Analytical and NumericalMethods to Study the MFPT and SR of a Stochastic Tumor-Immune Model

The Mean First-Passage Time (MFPT) and Stochastic Resonance (SR) of a stochastic tumor-immune model withnoise perturbation are discussed in this paper. Firstly, considering environmental perturbation, Gaussian whitenoise and Gaussian colored noise are introduced into a tumor growth model under immune surveillance. Asfollows, the long-time evolution of the tumor characterized by the Stationary Probability Density (SPD) and MFPTis obtained in theory on the basis of the Approximated Fokker-Planck Equation (AFPE). Herein the recurrenceof the tumor from the extinction state to the tumor-present state is more concerned in this paper. A moreefficient algorithmof Back-Propagation Neural Network (BPNN) is utilized in order to testify the correction of thetheoretical SPDandMFPT.With the existence of aweak signal, the functional relationship between Signal-to-NoiseRatio (SNR), noise intensities and correlation time is also studied. Numerical results show that both multiplicativeGaussian colored noise and additive Gaussian white noise can promote the extinction of the tumors, and themultiplicative Gaussian colored noise can lead to the resonance-like peak on MFPT curves, while the increasingintensity of the additiveGaussian white noise results in theminimum of MFPT. In addition, the correlation timesare negatively correlated with MFPT. As for the SNR, we find the intensities of both the Gaussian white noise andthe Gaussian colored noise, as well as their correlation intensity can induce SR. Especially, SNR is monotonouslyincreased in the case ofGaussian white noisewith the change of the correlation time.At last, the optimal parametersin BPNN structure are analyzed for MFPT from three aspects: the penalty factors, the number of neural networklayers and the number of nodes in each layer.

Optimal and Memristor-Based Control of A Nonlinear Fractional Tumor-Immune Model

In this article,the reduced differential transform method is introduced to solve the nonlinear fractional model of Tumor-Immune.The fractional derivatives are described in the Caputo sense.The solutions derived using this method are easy and very accurate.The model is given by its signal flow diagram.Moreover,a simulation of the system by the Simulink of MATLAB is given.The disease-free equilibrium and stability of the equilibrium point are calculated.Formulation of a fractional optimal control for the cancer model is calculated.In addition,to control the system,we propose a novel modification of its model.This modification is based on converting the model to a memristive one,which is a first time in the literature that such idea is used to control this type of diseases.Also,we study the system’s stability via the Lyapunov exponents and Poincare maps before and after control.Fractional order differential equations(FDEs)are commonly utilized to model systems that have memory,and exist in several physical phenomena,models in thermoelasticity field,and biological paradigms.FDEs have been utilized to model the realistic biphasic decline manner of elastic systems and infection of diseases with a slower rate of change.FDEs are more useful than integer-order in modeling sophisticated models that contain physical phenomena.

CONTROL STRATEGIES FOR A TUMOR-IMMUNE SYSTEM WITH IMPULSIVE DRUG DELIVERY UNDER A RANDOM ENVIRONMENT

This paper mainly studies the stochastic character of tumor growth in the presence of immune response and periodically pulsed chemotherapy.First,a stochastic impulsive model describing the interaction and competition among normal cells,tumor cells and immune cells under periodically pulsed chemotherapy is established.Then,sufficient conditions for the extinction,non-persistence in the mean,weak and strong persistence in the mean of tumor cells are obtained.Finally,numerical simulations are performed which not only verify the theoretical results derived but also reveal some specific features.The results show that the growth trend of tumor cells is significantly affected by the intensity of noise and the frequency and dose of drug deliveries.In clinical practice,doctors can reduce the randomness of the environment and increase the intensity of drug input to inhibit the proliferation and growth of tumor cells.

Dynamics analysis in a tumor-immune system with chemotherapy

An ordinary differential equation(ODE)model of tumor growth with the effect of tumor-immune interaction and chemotherapeutic drug is presented and studied.By analyzing the existence and stability of equilibrium points,the dynamic behavior of the system is discussed elaborately.The chaotic dynamics can be obtained in our model by equilibria analysis,which show the existence of chaos by calculating the Lyapunov exponents and the Lyapunov dimension of the system.Moreover,the action of the infusion rate of the chemotherapeutic drug on the resulting dynamics is investigated,which suggests that the application of chemotherapeutic drug can effectively control tumor growth.However,in the case of high-dose chemotherapeutic drug,chemotherapy-induced effector immune cells damage seriously,which may cause treatment failure.

Global dynamics of a tumor-immune model with an immune checkpoint inhibitor

In this paper,a mathematical ordinary differential tumor-immune model is proposed based on an immune checkpoint inhibitor,which is an innovative method for tumor immunotherapies.Two important factors in tumor-immune response are the programmed cell death protein 1(PD-1)and its ligand PD-L1.The model consists of three populations:tumor cells,activated T cells and anti-PD-1.By analyzing the dynamics of the model,it is found that there is always a unique tumor-free equilibrium and at most two tumor interior equilibria.The nonexistence of nontrivial positive periodic orbits is established by using the new Dulac function,and then a global dynamics of the model is obtained.The conclusions of our analysis show that increasing the possibility of T cells killing tumor cells(p),early detection of tumor cells,or the use of PD-1 inhibitors to activate T cells are effective in eliminating tumor cells.

