2,3,4}\{\rho _{i}^{2}\})) + V^{3} G \psi ]. \end{aligned}$$

Thus, we have

$$\begin{aligned}{} & {} L[e^{nt}(1+V)^{\psi }] \le m e^{nt} \\{} & {} \quad \Rightarrow \mathbb {E} [e^{nt}(1+V)^{\psi }] \\{} & {} \quad \le (1+V(0))^{\psi } + \frac{m}{n}(e^{nt}-1). \end{aligned}$$

Therefore, we get

$$\begin{aligned}{} & {} \limsup \limits _{t\rightarrow \infty }\mathbb {E}(V(t)^{\psi }) \\{} & {} \quad \le \limsup \limits _{t\rightarrow \infty }\mathbb {E}((1+V(t))^{\psi }) \\{} & {} \quad \le \frac{m}{n}. \end{aligned}$$

Since

$$\begin{aligned} (T+T_{8}+M+D)^{\psi }\le & {} 4^{\psi }(T^{2}+T^{2}_{8}+M^{2}+D^{2})^{\frac{\psi }{2}} \\= & {} 4^{\psi }|X(t)|^{\psi }, \end{aligned}$$

then

$$\begin{aligned}{} & {} \limsup \limits _{t\rightarrow \infty }\mathbb {E}|X(t)|^{-\psi }\\{} & {} \quad \le 4^{\psi }\limsup \limits _{t\rightarrow \infty }\mathbb {E}(V(t)^{\psi }) \\{} & {} \quad \le \frac{4^{\psi }m}{n}= K (\text {constant}). \end{aligned}$$

By previous theorem, we have

$$\begin{aligned}{} & {} \limsup \limits _{t\rightarrow + \infty }\mathbb {P}\{|X(t)| > \mu \} \le \kappa \\{} & {} \Rightarrow \limsup \limits _{t\rightarrow + \infty }\mathbb {P}\{|X(t)| \le \mu \} \ge 1-\kappa . \end{aligned}$$

Also, we have

$$\begin{aligned} \limsup \limits _{t\rightarrow + \infty }\mathbb {E}|X(t)|^{-\psi } \le K. \end{aligned}$$

For any $\kappa > 0$ and $\varphi =\big (\frac{\kappa }{K}\big )^{\frac{1}{\theta }}$, we get

$$\begin{aligned} \mathbb {P}\{|X(t)| < \mu \}= & {} \mathbb {P} \bigg \{\frac{1}{|X(t)|} > \frac{1}{\varphi } \bigg \} \\\le & {} \varphi ^{\psi }\mathbb {E}|X(t)|^{-\psi } = \kappa , \end{aligned}$$

which means that

$$\begin{aligned} \liminf \limits _{t\rightarrow + \infty }\mathbb {P}\{|X(t)| \ge \varphi \}\ge & {} 1-\kappa . \end{aligned}$$

Therefore, solution of the stochastic model (4) is stochastically permanence. $\square$

Theorem 5

Let $(T(t),~T_{8}(t),~M(t),~D(t))$ be the solution of the stochastic system (4) initiating for any value $(T(0),~T_{8}(0),~M(0),~D(0)) \in \mathbb {R}^{4}_{+}$. Then, the tumor cells extinct exponentially with probability one, that is, $\lim \limits _{t\rightarrow \infty } T(t) = 0$ almost surely if $r_{T} \le \rho ^{2}_{1} (\frac{1}{2}-T^{*})$.

Proof

Let us assume that $u=\ln T$. Applying It$\hat{o}$’s formula, we get

$$\begin{aligned} du= & {} \bigg [r_{T}(1-b_{T}T)-\frac{(\alpha _{T}M+\gamma _{T}T_{8})}{g_{T}+M}\\{} & {} -\frac{1}{2}\rho _{1}^{2}\bigg (1-\frac{T^{*}}{T} \bigg )^{2} \bigg ] dt \\{} & {} + \rho _{1} \bigg (1-\frac{T^{*}}{T} \bigg )dB_{1}. \end{aligned}$$

After some algebraic calculations, above equation leads to

$$\begin{aligned} d(\ln T)\le & {} [r_{T}(1-b_{T}T) -\frac{1}{2}\rho ^{2}_{1}+\rho ^{2}_{1}T^{*}]dt \nonumber \\{} & {} +\rho _{1}\bigg (1-\frac{T^{*}}{T}\bigg ) d B_{1}. \end{aligned}$$

(13)

Now, we take $P(T)=r_{T}(1-b_{T}T) -\frac{1}{2}\rho ^{2}_{1}+\rho ^{2}_{1}T^{*}$. Here, we intend to get the supremum value of P(T), we get $P^{'}(T)=-r_{T}b_{T} < 0$. Therefore, P(T) is a decreasing function of T(t) on the interval $[0,~\infty )$ and hence, supremum value of P(T) is

$$\begin{aligned} P(T)_{\sup t \ge 0}= & {} P(0) = r_{T}-\frac{1}{2}\rho ^{2}_{1}+\rho ^{2}_{1}T^{*}. \end{aligned}$$

From equation (13), we get

$$\begin{aligned} d(\ln T)\le & {} [r_{T} -\frac{1}{2}\rho ^{2}_{1}+\rho ^{2}_{1}T^{*}]dt\\{} & {} + \rho _{1}\bigg (1-\frac{T^{*}}{T} \bigg ) dB_{1}\\ \Rightarrow \ln T\le & {} \ln T_{0} + [r_{T} -\frac{1}{2}\rho ^{2}_{1}+\rho ^{2}_{1}T^{*}] t \\{} & {} + \rho _{1}\bigg (1-\frac{T^{*}}{T} \bigg ) B_{1}. \end{aligned}$$

From strong law of large numbers for local martingles, it follows that

$$\begin{aligned}{} & {} \lim \limits _{t\rightarrow \infty } \sup \frac{\ln T(t)}{t} \\{} & {} \le \lim \limits _{t\rightarrow \infty } \sup \bigg [\frac{\ln T_{0}}{t} +(r_{T} -\frac{1}{2}\rho ^{2}_{1}+\rho ^{2}_{1}T^{*})\\{} & {} \quad +\frac{\rho _{1}}{t}\bigg (1-\frac{T^{*}}{T} \bigg ) B_{1} \bigg ] \\{} & {} = r_{T} -\frac{1}{2}\rho ^{2}_{1}+\rho ^{2}_{1}T^{*}. \end{aligned}$$

Now, using the condition $r_{T} \le \rho _{1}^{2}\big (\frac{1}{2}-T^{*}\big )$, we get

$$\begin{aligned} \lim \limits _{t\rightarrow \infty } \sup \frac{\ln T(t)}{t}\le & {} 0 \\ \Rightarrow \lim \limits _{t\rightarrow \infty } T(t)= & {} 0,~~~~ \text {almost surely.} \end{aligned}$$

