The effect of a psychological scare on the dynamics of the tumor-immune interaction with optimal control strategy

Contracting cancer typically induces a state of terror among the individuals who are affected. Exploring how chemotherapy and anxiety work together to affect the speed at which cancer cells multiply and the immune system’s response model is necessary to come up with ways to stop the spread of cancer. This paper proposes a mathematical model to investigate the impact of psychological scare and chemotherapy on the interaction of cancer and immunity. The proposed model is accurately described. The focus of the model’s dynamic analysis is to identify the potential equilibrium locations. According to the analysis, it is possible to establish three equilibrium positions. The stability analysis reveals that all equilibrium points consistently exhibit stability under the defined conditions. The bifurcations occurring at the equilibrium sites are derived. Specifically, we obtained transcritical, pitchfork, and saddle-node bifurcation. Numerical simulations are employed to validate the theoretical study and ascertain the minimum therapy dosage necessary for eradicating cancer in the presence of psychological distress, thereby mitigating harm to patients. Fear could be a significant contributor to the spread of tumors and weakness of immune functionality.


Introduction
Models are instruments utilized in medicine and science to interpret results, develop hypotheses, and plan experiments to verify them [1].
An exemplary illustration of this methodology was the endeavour to elucidate species diversity through competition models [10][11][12][13][14][15][16].Mathematicsematical modeling is a highly versatile instrument in the field of infectious disease epidemiology, enabling the detection of epidemic patterns, extrapolation of epidemic behaviors, and evaluation of the impact of interventions, including pharmacological treatment, immunization, quarantine, social distance, and hygiene practices, among others [17][18][19][20][21][22].An example of a disease model is cancer, which is characterized by the proliferation of malignant cells that infiltrate other anatomical structures and currently ranks as the second most prevalent cause of mortality globally, surpassed only by cardiovascular disease.Developing novel treatment options is a burgeoning study field for scientists seeking to manage cancer effectively.
Nevertheless, comprehending the *Corresponding Author intricacies of tumor cell proliferation and their intricate interplay with the immune system is crucial in order to devise novel therapeutic approaches.
Several scientists have extensively researched the mathematical modeling of tumor evolution, its interaction with different cells, and the process of tumor growth.They have achieved this by creating multiple models over the past few decades [28][29][30][31][32][33].Cancer is amenable to a variety of treatment modalities, including chemotherapy, radiotherapy, and surgery.Chemotherapy, one of the cancer treatments, is a systematic approach that targets and eliminates cancer cells at the site of the tumor while minimizing its impact on effector-normal cells.This eliminates the ability of the tumor cells to metastasize to other anatomical sites [34][35][36].For instance, De Pillis and his associates examined multiple mathematical models to quantify the effects of chemotherapy [37].In addition, Pillis et al. devised a cancer treatment model in which they discovered that combining chemotherapy and immunotherapy can completely eradicate the tumor instead of using either therapy alone [38].On the other hand, The initial mathematical model that incorporated the influence of fear in a predator-prey system involving two species was presented by Wang et al. in 2016 [39].Prey animals may alter their grazing location to a more secure area and relinquish their most productive feeding sites due to predator-induced anxiety.
The user's text is incomplete and lacks information [40][41][42][43].Further, There has been a recent increase in research focusing on the importance of mathematical models for studying how fear-induced behavioral changes impact the spread of diseases [44][45][46][47][48].A medical study has demonstrated that psychological stress contributes to the dissemination of cancer cells throughout the patient's body.Psychological stress causes significant dilation and intensification of blood vessels, hence promoting the migration of cancer cells and facilitating the metastasis of the disease [49].Researchers have discovered that stress-induced hormones exacerbate the proliferation of cancer cells inside the "lymphatic system," thus facilitating their dissemination to other locations, thereby promoting the metastasis of the disease throughout the human body [50].
The present study proposes a psychological scare-cancer-immune-normal-chemotherapy model (PSCINC) regulated by systems of ordinary differential equations, drawing inspiration from the model presented in [51].We have enhanced the model of De Pillis et al. by replacing the linear functional response with the Holling type II functional response.This modification allows us to accurately depict the eradication of tumor cells by the immune system, considering the possibility of a weakened immune system due to the presence of psychological scare of cancer.Further, there is a lack of study about the influence of fear on the immune-cancer model.Hence, we deem it imperative to examine this phenomenon, as it contributes to reducing the occurrence of catastrophic circumstances.Further, there is a lack of study about the influence of fear on the immune-cancer model.Hence, we deem it imperative to examine this phenomenon, as it contributes to reducing the occurrence of catastrophic circumstances.Therefore, this study is dedicated to discussing the impact of anxiety on immune cancer patients, which could be a significant contributor to the spread of tumors and weakness of immune functionality.The subsequent sections of this document are organized as follows: section 2 examines the assumptions of the proposed model.The presence of potential equilibrium points is determined in section 3. Next, section 4 discusses the stability conditions of the steady states.The discussion in section 5 focuses on the global stability of equilibriums.In addition, section 6 acknowledges the local bifurcation conditions in close proximity to the fixed points.In section 7, numerical examinations are conducted to validate our analytical findings.