Numerical Approximation of Hopf Bifurcation for Tumor-Immune System Competition Model with Two Delays

This paper is concernedwith theHopf bifurcation analysis of tumor-immune system competition model with two delays.First,we discuss the stability of state points with different kinds of delays.Then,a sufficient condition to the existence of the Hopf bifurcation is derived with parameters at different points.Furthermore,under this condition,the stability and direction of bifurcation are determined by applying the normal form method and the center manifold theory.Finally,a kind of Runge-Kutta methods is given out to simulate the periodic solutions numerically.At last,some numerical experiments are given to match well with the main conclusion of this paper.

SOX11 as a potential prognostic biomarker in hepatocellular carcinoma linked to immune infiltration and ferroptosis

Objective:SOX11 is expressed in numerous malignancies,including hepatocellular carcinomas(HCC),but its oncogenic function has not been elucidated.Here,we performed a comprehensive bioinformatics analysis of the Liver Hepatocellular Carcinoma(LIHC)dataset to investigate the function of SOX11 in tumorgenesis.Methods:SOX11 expression data from The Cancer Genome Atlas(TCGA)and Gene Expression Omnibus(GEO)were validated by immunohistochemistry(IHC).Co-expression,differential expression,and functional analyses utilized TCGA-LIHC,Timer 2.0,Metascape,GTEx,and LinkedOmics databases.Associations with immune infiltration,ferroptosis,and immune checkpoint genes were assessed.Genetic changes were explored via CBioPortal.Logistic regression,receiver operating characteristic curve(ROC),Kaplan-Meier analysis,and nomogram modeling evaluated associations with HCC clinicopathological features.SOX11’s impact on proliferation and migration was studied in HepG2 and HuH7 cell lines.Results:SOX11 was significantly elevated in HCC tumors compared to controls.SOX11-associated genes exhibited differential expression in pathways involving extracellular membrane ion channels.Significant associations were found between SOX11 levels,immune infiltration,ferroptosis,and immune checkpoint genes in HCC tissue.SOX11 levels correlated with HCC stage,histologic grade,and tumor status,and independently predicted overall and disease-specific survival.SOX11 expression effectively distinguished between tumor and normal liver tissue.Spearman correlations highlighted a significant relationship between SOX11 and ferroptosis-associated genes.Decreased SOX11 levels in HepG2 and HuH7 cells resulted in reduced proliferation and migration.Conclusions:SOX11 was found to represent a promising biomarker within HCC diagnosis and prognosis together with being a possible drug-target.

Tumor-intrinsic metabolic reprogramming and how it drives resistance to anti-PD-1/PD-L1 treatment

The development of immune checkpoint blockade (ICB) therapies has been instrumental in advancing the field of immunotherapy. Despite the prominence of these treatments, many patients exhibit primary or acquired resistance, rendering them ineffective. For example, anti-programmed cell death protein 1 (anti-PD-1)/anti-programmed cell death ligand 1 (anti-PD-L1) treatments are widely utilized across a range of cancer indications, but the response rate is only 10%-30%. As such, it is necessary for researchers to identify targets and develop drugs that can be used in combination with existing ICB therapies to overcome resistance. The intersection of cancer, metabolism, and the immune system has gained considerable traction in recent years as a way to comprehensively study the mechanisms that drive oncogenesis, immune evasion, and immunotherapy resistance. As a result, new research is continuously emerging in support of targeting metabolic pathways as an adjuvant to ICB to boost patient response and overcome resistance. Due to the plethora of studies in recent years highlighting this notion, this review will integrate the relevant articles that demonstrate how tumor-derived alterations in energy, amino acid, and lipid metabolism dysregulate anti-tumor immune responses and drive resistance to anti-PD-1/PD-L1 therapy.

Chaotic dynamics of a delayed tumor immune interaction model

Due to the unpredictable growth of tumor cells,the tumor-immune interactive dynamics continues to draw attention from both applied mathematicians and oncologists.Math-ematical modeling is a powerful tool to improve our understanding of the complicated biological system for tumor growth.With this goal,we report a mathematical model which describes how turmor cells evolve and survive the brief encounter with the immune system mediated by immune effector cells and host cells which includes discrete time delay.We analyze the basic mathematical properties of the considered model such as positivity of the system and the boundedness of the solutions.By analyzing the distri-bution of eigenvalucs,local stability analysis of the biologically feasible equilibria and the existence of Hopf bifurcation are obtained in which discrete time delay is used as a bifurcation parameter.Based on the normal form theory and center manifold theorem,we obtain explicit expressions to determine the direction of Hopf bifurcation and the stability of Hopf bifurcating periodic solutions.Numerical simulations are carried out to illustrate the rich dynamical behavior of the delayed tumor model.Our model simula-tions demonstrate that the delayed tumor model exhibits regular and irregular periodic oscillations or chaotic behaviors,which indicate the scenario of long-term tumor relapse.

The pulse vaccination effects in mammary carcinoma

Induction of antitumor immunity by vaccination is one of the major current immunother- apy strategies. We present a mathematical model of the competition between immune cells and mammary carcinogenesis under the effect of Triplex vaccine. The model describes both humoral and cell-mediated immune responses against cancer cells. The control of the cancer cells growth occurs through the application of the pulse vaccination. Here we determine the relationship between the strength of the vaccine and the time required to eradicate cancer cells, and we present some simulations to illustrate our theoretical results, namely, the total cancer cells depletion, which is influenced by competition occurs among the immune and cancer cells.