Therefore, tumor cells become extinct. $\square$

5 Stochastic stability of the singular point

By introducing new variables $v_{1}=T-T^{*},~v_{2}=T_{8}-T_{8}^{*},~v_{3}=M-M^{*},~v_{4}=D-D^{*}$, the stochastic differential system (4) can be centered around its interior singular point $E^{*}=(T^{*},~T_{8}^{*},~M^{*},~D^{*})$. As the system (4) consists of nonlinear equations, then it is very difficult to derive asymptotically stability in mean square sense (with probability 1) by constructing Lyapunov function method. We find out a set of stochastic differential equation by linearizing the drift coefficient h around the singular point $E^{*}=(T^{*},~T_{8}^{*},~M^{*},~D^{*})$. Then, the linearized system leads to

$$\begin{aligned} dv(t)= & {} h(v(t))dt+n(v(t))dB, \end{aligned}$$

(14)

where

$$\begin{aligned} v(t)= & {} col(v_{1}(t),~v_{2}(t),~v_{3}(t),~v_{4}(t)), \end{aligned}$$

and

$$\begin{aligned} h(v(t))= & {} \left[ \begin{array}{cccc} h_{11}v_{1} &{} h_{12}v_{2} &{} h_{13}v_{3} &{} h_{14}v_{4} \\ h_{21}v_{1} &{} h_{22}v_{2} &{} h_{23}v_{3} &{} h_{24}v_{4} \\ h_{31}v_{1} &{} h_{32}v_{2} &{} h_{33}v_{3} &{} h_{34}v_{4} \\ h_{41}v_{1} &{} h_{42}v_{2} &{} h_{43}v_{3} &{} h_{44}v_{4} \end{array} \right] ,\\ n(v(t))= & {} \left[ \begin{array}{cccc} \rho _{1}v_{1} &{} 0 &{} 0 &{} 0 \\ 0 &{} \rho _{2}v_{2} &{} 0 &{} 0 \\ 0 &{} 0 &{} \rho _{3}v_{3} &{} 0 \\ 0 &{} 0 &{} 0 &{} \rho _{4}v_{4} \end{array} \right] , \end{aligned}$$

with

$$\begin{aligned} h_{11}= & {} r_{T}-2r_{T}b_{T}T^{*}-\frac{(\alpha _{T}M^{*}+\gamma _{T}T_{8}^{*})}{g_{T}+M^{*}}, \\ h_{12}= & {} -\frac{\gamma _{T}T^{*}}{g_{T}+M^{*}}, \\ h_{13}= & {} -\frac{(g_{T}\alpha _{T}-\gamma _{T}T_{8}^{*})T^{*}}{(g_{T}+M^{*})^{2}}, \\ h_{14}= & {} 0, \\ h_{21}= & {} 0, \\ h_{22}= & {} -\frac{\alpha _{8}D^{*}}{(g_{8}+T_{8}^{*})^{2}}-\delta _{8}, \\ h_{23}= & {} 0, \\ h_{24}= & {} \frac{\alpha _{8}}{g_{8}+T_{8}^{*}}, \\ h_{31}= & {} -\frac{\alpha _{m}T_{8}^{*}}{(g_{m}+T_{8}^{*})(g_{m1}+T^{*})^{2}}-\gamma _{m}M^{*}, \\ h_{32}= & {} \frac{\alpha _{m}g_{m}}{(g_{m}+T_{8}^{*})^{2}(g_{m1}+T^{*})}, \\ h_{33}= & {} -\gamma _{m}T^{*}-\delta _{m}, \\ h_{34}= & {} 0, \\ h_{41}= & {} \frac{g_{d}\alpha _{d}}{(g_{d}+T^{*})^{2}}, \\ h_{42}= & {} 0, \\ h_{43}= & {} 0, \\ h_{44}= & {} -\delta _{d}. \end{aligned}$$

It is to be observed that, in equation (14) the interior singular point $E^{*}=(T^{*},~T_{8}^{*},~M^{*},~D^{*})$ corresponds to the trivial solution $(v_{1},~v_{2},~v_{3},~v_{4})=(0,~0,~0,~0)$. Let us consider that the set $\sum = \{\mathbb {R}^{4} \times (t\ge t_{0}),~t_{0} \in \mathbb {R}^{+}\}$. If $W \in \mathbb {C}^{2}(\sum )$ is a continuous function with respect to t and a twice continuously differentiable function with respect to v, then we can assert the following lemma as presented by Gard T.C. [40].

Lemma 1

Consider a function $W(v,t) \in \mathbb {C}^{2}(\sum )$ satisfying the inequalities

$$\begin{aligned} c_{1}|v|^{\pi } \le W(v,~t) \le c_{2}|v|^{\pi }, \end{aligned}$$

(15)

and

$$\begin{aligned} LW(v,~t) \le -c_{3}|v|^{\pi },~c_{i}> 0,~\pi > 0~(i=1,~2,~3).\nonumber \\ \end{aligned}$$

(16)

Then, the trivial solution of (14) is exponentially $\pi$-stable, for all $t \ge 0$.

The trivial solution of (14) is said to be exponentially mean square stability if $\pi =2$. Furthermore, this trivial solution exhibits globally asymptotically stable in the sense of probability. With reference to (14), LW(v,  t) is given by the following expression:

$$\begin{aligned}{} & {} LW(v,~t) = \frac{\partial W(v,~t)}{\partial t} + h^{T}(v(t))\frac{\partial W(v,~t)}{\partial v}\nonumber \\{} & {} +\frac{1}{2}Tr\bigg [n^{T}(v(t))\frac{\partial ^{2} W(v,~t)}{\partial v^{2}}n(v(t)) \bigg ], \end{aligned}$$

(17)

where

$$\begin{aligned}{} & {} \frac{\partial W}{\partial v} = col \bigg (\frac{\partial W}{\partial v_{1}},~\frac{\partial W}{\partial v_{2}},~\frac{\partial W}{\partial v_{3}},\frac{\partial W}{\partial v_{4}} \bigg ),\\{} & {} \quad \frac{\partial ^{2} W(v,~t)}{\partial v^{2}} = \bigg (\frac{\partial ^{2}W}{\partial v_{i}\partial v_{j}} \bigg )_{i,j=1,2,3,4}. \end{aligned}$$

Now, we shall investigate our main result in the following theorem.

Theorem 6

If the following conditions

$$\begin{aligned} \begin{array}{llll} (i)~\rho _{1}^{2}&{}<&{} 2 \\ &{}&{} \bigg (-r_{T}+2r_{T}b_{T}T^{*}+ \frac{(\alpha _{T}M^{*}+\gamma _{T}T_{8}^{*})}{g_{T}+M^{*}}-\frac{1}{4(\eta _{1}^{*}+\eta _{2}^{*})} \bigg ), \\ (ii)~\rho _{2}^{2} &{}<&{} 2\bigg (\frac{\gamma _{T}T^{*}}{g_{T}+M^{*}}+ \frac{\alpha _{8}D^{*}}{(g_{8}+T_{8}^{*})^{2}}+\delta _{8}-\frac{1}{4\eta _{2}^{*}} \bigg ), \\ (iii)~\rho _{3}^{2} &{}<&{} 2(\gamma _{m}T^{*}+\delta _{m}), \\ (iv)~\rho _{4}^{2} &{}<&{} 2(\delta _{d}-A^{2}-B^{2}), ~~ \text {with}\\ A &{}=&{} \eta _{2}^{*}\frac{\alpha _{8}}{g_{8}+T_{8}^{*}}+ \frac{g_{d}\alpha _{d}}{(g_{d}+T^{*})^{2}}, ~B = \eta _{2}^{*}\frac{\alpha _{8}}{g_{8}+T^{*}_{8}}, \end{array} \end{aligned}$$

hold, then the zero solution of stochastic differential system (4) is asymptotically mean square stable.