Assumptions of the model
Let's examine a system of differential equations (PSCINC) that involves immune cells I (t), tumor cells C(t), normal cells N (t), and chemotherapy treatment H(t) represented as In the first equation of the PSCINC model, the term α 1+eC stands for the regular production of immune cells in the body, which is affected by the presence of cancer cells by the psychological scare factor e.
Therefore,e the birth-term changes by producing fear function.
The fear function is incorporated by the decreasing function φ (e, C) = 1 1+eC , which was initially introduced by Wang et al. [46].From the biological point of view, φ (e, C) is appropriate since The Michaelis-Menten term p   Decay rate of the chemo-drug 0.05 [53] Proof.By integrating the second and third functions of the (PSCINC) model for C (t) and N (t) with a positive initial condition (I (0) , C (0) , N (0), H(0)), we obtain From the first equation of the (PSCINC) model, we have Therefore, after eliminating the non-negative terms, this produces 0000-0003-4022-8053 Consequently, by integrating the equation shown above for I(t), these yields

. All the solutions of the (PSCINC) model are uniformly bounded if the following condition is hold
+ be an initial condition for the (PSCINC), then, by using the Bernoulli method, we get Now, by using the standard comparison theory [48] and the above bound for the cancer cells, we get Therefore, the corresponding domain region for the (PSCINC) model is □

Equilibria analysis
This section will delve into finding the possible equilibrium and analyzing the system's stability, specifically its stability in the vicinity of equilibrium.To accomplish this, we compute dI dt = dC dt = dN dt = dH dt = 0 and get the following equilibrium in two cases: (1) No treatment case: in this case, we have two equilibrium points given by (a) The cancer-free or healthy point A 0 = (I 0 , 0, N 0 ), where where and C 1 is the root of the following equation where, Clearly, f 1 (0) = (αβ 1 p 3 − β 2 r 4 r 6 ), and Therefore, by the intermediate value theorem [55], f 1 (C) has a positive root, say C 1 in the interval (0, k 1 ) if one of the following conditions is satisfied Now, for I 1 and N 1 to be positive, the following two conditions must be satisfied: (2) After treatment case: in this case, we have one positive equilibrium point and C 2 is the root of the following equation where Clearly, Therefore, by the intermediate value theorem, f 2 (C) has a positive root, say C 2 in the interval (0, k 1 ) if one of the following conditions is satisfied For I 2 and N 2 to be positive, the following two conditions must be satisfied: Since N = 0 indicates that the patients are deceased, we exclude cases where N = 0 from consideration.In order to analyze the linear stability of the system at the three equilibrium points mentioned above, it is necessary to calculate the Jacobian matrix of the system, and the Jacobian is here.
• The Jacobian matrix at A 0 = (I 0 , 0, N 0 ) is given as: Then, the eigenvalues of J (A 0 ) are and λ 0 3 < 0. Therefore, A 0 is asymptotic stable whenever if is given as: where So, the eigenvalues of J (A 2 ) are the roots of the following equation where: 12 a [1] 21 12 a [1] 21 a [1] 33 11 a [1] 23 a 12 a [1] 21 a [1] 33 Thus, according to the Routh-Hurwitz rule [56], A 1 will be asymptotically stable if where, So, the eigenvalues of J (A 2 ) are the roots of the following equation where, Thus, according to the Routh-Hurwitz rule, A 2 will be asymptotically stable on the condition that D 1 > 0, D 3 > 0 and

Global stability at the cancer-free steady state
To reach a healthy state, in this section, we will examine the global stability surrounding A 0 to explore the dynamics of the (PSCINC) system at regions far from the equilibrium point A 0 .
Theorem 3. A 0 is a GAS provided the following conditions hold: Proof.Let's define a Lyapunov function [57] for the (PSCINC) model at A 0 as follows: Therefore, dL dt = (I − I 0 ) Therefore, dL /dt < 0, and hence L(t) is a Lyapunov function under condition 10. □ Thus, the cancer-free steady state A 0 fulfills the requirements for local stability, rendering the point globally stable.From a biological perspective, chemotherapy refers to the process of selectively eliminating tumor cells if conditions (10) are met.