Proof

To prove the theorem, we consider the positive definite Lyapunov function

$$\begin{aligned} W(v(t),t)= & {} \frac{1}{2}[\eta _{1}v^{2}_{1} +\eta _{2}(v_{1}+v_{2})^{2}+\eta _{3}v^{2}_{3} +v^{2}_{4}],\nonumber \\ \end{aligned}$$

(18)

where $\eta _{i}~(i=1,2,3)$ are real positive constants to be selected later. It is easy to prove that the equations (15) and (16) hold for $\pi =2$. Using equation (17) to the stochastic system (4), we have

$$\begin{aligned} \begin{array}{llll} &{}&{} LW(v,t) = (h_{11}v_{1}+h_{12}v_{2}+h_{13}v_{3}+h_{14}v_{4})\eta _{1}v_{1} \\ &{}&{}\quad + \big ((h_{11}+h_{21})v_{1}+(h_{12}+h_{22})v_{2}\\ &{}&{}\quad +(h_{13}+h_{23})v_{3}+(h_{14}+h_{24})v_{4}\big )\eta _{2}(v_{1}+v_{2}) \\ &{}&{}\quad +(h_{31}v_{1}+h_{32}v_{2}+h_{33}v_{3}+h_{34}v_{4})\eta _{3}v_{3} \\ &{}&{}\quad + (h_{41}v_{1}+h_{42}v_{2}+h_{43}v_{3}+h_{44}v_{4})v_{4} \\ &{}&{}\quad + \frac{1}{2}Tr\bigg [n^{T}(v(t))\frac{\partial ^{2} W(v,~t)}{\partial v^{2}}n(v(t)) \bigg ]. \end{array} \end{aligned}$$

After some algebraic calculations, we have

$$\begin{aligned} \begin{array}{llll} &{}&{} \!\!\!\!\! LW(v,t) = \eta _{1}v_{1}\bigg [\bigg (r_{T}-2r_{T}b_{T}T^{*}-\frac{(\alpha _{T}M^{*} +\gamma _{T}T_{8}^{*})}{g_{T}+M^{*}} \bigg )v_{1}\\ &{}&{} \quad -\frac{\gamma _{T}T^{*}}{g_{T}+M^{*}} v_{2}-\frac{(g_{T}\alpha _{T} -\gamma _{T}T_{8}^{*})T^{*}}{(g_{T}+M^{*})^{2}} v_{3}\bigg ] \\ &{}&{} \quad + \eta _{2}(v_{1}+v_{2}) \bigg [\bigg (r_{T}-2r_{T}b_{T}T^{*}-\frac{(\alpha _{T}M^{*} +\gamma _{T}T_{8}^{*})}{g_{T}+M^{*}}\bigg )v_{1}\\ &{}&{} \quad -\bigg (\frac{\gamma _{T}T^{*}}{g_{T}+M^{*}}+ \frac{\alpha _{8}D^{*}}{(g_{8}+T_{8}^{*})^{2}}+\delta _{8}\bigg )v_{2} \\ &{}&{} \quad -\frac{(g_{T}\alpha _{T}-\gamma _{T}T_{8}^{*})T^{*}}{(g_{T}+M^{*})^{2}}v_{3} + \frac{\alpha _{8}}{g_{8}+T_{8}^{*}} v_{4} \bigg ] \\ &{}&{} \quad + \eta _{3}v_{3}\bigg [-\bigg (\frac{\alpha _{m}T_{8}^{*}}{(g_{m}+T_{8}^{*})(g_{m1}+T^{*})^{2}}+\gamma _{m}M^{*} \bigg )v_{1} \\ &{}&{} \quad + \frac{\alpha _{m}g_{m}}{(g_{m}+T_{8}^{*})^{2}(g_{m1}+T^{*})} v_{2} - (\gamma _{m}T^{*}+\delta _{m})v_{3}\bigg ] \\ &{}&{} \quad + v_{4} \bigg [\frac{g_{d}\alpha _{d}}{(g_{d}+T^{*})^{2}}v_{1}-\delta _{d}v_{4} \bigg ] \\ &{}&{} \quad + \frac{1}{2}Tr\bigg [n^{T}(v(t))\frac{\partial ^{2} W(v,~t)}{\partial v^{2}}n(v(t)) \bigg ]. \end{array} \end{aligned}$$

(19)

Also, we can write

$$\begin{aligned} \frac{\partial ^{2}W}{\partial v^{2}}= \left[ \begin{array}{cccc} \eta _{1}+\eta _{2} &{} \eta _{2} &{} 0 &{} 0 \\ \eta _{2} &{} \eta _{2} &{} 0 &{} 0 \\ 0 &{} 0 &{} \eta _{3} &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \end{array} \right] , \end{aligned}$$

which implies

$$\begin{aligned}{} & {} n^{T}(v(t))\frac{\partial ^{2} W(v,~t)}{\partial v^{2}}n(v(t))\\{} & {} \quad = \left[ \begin{array}{cccc} (\eta _{1}+\eta _{2})\rho _{1}^{2}v_{1}^{2} &{} \eta _{2}\rho _{1}\rho _{2}v_{1}v_{2} &{} 0 &{} 0 \\ \eta _{2}\rho _{1}\rho _{2}v_{1}v_{2} &{} \eta _{2}\rho _{2}^{2}v_{2}^{2} &{} 0 &{} 0 \\ 0 &{} 0 &{} \eta _{3}\rho _{3}^{2}v_{3}^{2} &{} 0 \\ 0 &{} 0 &{} 0 &{} \rho _{4}^{2}v_{4}^{2} \end{array} \right] . \end{aligned}$$

Then, we have

$$\begin{aligned}{} & {} \frac{1}{2}Tr\bigg [n^{T}(v(t))\frac{\partial ^{2} W(v,~t)}{\partial v^{2}}n(v(t)) \bigg ]\nonumber \\{} & {} = \frac{1}{2}[(\eta _{1}+\eta _{2})\rho _{1}^{2}v_{1}^{2}+\eta _{2}\rho _{2}^{2}v_{2}^{2}\nonumber \\{} & {} \quad +\eta _{3}\rho _{3}^{2}v_{3}^{2}+\rho _{4}^{2}v_{4}^{2}]. \end{aligned}$$

(20)

Therefore, the expression (19) leads to

$$\begin{aligned} \begin{aligned}&LW(v,t) = -(\eta _{1}+\eta _{2}) \\&\bigg [-r_{T}+2r_{T}b_{T}T^{*}+\frac{(\alpha _{T}M^{*}+\gamma _{T}T_{8}^{*})}{g_{T}+M^{*}}-\frac{\rho ^{2}_{1}}{2} \bigg ]v^{2}_{1} \\&\quad + \bigg [\bigg (r_{T}-2r_{T}b_{T}T^{*}-\frac{\alpha _{T}M^{*}+\gamma _{T}T_{8}^{*}}{g_{T}+M^{*}}\\&\quad -\bigg (\frac{\gamma _{T}T^{*}}{g_{T}+M^{*}}+\frac{\alpha _{8}D^{*}}{(g_{8}+T_{8}^{*})^{2}} +\delta _{8} \bigg )\bigg )\eta _{2}\bigg . \\&\quad \bigg .-\frac{\gamma _{T}T^{*}}{g_{T}+M^{*}}\eta _{1}\bigg ]v_{1}v_{2} \\&\quad + \bigg [\frac{(\gamma _{T}T_{8}^{*}-g_{T}\alpha _{T})T^{*}}{(g_{T}+M^{*})^{2}} (\eta _{1}+\eta _{2}) \\&\quad \bigg .- \bigg (\frac{\alpha _{m}T_{8}^{*}}{(g_{m}+T_{8}^{*})(g_{m1}+T^{*})^{2}} +\gamma _{m}M^{*} \bigg )\eta _{3} \bigg ]v_{1}v_{3} \\&\quad + \bigg [\eta _{2}\frac{\alpha _{8}}{g_{8}+T_{8}^{*}} + \frac{g_{d}\alpha _{d}}{(g_{d}+T^{*})^{2}} \bigg ]v_{1}v_{4} \\&\quad - \eta _{2}\bigg [\frac{\gamma _{T}T^{*}}{g_{T}+M^{*}}+\frac{\alpha _{8}D^{*}}{(g_{8}+T_{8}^{*})^{2}} +\delta _{8}-\frac{\rho _{2}^{2}}{2}\bigg ]v_{2}^{2} \\&\quad + \bigg [\eta _{3}\frac{\alpha _{m}g_{m}}{(g_{m}+T_{8}^{*})^{2}(g_{m1}+T^{*})} -\eta _{2} \bigg .\\&\quad \bigg .\frac{(g_{T}\alpha _{T}-\gamma _{T}T_{8}^{*})T^{*}}{(g_{T}+M^{*})^{2}} \bigg ]v_{2}v_{3} \\&\quad + \Big [\eta _{2}\frac{\alpha _{8}}{g_{8}+T^{*}_{8}} \Big ]v_{2}v_{4} \\&\quad - \eta _{3}\Big [\gamma _{m}T^{*}+\delta _{m}-\frac{\rho ^{2}_{3}}{2} \Big ]v^{2}_{3}-\Big [\delta _{d}-\frac{\rho _{4}^{2}}{2} \Big ]v_{4}^{2}. \end{aligned} \end{aligned}$$

(21)

If we choose $\eta _{1}^{*},~\eta _{2}^{*}$ and $\eta _{3}^{*}$ in such way that

$$\begin{aligned}{} & {} \bigg (r_{T}-2r_{T}b_{T}T^{*}-\frac{\alpha _{T}M^{*}+\gamma _{T}T_{8}^{*}}{g_{T}+M^{*}}\\{} & {} \quad -\bigg (\frac{\gamma _{T}T^{*}}{g_{T}+M^{*}}+\frac{\alpha _{8}D^{*}}{(g_{8}+T_{8}^{*})^{2}} +\delta _{8} \bigg )\bigg )\eta _{2}^{*}\\{} & {} -\frac{\gamma _{T}T^{*}}{g_{T}+M^{*}}\eta _{1}^{*} = 0,\\{} & {} \quad \frac{(\gamma _{T}T_{8}^{*}-g_{T}\alpha _{T})T^{*}}{(g_{T}+M^{*})^{2}}(\eta _{1}^{*}+\eta _{2}^{*})\\{} & {} \quad -\bigg (\frac{\alpha _{m}T_{8}^{*}}{(g_{m}+T_{8}^{*})(g_{m1}+T^{*})^{2}}+\gamma _{m}M^{*} \bigg )\eta _{3}^{*} = 0, \end{aligned}$$

and

$$\begin{aligned}{} & {} \eta _{3}^{*}\frac{\alpha _{m}g_{m}}{(g_{m}+T_{8}^{*})^{2}(g_{m1}+T^{*})} -\eta _{2}^{*}\frac{(g_{T}\alpha _{T}-\gamma _{T}T_{8}^{*})T^{*}}{(g_{T}+M^{*})^{2}} \\{} & {} \quad = 0. \end{aligned}$$

Then, the expression (21) becomes

$$\begin{aligned} \begin{aligned}&LW(v,t) = -(\eta _{1}^{*}+\eta _{2}^{*}) \\&\bigg [-r_{T}+2r_{T}b_{T}T^{*}+\frac{(\alpha _{T}M^{*}+\gamma _{T}T_{8}^{*})}{g_{T}+M^{*}}-\frac{\rho ^{2}_{1}}{2} \bigg ]v^{2}_{1} \\&\quad + Av_{1}v_{4}-\eta _{2}^{*}\bigg [\frac{\gamma _{T}T^{*}}{g_{T}+M^{*}} +\frac{\alpha _{8}D^{*}}{(g_{8}+T_{8}^{*})^{2}}\\&\quad +\delta _{8}-\frac{\rho _{2}^{2}}{2}\bigg ]v_{2}^{2} +Bv_{2}v_{4} \\&\quad - \eta _{3}^{*}\Big [\gamma _{m}T^{*}+\delta _{m}-\frac{\rho ^{2}_{3}}{2} \Big ]v^{2}_{3} -\Big [\delta _{d}-\frac{\rho _{4}^{2}}{2} \Big ]v_{4}^{2}, \end{aligned} \end{aligned}$$

(22)

where

$$\begin{aligned} \begin{array}{llll} A &{}=&{} \eta _{2}^{*}\frac{\alpha _{8}}{g_{8}+T_{8}^{*}}+ \frac{g_{d}\alpha _{d}}{(g_{d}+T^{*})^{2}}, \\ B &{}=&{} \eta _{2}^{*}\frac{\alpha _{8}}{g_{8}+T^{*}_{8}}. \end{array} \end{aligned}$$

Using standard inequality, we have

$$\begin{aligned} \begin{array}{llll} Av_{1}v_{4} &{}\le &{} \frac{1}{4}v_{1}^{2}+A^{2}v^{2}_{4}, \\ Bv_{2}v_{4} &{}\le &{} \frac{1}{4}v_{2}^{2}+B^{2}v^{2}_{4}. \end{array} \end{aligned}$$

Thus, the expression (22) leads to

$$\begin{aligned}&LW(v,t) \le -\bigg [(\eta _{1}^{*}+\eta _{2}^{*})\bigg (-r_{T}+2r_{T}b_{T}T^{*}\nonumber \\&\quad +\frac{(\alpha _{T}M^{*}+\gamma _{T}T_{8}^{*})}{g_{T}+M^{*}}-\frac{\rho ^{2}_{1}}{2} \bigg )-\frac{1}{4} \bigg ]v^{2}_{1} \nonumber \\&\quad - \bigg [\eta _{2}^{*}\bigg (\frac{\gamma _{T}T^{*}}{g_{T}+M^{*}}+\frac{\alpha _{8}D^{*}}{(g_{8}+T_{8}^{*})^{2}} +\delta _{8}-\frac{\rho _{2}^{2}}{2}\bigg )-\frac{1}{4}\bigg ]v_{2}^{2} \nonumber \\&\quad - \eta _{3}^{*}\Big [\gamma _{m}T^{*}+\delta _{m}-\frac{\rho ^{2}_{3}}{2} \Big ]v^{2}_{3}\nonumber \\&\quad -\Big [\delta _{d}-\frac{\rho _{4}^{2}}{2}-A^{2}-B^{2} \Big ]v_{4}^{2}. \end{aligned}$$

(23)

Then, it can be expressed as

$$\begin{aligned} \begin{array}{llll} LW(v,t)\le & {} -v^{T}Qv, \end{array} \end{aligned}$$

(24)

where

$$\begin{aligned} Q= \left[ \begin{array}{cccc} q_{11} &{} q_{12} &{} q_{13} &{} q_{14} \\ q_{21} &{} q_{22} &{} q_{23} &{} q_{24} \\ q_{31} &{} q_{32} &{} q_{33} &{} q_{34} \\ q_{41} &{} q_{42} &{} q_{43} &{} q_{44} \end{array} \right] , \end{aligned}$$

with

$$\begin{aligned} q_{11}= & {} (\eta _{1}^{*}+\eta _{2}^{*})\bigg (-r_{T}+2r_{T}b_{T}T^{*}\\{} & {} +\frac{(\alpha _{T}M^{*}+\gamma _{T}T_{8}^{*})}{g_{T}+M^{*}}-\frac{\rho ^{2}_{1}}{2} \bigg )-\frac{1}{4}, \\ q_{12}= & {} q_{21}=0, \\ q_{13}= & {} q_{31}=0, \\ q_{14}= & {} q_{41}=0, \\ q_{22}= & {} \eta _{2}^{*}\bigg (\frac{\gamma _{T}T^{*}}{g_{T}+M^{*}} +\frac{\alpha _{8}D^{*}}{(g_{8}+T_{8}^{*})^{2}}+\delta _{8} -\frac{\rho _{2}^{2}}{2}\bigg )-\frac{1}{4}, \\ q_{23}= & {} q_{32}=0, \\ q_{24}= & {} q_{42}=0, \\ q_{33}= & {} \eta _{3}^{*}\Big [\gamma _{m}T^{*}+\delta _{m}-\frac{\rho ^{2}_{3}}{2} \Big ], \\ q_{34}= & {} q_{43}=0, \\ q_{44}= & {} \delta _{d}-\frac{\rho _{4}^{2}}{2}-A^{2}-B^{2}. \end{aligned}$$