Local bifurcation
This section examines the local bifurcation conditions close to steady states by applying Sotomayor's rule for local bifurcation [58,59].
(2) A transcritical bifurcation (TB) if (3) A pitchfork bifurcation (PB) if condition (11) is violated where the notation in (11) will be introduced during the proof.
T represent the eigenvectors corresponding to the zero eigenvalue of J * (A 0 ) and J * T (A 0 ) respectively.Direct computation gives and Now, let h = (h 1 (I, C), h 2 (I, C, N ), h 3 (C, N )) T , then differentiating h with respect to m 1 gives: Hence, That means the (SNB) cannot happen at m 1 * .Subsequently, since This means the required conditions for (TB) are satisfied under condition (11).Finally, if condition (11) is not satisfied, then. For 22 a [1] 33 a [1] 11 12 a [1] 21 32 a [1] 33 where γ 2 * > 0, and the formulas of a [2] ij are given in (8)

Optimal control
This section focuses on analyzing the model following the administration of chemotherapy treatment at a certain time.From a biomedical standpoint, we have included the notion of optimum control in the model.For this purpose, we should look into the problem with a control strategy that can lessen the health hazard for the patient.Therefore, we propose and analyze the optimal control problem applicable to model (PSCINC) to determine the optimal dose of chemotherapy to control the tumor.We decide on control inputs v of cellular chemotherapy, included in fourth equation of the (PSCINC) model, to be supplied from an external source at different times.So, let us assume that the time-dependent form of our considered model is given in (1) with the following initial conditions for the model set: So, let us assume that the time-dependent form of our considered model is given in (1) with the following initial conditions for the (PSCINC) system set: The objective function, which is to be minimized, is defined as follows: (15) The constants ε 1 represent the weight factors of the respective terms.These are utilized to equalize the magnitude of the phrases.The ideal selection of control variable ν will effectively reduce tumor density and maximize immune density simultaneously, while also minimizing any unfavorable side effects within a set time frame.The initial component of the integrand function represents the overall quantity of tumor cells, the subsequent component of the integrand function represents the overall quantity of immune cells, and the last component of the integrand function indicates the efficacy of the administered medications on the organism.Here, we employ an optimum control problem to the model to minimize the administration of chemotherapeutic drugs, aiming to mitigate side effects and shorten the patient's recovery period.Here, we set up an optimal control ν * such that where ∆ = {ν : measurable, 0 ≤ ν ≤ 1, t ∈ [0, t f ]} is the admissible control set.

The existence of optimal control
In this sub-section, we analyze the existence of an optimal control of the (PSCINC) model ( 1).The property of super solutions Ī, C, N , and H of the model ( 1) is that trajectories given by are bounded.In vector form, we can express the above system (17) as: Since this is a linear system with bounded coefficients and the time frame is limited, so, we can conclude that the solutions Ī, C, N , and H, of the above system are bounded.Using the theorem proposed by Lukes [60], we found that the admissible control class and the corresponding state equations with assumed initial conditions are non-empty.Also, by the definition of the set ∆, it is clear that the control set ∆ is convex and closed.Since the state solutions are bounded, hence, the right-hand sides of the state system (1) are continuous and bounded by a sum of the bounded controls and the states.Now, we examine the convexity of the integrand of Ω (ν) on ∆ and that it is bounded below by τ 1 ν 2 (t) − τ 2 with τ 1 , τ 2 > 0. Let p, q be distinct elements of Ω and 0 ≤ Y ≤ 1.We have to show that Ω (p To establish it, we proceed as follows: This shows that τ 1 ν 2 (t) − τ 2 is a lower bound of Ω (τ, µ).This verifies that there exists an optimal control ν * for which Ω (ν * )=min Ω (ν * ) = min {Ω (ν) : ν ∈ ∆} From the above analysis and conclusion, we state the following theorem.

Characterization of the optimal control
For applying the Pontryagin maximum principle [46], we introduced the four co-state variables 2,3,4).The Hamiltonian function is given by With substitution from (1) into (18), we get The Hamiltonian equations are: where, ξ i (t) , i = 1, 2, 3, 4 are the adjoint functions to be determined suitably.The form of the adjoint equations and transversality conditions are standard results from Pontryagin's Maximum Principle [61].The adjoint system can be written in the form: The transversality conditions are ξ i (t f ) = 0, for i = 1, 2, 3, 4.
The condition dictate the necessary optimum control functions is ∂h * ∂ν = 0. Hence, we get By using the bounds for the control ν * (t) from ( 20), we get In compact notation, we have Based on the analysis and conclusion presented above, the subsequent theorem is derived.