Therefore, we have all $q_{ij} \ge 0; ~(i,~j=1,~2,~3,~4)$ for $q_{11}> 0,~q_{22}> 0,~q_{33} > 0$ and $q_{44} > 0$. Thus, from the previous inequalities, we get

$$\begin{aligned} \rho _{1}^{2}< & {} 2\bigg (-r_{T}+2r_{T}b_{T}T^{*} + \frac{(\alpha _{T}M^{*}+\gamma _{T}T_{8}^{*})}{g_{T}+M^{*}}\nonumber \\{} & {} -\frac{1}{4(\eta _{1}^{*}+\eta _{2}^{*})} \bigg ), \end{aligned}$$

(25)

$$\begin{aligned} \rho _{2}^{2}< & {} 2\bigg (\frac{\gamma _{T}T^{*}}{g_{T}+M^{*}}+ \frac{\alpha _{8}D^{*}}{(g_{8}+T_{8}^{*})^{2}} +\delta _{8}-\frac{1}{4\eta _{2}^{*}} \bigg ), \end{aligned}$$

(26)

$$\begin{aligned} \rho _{3}^{2}< & {} 2(\gamma _{m}T^{*}+\delta _{m}), \end{aligned}$$

(27)

$$\begin{aligned} \rho _{4}^{2}< & {} 2(\delta _{d}-A^{2}-B^{2}). \end{aligned}$$

(28)

Therefore, Q is real symmetric and positive definite matrix and hence, its eigenvalues $\lambda _{1},\lambda _{2},\lambda _{3}$ and $\lambda _{4}$ will be positive real quantities if the conditions (25), (26), (27) and (28) hold. If $\lambda _{m}$ is the $\min \{\lambda _{1},\lambda _{2},\lambda _{3},\lambda _{4}\}$, then we have $LW(v,t) \le -\lambda _{m}|v(t)|^{2}$ and we calculate that the zero solution of stochastic differential system (4) is an asymptotic mean square stable. $\square$

6 Parameter estimation

The system parameters play a crucial role in determining the behavior and analysis of the mathematical model. In this section, we will elaborate on how we estimated specific parameters of our system (1) using information available from existing literature. Our methodology for parameter estimation is delineated as follows.

Decay rate of CD8+ T cells, symbolized as  $\delta _{8}$: The calculated half-life of CD8+ T cells, represented as $\delta _{8}$, is around 3.9 days as per the research in [41, 42]. We can derive the death rate $\delta _{8}$ of CD8+ T cells using the equation

$$\begin{aligned} \frac{1}{2} T_{8}(0)= & {} T_{8}(0) e^{(-\delta _{8} t_{1/2}^{T_{8}})}. \end{aligned}$$

This equation yields the value of $\delta _{8}$ as

$$\begin{aligned} \delta _{8}= & {} \frac{\ln 2}{3.9}~\text {day}^{-1} \\\approx & {} 0.178~\text {day}^{-1}. \end{aligned}$$

Decay rate of macrophages, symbolized as  $\delta _{m}$: The parameter $\delta _{m}$ can be computed utilizing the information presented by Wacker et al. [43] & Khajanchi S. [44], who reported a half-life of 12.4 days for macrophages. This can be formulated as

$$\begin{aligned} \delta _{m}= & {} \frac{\ln 2}{12.4~\text {day}} \\\approx & {} 0.056~\text {day}^{-1}. \end{aligned}$$

Decay rate of dendritic cells, labeled as  $\delta _{d}$: Holt et al. [45] suggest that the half-life of dendritic cells is 3 to 4 days. We take a median half-life of 4 days for our computations. Consequently, the decay rate $\delta _{d}$ can be determined as follows

$$\begin{aligned} \delta _{d}= & {} \frac{\ln 2}{4~\text {day}} \\\approx & {} 0.17~\text {day}^{-1}. \end{aligned}$$

Activation rate of dendritic cells, denoted as  $\alpha _{d}$: The parameter denoting the half-saturation constant for tumor cells is denoted as $g_{d}$. This can be mathematically represented as

$$\begin{aligned} \frac{T}{g_{d} + T} = \frac{1}{2}. \end{aligned}$$

From the research paper conducted by Coventry et al. [46], which observed the density of dendritic cells in breast cancer patients as $D = 4\times 10^{-4}$ cells, and by incorporating the equilibrium state of the fourth equation in (1), we can deduce the equation

$$\begin{aligned} \alpha _{d}\frac{T}{g_{d}+T} = \delta _{d}D, \end{aligned}$$

which implies

$$\begin{aligned} \alpha _{d} = 1.36\times 10^{-4}~\text {cell}/\text {day}. \end{aligned}$$

Constant source rate of dendritic cells, denoted as  $s_{d}$: The constant source rate $s_{d}$ is influenced by the presence of tumor cells. In a healthy individual, the presence of tumor cells results in the absence of dendritic cell production. Therefore, at a steady state, we can express this relation as

$$\begin{aligned} s_{d}= & {} \delta _{d}D \\= & {} 0.68\times 10^{-4}~\text {cell}/\text {day}. \end{aligned}$$

Natural degradation rate of Tregs, denoted as  $\delta _{g}$: According to Q. Tang [47], the half-life of regulatory T-cells is 32 h, approximately equivalent to 1.3 days. Therefore, we can calculate the death rate $\delta _{g}$ as follows

$$\begin{aligned} \delta _{g}= & {} \frac{\ln 2}{1.3}~\text {day}^{-1} \\\approx & {} 0.53~\text {day}^{-1}. \end{aligned}$$

Activation rate of Tregs, denoted as  $\alpha _{g}$: As reported by Q. Tang [47], the estimated average count of circulating regulatory T-cells in an adult human is approximately $0.25 \times 10^{9}~pg\cdot ml^{-1}$. Additionally, assuming a density of $2\times 10^{7}$ CD8+ T cells/ml, we derive the following equation by considering the steady state of the fifth equation in (1)

$$\begin{aligned} \alpha _{g}T_{8} - \delta _{g}T_{g}= & {} 0, \end{aligned}$$

which implies

$$\begin{aligned} \alpha _{g}= & {} \frac{\delta _{g}T_{g}}{T_{8}}. \end{aligned}$$

Substituting the values, we get

$$\begin{aligned} \alpha _{g}= & {} \frac{0.53 \times 0.25 \times 10^{9} (\text {day}^{-1}.\text {pg}.\text {ml}^{-1})}{2 \times 10^{7}(\text {cell}.\text {ml}^{-1})}. \end{aligned}$$

This simplifies to

$$\begin{aligned} \alpha _{g}= & {} 6.62~\text {pg}.\text {day}^{-1}.\text {cell}^{-1}. \end{aligned}$$

Degradation rate of interleukin-10, denoted as $\delta _{10}$: Huhn et al. [48] reported the half-life of interleukin-10 (IL-10) as 4.5 h, approximately equivalent to 0.1875 days. Hence, we can calculate the degradation rate $\delta _{10}$ as follows

$$\begin{aligned} \delta _{10}= & {} \frac{\ln 2}{0.1875}~\text {day}^{-1} \approx 3.696~\text {day}^{-1}. \end{aligned}$$

Activation rate of interleukin-10, denoted as $\alpha _{10}$: Toossi et al. [49] observed that $10^6$ alveolar macrophages produced 3,200 pg/mL of interleukin-10 ($I_{10}$). By utilizing the steady-state equation from the sixth equation in (1), we can express this as

$$\begin{aligned} \alpha _{10}M - \delta _{10}I_{10}= & {} 0, \end{aligned}$$

which implies

$$\begin{aligned} \alpha _{10}= & {} \frac{3.696~ \times 3200}{10^{6}}~(\text {day}^{-1}.\text {pg}.\text {ml}^{-1})/ (\text {cell}.\text {ml}^{-1}). \end{aligned}$$

Simplifying the above expression, we get

$$\begin{aligned} {\alpha _{10}}= & {} {0.01182~\text {pg}.\text {day}^{-1}.\text {cell}^{-1}}. \end{aligned}$$

Death rate of TGF-$\beta$, denoted as $\delta _{\beta }$: Assuming the estimated median half-life of TGF-$\beta$ is approximately 20 h, equivalent to 0.83 days, we can calculate the death rate of TGF-$\beta$ as follows

$$\begin{aligned} \delta _{\beta }= & {} \frac{\ln 2}{0.83}~\text {day}^{-1} ~~~\approx 0.832~\text {day}^{-1}. \end{aligned}$$

Source rate of TGF-$\beta$, denoted as $s_{\beta }$: Peterson et al. [50] reported a TGF-$\beta$ $(T_{\beta })$ density of 609 pg/ml. It is observed that in a healthy person the concentration of TGF-$\beta$ is 10 times less than a cancer patient. Assuming a cerebral spinal fluid volume of 150 ml, in the absence of cancer-induced TGF-$\beta$ production, we can observe at steady state that

$$\begin{aligned} s_{\beta }= & {} \delta _{\beta }T_{\beta } \\= & {} 0.832~\text {day}^{-1} \times 150~\text {ml} \times 60.9~\text {pg}\cdot \text {ml}^{-1} \\= & {} 7.6\times 10^{3}~\text {pg}\cdot \text {day}^{-1}. \end{aligned}$$

Release rate of TGF-$\beta$ by tumor cells, denoted as  $\alpha _{\beta }$: The average concentration of the immuno-suppressive cytokine TGF-$\beta$ ($T_{\beta }$) is determined to be $609~\text {pg}/ \text {ml} \times 150~\text {ml} = 91350~\text {pg}$ based on the findings by Peterson et al. [50], which pertains to high-grade glioblastoma patients. By applying the seventh equation from the system (1) at an equilibrium state, we can derive that

$$\begin{aligned} \alpha _{\beta }= & {} \frac{\delta _{\beta }T_{\beta } - s_{\beta }}{T} \\= & {} \frac{91350~\text {pg} \times 0.832~\text {day}^{-1}- 7.6\times 10^{3}~\text {pg}\cdot \text {day}^{-1}}{10^{6}~\text {cell}} \\= & {} 0.0684~\text {pg}.\text {day}^{-1}.\text {cell}^{-1}. \end{aligned}$$

Decay rate of interleukin-12, denoted as $\delta _{12}$: The half-life of interleukin-12 $(I_{12})$ is 30 h, equivalent to 1.25 days, which is reported by Carreno et al. [51]. Therefore, the decay rate of interleukin-12 $(I_{12})$ is

$$\begin{aligned} \delta _{12}= & {} \frac{0.693}{1.25~\text {day}} \\\approx & {} 0.55~\text {day}^{-1}. \end{aligned}$$

Release rate of IL-12 by dendritic cells, denoted as  $\alpha _{12}$: In a breast cancer patient, the concentration of interleukin-12 in the blood serum is about $1.5 \times 10^{-10}$ pg/ml [52]. Meanwhile, the concentration of dendritic cells is $4 \times 10^{-4}$ cell/ml [46]. Thus, at the steady state of the eighth equation of (1), we can derive that

$$\begin{aligned} \alpha _{12}= & {} \frac{\delta _{12}I_{12}}{D} \\= & {} \frac{1.5 \times 10^{-10}~\text {pg}\cdot \text {ml}^{-1}\times 0.55~\text {day}^{-1}}{4 \times 10^{-4}~\text {cell}\cdot \text {ml}^{-1}} \\= & {} 2.06\times 10^{-7}\text {pg}.\text {day}^{-1}.\text {cell}^{-1}. \end{aligned}$$

Decay rate of IFN-$\gamma$, denoted as $\delta _{\gamma }$: The median half-life of IFN-$\gamma$ is calculated to be 6.8 h $\approx$ 0.283 days [53]. Therefore, the decay rate of IFN-$\gamma$ is

$$\begin{aligned} \delta _{\gamma }= & {} \frac{\ln 2}{0.283}~\text {day}^{-1} \\\approx & {} 2.45~\text {day}^{-1}. \end{aligned}$$

Activation rate of IFN-$\gamma$ by CD8+ T cells, denoted as $\alpha _{\gamma }$: Kim et al. [54] reported that CD8+ T cells produce 200 pg/ml of IFN-$\gamma$. Based on this data, we assume that the concentration of CD8+ T cells is $2 \times 10^{7}$ cell/ml. Therefore, by using the steady state of the ninth equation in (1), we can calculate

$$\begin{aligned} \alpha _{\gamma }= & {} \frac{\delta _{\gamma }I_{\gamma }}{T_{8}} \\= & {} \frac{200~\text {pg}\cdot \text {ml}^{-1}\times 2.45~\text {day}^{-1}}{2\times 10^{7}\text {cell}\cdot \text {ml}^{-1}}\\= & {} 2.45\times 10^{-5}~\text {pg}.\text {day}^{-1}.\text {cell}^{-1}. \end{aligned}$$

Table 1 presents a summary of all the model parameters.

7 Numerical results

This section provides extensive numerical illustrations of the theoretical analysis obtained from both deterministic and stochastic tumor-immune interaction models under various situations. We illustrate our tumor-immune interaction model numerically using MATLAB by selecting appropriate parameter values, which are given in Table 1.

Table 1 The parameter values employed for simulating the tumor-immune interaction model are as follows

Fig. 1

Time series evolution for tumor-free singular point $E^{1}$ of deterministic system (2) shows stable behavior with respect to the initial value $(0.0001, ~2.94 \times 10^{-9}, ~3000.57, ~0.00038)$ and parameters value are taken from Table 1

Fig. 2

Time series evolution for low tumor-presence singular point $E^{2}$ of deterministic system (2) shows unstable behavior with respect to the initial value $(0.12, ~2 \times 10^{-10}, ~3000, ~0.0002)$ and parameters value are taken from table 1

Fig. 3

Time series evolution for high tumor-presence singular point $E^{3}$ of deterministic system (2) shows stable behavior with respect to the initial value $(400000,~ 5.56441 \times 10^{-9}, ~0.005, ~0.001)$ and parameters value are taken from table 1

Fig. 4

Time series evolution of the stochastic system (4) for $\rho _{1}=\rho _{2}=\rho _{3}=\rho _{4}=0.1$ with initial density $T(0)=0.001,~T_{8}(0)=2.94\times 10^{-12},~ M(0)=3000,~D(0)=0.0003$ and other parameters value are taken from table 1

Fig. 5

Time series evolution of the stochastic system (4) for $\rho _{1}=1.4,~\rho _{2}=\rho _{3}=\rho _{4}=0.5$ with initial density $T(0)=0.001,~T_{8}(0)=2.94\times 10^{-12},~ M(0)=3000,~D(0)=0.0003$ and other parameters value are taken from Table 1