Theorem 8.
For optimal control ν * and corresponding state variable solutions I * (t) , C * (t), N * (t) and H * (t) that minimize over ∆, there exist specific adjoint variables ξ i (t), i = 1, 2, 3, 4 satisfying the following system: subject to the transversality conditions Furthermore, the subsequent properties are valid:

Numerical Analysis
Numerical verification is essential for completing analytical studies.In this section, we visually confirmed the accuracy of our analytical findings for the (PSCINC) system using the software MATLAB.This verification holds significant practical significance.The simulations were conducted using the parameter values specified below [53].

Case II: no treatment case
Here, we examine the behavior of the (PSCINC) model in the absence of treatment and the psychological scare.
Figure 3 illustrates the performance of the (PSCINC) model where ν = 0 and e = 0.All initial conditions lead to the convergence of the system to a treatment-free equilibrium point A 1 = (I 1 , C 1 , N 1 , 0) = (0.25, 0.13, 0.9, 0).In addition, the population of immune cells steadily diminishes as the number of tumor cells gradually increases.Furthermore, this case clearly demonstrates that eradicating tumor cells is unattainable without a well-defined therapeutic strategy.

Case III: psychological scare case
The objective of this case is to demonstrate the impact of anxiety on the interaction between cancer cells and immune cells in the absence of chemotherapy drugs.

Case IV: a treatment case
In this instance, we will examine the intricacies of the (PSCINC) system when subjected to chemo-drug.The administration of chemotherapy leads to a substantial decrease in tumor cells within the body compared to past instances.In addition, chemotherapy also adversely affects the immune cells, decreasing the quantity of immune cells compared to the previous cases.Considering those mentioned above, additional doses are necessary to achieve a state devoid of tumors.

Case V: a minimum dosage of chemo-drug
This case aims to examine the effects of modifying the number of chemotherapy doses required to achieve a healthy state.Figure 5 clarifies the performance of the (PSCINC) model with various values of ν.The solution of the (PSCINC) system asymptotically converges to A 2 when v is less than 0.14.Conversely, the system tends towards a cancer-free state A 0 when ν = 0.14.Thus, a value of ν = 0.14 is the minimum dosage of chemotherapy necessary to achieve a condition devoid of cancer.A stability analysis was conducted on the system under consideration to investigate the model's dynamic behavior.Our research indicates that the constant state devoid of tumors is stable globally under particular conditions.This suggests that the prescribed treatment can eliminate tumor cells from the body for a specific tumor growth rate.The numerical simulations validate the analytical findings.Precisely, the threshold values for the transcritical bifurcation are calculated, indicating the point at which cancer transitions from persisting to eradicating.Additionally, numerical analysis reveals that when the tumor size is modest, the prescribed chemotherapy drug can effectively eliminate tumor cells from the body with a minimal minimum dose.Nonetheless, a constraint of our model is that prolonged treatment and a substantial dosage of medications are necessary to eradicate large tumors, both of which can be detrimental to the patient's health.Our upcoming research will focus on augmenting the immune system by regular vitamin intake or the utilization of stem cells.
Figure 4 explains the performance of the (PSCINC) model where ν = 0 with various values of e.The relationship between rising anxiety and declining immune function is evident.As a result, the tumor cells significantly grow; therefore, external treatment is needed.

Figure 4 .
Figure 4.The dynamics of the (P SCIN C) model with ν = 0 and various value of e.

Figure 5 .
Figure 5.The dynamics of the (PSCINC) model with treatment case.

Figure 6 .
Figure 6.The dynamics of the (PSCINC) model with various values of ν 8. Conclusion It has been looked at how an ODE mathematical model for tumor growth works, which includes how immune cells interact with tumor cells and how psychological scares and chemotherapy drugs work.The fundamental attributes of the model's 1IC β 1 +C signifies the existence of tumor cells that provoke the immune system's response.p 2 IC indicates the immune cells' decay rate due to tumor cells.d 1 I denotes the effector cells' death rate.d 2 IH designates the decay rate of effector cells due to chemo-drug.In the second equation, the (m 1 C (1 − k 1 C)) represents the tumor growth term.The term p 3 IC β 2 +C stands for the eradication of cancerous cells by the body's immune system.γ 1 CN indicates the tumor cells' decay rate due to effector cells.d 3 HC designates the decay rate of cancer cells due to chemo-drug.In the third equation, m 2 N (1 − k 2 N ) denotes the normal cells' growth.γ 2 CN represents the rate of disintegration of normal cells caused by the presence of tumor cells.In the last equation, ν is the infusion of chemotherapy drugs externally, and d 4 H is the decay rate of the chemo-drug.