Fig. 6

Time series evolution of the stochastic system (4) for different values of intensity of population fluctuation with initial density $T(0)=0.001,~T_{8}(0)=2.94\times 10^{-12},~ M(0)=3000,~D(0)=0.0003$ and other parameters value taken from Table 1

Fig. 7

Three dimensional phase portrait diagram of the stochastic system (4) with initial density $T(0)=0.001,~T_{8}(0)=2.94\times 10^{-12},~ M(0)=3000,~D(0)=0.0003$ and other parameters value taken from Table 1

Fig. 8

Relative frequency density of tumor cells, CD8+ T cells, macrophages and dendritic cells of stochastic system (4) for $\rho _{1}=\rho _{2}=\rho _{3}=\rho _{4}=0.1$ and other parameters value taken from Table 1

7.1 Numerical results for deterministic model

To perform local stability analysis of the deterministic model (2), we run our simulation for 150 days and take different initial values. Using parameters value from Table 1, we obtained the tumor-free singular point $E^{1}~(0,~2.94586\times 10^{-11},$ $~9678.57,~0.0004)$ and the corresponding eigenvalues are $-0.865106,~-0.178,$ $~-0.17,~-0.056$. We observe that all the eigenvalues are real and have negative real part. Thus, our deterministic system (2) is stable around the tumor-free singular point $E^{1}$ (see the Fig. 1). Numerically, we have also checked the condition(s) for the Theorem 1 and calculated that $r_{T} - \frac{\alpha _{T}M^{1} + \gamma _{T}T_{8}^{1}}{g_{T} + M^{1}} = -0.865108 < 0$, which assured the local asymptotic stability of the tumor-free steady state $E^{1}$. Theorem 2.1 has significant implications for understanding the dynamics of tumor growth and immune response in biological systems. If the conditions outlined in the theorem are satisfied, indicating that the tumor-free steady state is locally asymptotically stable, it suggests that in the absence of tumors, the biological system tends to remain in a stable equilibrium state. This stability implies that the immune response is effective in controlling of tumor growth. We obtained two interior steady states, namely low tumor-presence singular point $E^{2}(0.179273,~2.94586\times 10^{-11},~3886.16,~0.0004)$ and high tumor-presence singular point $E^{3}(800000,~5.56441\times 10^{-11},~0.00145511,$ 0.00075). The eigenvalues around the low tumor-presence singular point $E^{2}$ are $-0.300816,~-0.178,~-0.17,~0.161346$, which depicts that the our deterministic system (2) is unstable around $E^{2}$ (see the Fig. 2). Also, $-372480,~-0.5822,~-0.178,~-0.17$ are the eigenvalues around the high tumor-presence singular point $E^{3}$. It can be noticed that all of the eigenvalues are real and have negative real part. Therefore, our deterministic system (2) is stable around the high tumor-presence singular point $E^{3}$ (see the Fig. 3).

7.2 Numerical results for stochastic model

Now, we investigate extensive numerical results obtained from the stochastic tumor-immune interaction system (4). We employ Milstein’s scheme [55] to evaluate the strong approximation solution of (4) with an initial condition based on the It$\hat{o}$ process. This scheme is derived from the Euler-Maruyama technique and involves incorporating a corrective component into the stochastic increment. For numerical illustrations, we only change the values of $\rho _1$, $\rho _2$, $\rho _3$ and $\rho _4$ to investigate their influence on the dynamics of the tumor-immune interaction model system (4). Now, we discretized the time interval $[t_{0},~t_{f}]$ as follows:

$$\begin{aligned} t_{0}=0<t_{1}=t_{2}<........<t_{N}<t_{N+1}=t_{f}. \end{aligned}$$

We apply the Milstein’s scheme in our stochastic system (4) as follows

$$\begin{aligned} T(j+1)= & {} T(j) + \bigg [r_{T}T(j)(1-b_{T}T(j))\\{} & {} -\frac{\alpha _{T}M(j) +\gamma _{T}T_{8}(j))T(j)}{g_{T}+M(j)} \bigg ]\Delta t \\{} & {} + \rho _{1}(T(j)-T^{*})I_{1,j}\sqrt{\Delta t} \\{} & {} + \frac{1}{2} \rho ^{2}_{1}(T(j)-T^{*})(I_{1,j}^{2}-1)\Delta t, \\ T_{8}(j+1)= & {} T_{8}(j) + \bigg [\frac{\alpha _{8}D(j)}{g_{8}+T_{8}(j)} \\{} & {} -\delta _{8}T_{8}(j) \bigg ]\Delta t +\rho _{2}(T_{8}(j)-T_{8}^{*})I_{2,j}\sqrt{\Delta t} \\{} & {} + \frac{1}{2} \rho ^{2}_{2}(T_{8}(j)-T_{8}^{*})(I_{2,j}^{2}-1)\Delta t,\\ M(j+1)= & {} M(j) \\{} & {} \quad + \bigg [s_{m}+\frac{\alpha _{m}T_{8}(j)}{(g_{m}+T_{8}(j))(g_{m1}+T(j))}\\{} & {} -\gamma _{m}M(j)T(j)-\delta _{m}(j) \bigg ]\Delta t \\{} & {} + \rho _{3}(M(j)-M^{*})I_{3,j}\sqrt{\Delta t} \\{} & {} + \frac{1}{2} \rho ^{2}_{3}(M(j)-M^{*})(I_{3,j}^{2}-1)\Delta t, \\ D(j+1)= & {} D(j) + \bigg [ s_{d}+\frac{\alpha _{d}T(j)}{g_{d}+T(j)}-\delta _{d}D(j) \bigg ]\Delta t \\{} & {} +\rho _{4}(D(j)-D^{*})I_{4,j}\sqrt{\Delta t} \\{} & {} + \frac{1}{2} \rho ^{2}_{4}(D(j)-D^{*})(I_{4,j}^{2}-1)\Delta t, \end{aligned}$$

where $I_{i,~j}$ (i, j=1, 2, 3, 4) is the j-th realization of $I_{i}$ and $I_{i}$ is Gaussian random variable N(0, 1).

A suitable Lyapunov function is considered to demonstrate that the tumor-presence singular point is asymptotically mean square stable. Positive fluctuations $\rho _{1},\rho _{2},~\rho _{3}$ and $\rho _{4}$ are depended upon in our stochastic system. For this, $\rho _{1}=\rho _{2}=\rho _{3}=\rho _{4}=0.1$ are chosen and the real positive constants $\eta _{1}=\eta _{2}=\eta _{3}=0.5$ are set. Now, we have calculated $2\bigg (-r_{T}+2r_{T}b_{T}T^{*}+ \frac{(\alpha _{T}M^{*}+\gamma _{T}T_{8}^{*})}{g_{T}+M^{*}}-\frac{1}{4(\eta _{1}^{*}+\eta _{2}^{*})} \bigg ) = 0.6644 > \rho _{1}^{2}$, $2\bigg (\frac{\gamma _{T}T^{*}}{g_{T}+M^{*}}+ \frac{\alpha _{8}D^{*}}{(g_{8}+T_{8}^{*})^{2}}+\delta _{8}-\frac{1}{4\eta _{2}^{*}}\bigg ) = 383.357 > \rho _{2}^{2}$, $2(\gamma _{m}T^{*}+\delta _{m}) = 744.960 > \rho _{3}^{2}$ and $2(\delta _{d}-A^{2}-B^{2}) = 0.34 > \rho _{4}^{2}$. Therefore, all conditions of Theorem 6 have been numerically verified. In Fig. 4, we present a time series evolution of our stochastic system (4) with fluctuation density $\rho _{1}=\rho _{2}=\rho _{3}=\rho _{4}=0.1$ and the other parameters value are obtained from Table 1. It is observed that the tumor population reaches zero after 20 days. Again, it is observed that the tumor cell population becomes extinct after 30 days due to the high intensity of fluctuation (see Fig. 5). It is to be noted that the extinction criterion, as stated in Theorem 5, is satisfied as $r_{T}=0.5822 < 0.601425 = \rho ^{2}_{1}\bigg (\frac{1}{2}-T^{*} \bigg )$, which provides numerical verification for our analytical results. For this, we consider $\rho _{1}=1.4$, $\rho _{2}=\rho _{3}=\rho _{4}=0.5$ and $T^{*}=0.179273$ (low tumor density), while other parameters value are taken from Table 1. After carefully analyzing the results and observations, we conclude that when the intensity of fluctuation density is small, the tumor cells become extinct, reaching zero in a relatively short time. This deduction is based on the findings obtained from our analysis and serves to highlight the significant impact of fluctuation intensity on the dynamics of tumor cells. In Fig. 6, we present a comparison of our stochastic differential equation (4) under different values of fluctuation intensity. The red curve corresponds to the fluctuation value $\rho _{1}=\rho _{2}=\rho _{3}=\rho _{4}=0.001$, the blue curve represents $\rho _{1}=\rho _{2}=\rho _{3}=\rho _{4}=0.03$, the green curve depicts $\rho _{1}=\rho _{2}=\rho _{3}=\rho _{4}=0.07$ and the black curve indicates $\rho _{1}=\rho _{2}=\rho _{3}=\rho _{4}=0.15$. We observe that increasing the intensity value of population fluctuation leads to greater fluctuations in our cell population. Therefore, the behavior of our stochastic system (4) depends on the intensity value of cell population fluctuations. By adjusting the intensity of fluctuations, we can modulate the level of variability and randomness exhibited by the cell populations, thereby influencing the overall behavior of the system. The dependence of our stochastic system on fluctuation intensity underscores the importance of considering this parameter in the study of tumor growth dynamics and other stochastic processes. Furthermore, we have constructed a three-dimensional phase portrait diagram of the stochastic system (4) (see Fig. 7) for $\rho _{1}=\rho _{2}=\rho _{3}=\rho _{4}=0.1$, considering initial densities $T(0)=0.001,~T_{8}(0)=2.94\times 10^{-12},~ M(0)=3000,~D(0)=0.0003$ and utilizing other parameters value from Table 1. By analyzing this phase portrait, we can better understand the underlying dynamics and make informed interpretations about tumor growth and its response to varying environmental conditions.

Over the course of 200 days and through approximately 5000 simulations, the mean $(\mu _{X_{i}})$ and the standard deviation $(\sigma _{X_{i}})$ of the cell populations are

$$\begin{aligned} \mu _{T}= & {} 0.000144587,~~~~~\mu _{T_{8}} = 2.657\times 10^{-10},\\ \mu _{M}= & {} 9575.98,~~~~~\mu _{D} = 0.00398225 \end{aligned}$$

and

$$\begin{aligned} \sigma _{T}= & {} 0.00101432,~~~~~\sigma _{T_{8}} = 7.17345\times 10^{-11},\\ \sigma _{M}= & {} 1954.66,~~~~~\sigma _{D} = 0.000893139. \end{aligned}$$

We observe that the stochastic mean closely aligns with its deterministic low tumor-presence singular point $E^{2}$. This observation indicates that the dynamics of cell populations are notably influenced by stochastic fluctuations. To further investigate the behavior of our stochastic system (4), we conduct relative frequency density and normal distribution analysis, as illustrated in Fig. 8. The study of relative frequency density and normal distribution provides valuable insights into the probabilistic nature of the stochastic system. These analyses allow us to better comprehend the probability distribution of cell population values and assess how stochastic fluctuations contribute to the overall variability observed in the dynamics of the system.

8 Conclusion

This section provides a concise overview of the research and emphasizes its significant contributions to the field of tumor-immune interaction models using a stochastic system. In this study, a mathematical model based on biology is introduced to elucidate the dynamics of the tumor-immune interaction system. The model incorporates various cell types including tumor cells, CD8+ T cells, macrophages, dendritic cells, and several cytokines such as Tregs or regulatory T-cells, TGF-$\beta$, IL-10, IL-12, and IFN-$\gamma$. To develop the model, we utilized a system of coupled ordinary differential equations. To get a better understanding of the tumor-immune interaction system, we simplified the model using quasi-steady-state approximations for the concentration of cytokines. This reduced model effectively captures the detailed interactions between tumor cells, CD8+ T cells, macrophages, and dendritic cells.

Deterministic models often lead to inaccurate predictions of biological systems due to the influence of environmental fluctuations. Consequently, to study the effects of these fluctuations, we expanded our deterministic system into a stochastic system. In this study, we introduced randomly fluctuating terms without altering any parameters in the original deterministic model. These stochastic random perturbations follow a white noise type which is directly proportional to the distances of the interior singular point from the cell population. In this study, we demonstrate the existence and uniqueness of the solution for our stochastic system (4). Through this analysis, we establish the foundation for exploring the dynamic behavior of the system under stochastic conditions. To further characterize the behavior of the system, we construct the appropriate Lyapunov function. By It$\hat{o}$’s lemma, we establish both the stochastically ultimate bounded and stochastically permanence of our stochastic system (4). This key finding highlights the system’s ability to remain within certain bounded regions over time and persist in the presence of environmental fluctuations. Moreover, our investigation leads to the identification of an extinction criterion for the tumor cell population. We derive some conditions, as stated in Theorem 6, to ensure the asymptotic mean-square stability of the zero solution in the stochastic system (4). Also, we present several figures of the stochastic system (4) for different values of the intensity of population fluctuation. Through our observations, we have noticed that when the intensity fluctuation parameters in the stochastic system (4) are set to sufficiently small values, the tumor cell population ultimately reaches zero. The observed behavior emphasizes the sensitivity of tumor cell population dynamics to stochastic influences. Even small random fluctuations in the environment can significantly impact the trajectory of the tumor cell population, driving it toward extinction. Furthermore, as we increase the values of population fluctuations, we notice that the nature of the cell population becomes more fluctuated. Comparing the stochastic mean solution to its deterministic solution, we find that it closely aligns with the deterministic low tumor-presence singular point $E^{2}$. This observation indicates that stochastic noise has a significant impact on the behavior of the cell populations.

Overall, this research contributes to a deeper understanding of tumor-immune interaction dynamics and the intricate interplay between deterministic and stochastic components in biological system. The stochastic model allowed us to capture the inherent randomness and uncertainty present in these interactions, leading to a more accurate representation of real-world scenarios. We demonstrated that stochasticity plays a crucial role in shaping the outcome of these interactions, affecting both tumor progression and the effectiveness of immune responses. Our findings have significant implications for cancer research, where environmental fluctuation plays a crucial role. We are hopeful that our mathematical findings will prove beneficial for future researchers working in this field. The insights gained from our study can serve as a valuable contribution to the advancement of knowledge in this area of research. In the future, researchers can integrate stochastic fluctuations into their deterministic models without altering any parameters or introduce stochastic perturbations to key parameters identified in this research paper. Additionally, they can utilize parameter values derived from the findings of this study to ensure consistency and comparability across different investigations. This approach allows for a more comprehensive exploration of the impact of stochasticity on tumor-immune interaction dynamics, facilitating a deeper understanding of the underlying mechanisms and improving the predictive accuracy of mathematical models in this field.