The influence of time delay in a chaotic cancer model

Intelligent solution predictive networks for non-linear tumor-immune delayed model

In this article, we analyze the dynamics of the non-linear tumor-immune delayed (TID) model illustrating the interaction among tumor cells and the immune system (cytotoxic T lymphocytes, T helper cells), where the delays portray the times required for molecule formation, cell growth, segregation, and transportation, among other factors by exploiting the knacks of soft computing paradigm utilizing neural networks with back propagation Levenberg Marquardt approach (NNLMA). The governing differential delayed system of non-linear TID, which comprised the densities of the tumor population, cytotoxic T lymphocytes and T helper cells, is represented by non-linear delay ordinary differential equations with three classes. The baseline data is formulated by exploiting the explicit Runge-Kutta method (RKM) by diverting the transmutation rate of Tc to Th of the Tc population, transmutation rate of Tc to Th of the Th population, eradication of tumor cells through Tc cells, eradication of tumor cells through Th cells, Tc cells' natural mortality rate, Th cells' natural mortality rate as well as time delay. The approximated solution of the non-linear TID model is determined by randomly subdividing the formulated data samples for training, testing, as well as validation sets in the network formulation and learning procedures. The strength, reliability, and efficacy of the designed NNLMA for solving non-linear TID model are endorsed by small/negligible absolute errors, error histogram studies, mean squared errors based convergence and close to optimal modeling index for regression measurements.

Dissecting cell fate dynamics in pediatric glioblastoma through the lens of complex systems and cellular cybernetics

Cancers are complex dynamic ecosystems. Reductionist approaches to science are inadequate in characterizing their self-organized patterns and collective emergent behaviors. Since current approaches to single-cell analysis in cancer systems rely primarily on single time-point multiomics, many of the temporal features and causal adaptive behaviors in cancer dynamics are vastly ignored. As such, tools and concepts from the interdisciplinary paradigm of complex systems theory are introduced herein to decode the cellular cybernetics of cancer differentiation dynamics and behavioral patterns. An intuition for the attractors and complex networks underlying cancer processes such as cell fate decision-making, multiscale pattern formation systems, and epigenetic state-transitions is developed. The applications of complex systems physics in paving targeted therapies and causal pattern discovery in precision oncology are discussed. Pediatric high-grade gliomas are discussed as a model-system to demonstrate that cancers are complex adaptive systems, in which the emergence and selection of heterogeneous cellular states and phenotypic plasticity are driven by complex multiscale network dynamics. In specific, pediatric glioblastoma (GBM) is used as a proof-of-concept model to illustrate the applications of the complex systems framework in understanding GBM cell fate decisions and decoding their adaptive cellular dynamics. The scope of these tools in forecasting cancer cell fate dynamics in the emerging field of computational oncology and patient-centered systems medicine is highlighted.

A Review of Mathematical and Computational Methods in Cancer Dynamics

Cancers are complex adaptive diseases regulated by the nonlinear feedback systems between genetic instabilities, environmental signals, cellular protein flows, and gene regulatory networks. Understanding the cybernetics of cancer requires the integration of information dynamics across multidimensional spatiotemporal scales, including genetic, transcriptional, metabolic, proteomic, epigenetic, and multi-cellular networks. However, the time-series analysis of these complex networks remains vastly absent in cancer research. With longitudinal screening and time-series analysis of cellular dynamics, universally observed causal patterns pertaining to dynamical systems, may self-organize in the signaling or gene expression state-space of cancer triggering processes. A class of these patterns, strange attractors, may be mathematical biomarkers of cancer progression. The emergence of intracellular chaos and chaotic cell population dynamics remains a new paradigm in systems medicine. As such, chaotic and complex dynamics are discussed as mathematical hallmarks of cancer cell fate dynamics herein. Given the assumption that time-resolved single-cell datasets are made available, a survey of interdisciplinary tools and algorithms from complexity theory, are hereby reviewed to investigate critical phenomena and chaotic dynamics in cancer ecosystems. To conclude, the perspective cultivates an intuition for computational systems oncology in terms of nonlinear dynamics, information theory, inverse problems, and complexity. We highlight the limitations we see in the area of statistical machine learning but the opportunity at combining it with the symbolic computational power offered by the mathematical tools explored.

Is fractional-order chaos theory the new tool to model chaotic pandemics as Covid-19?

The deadly outbreak of the second wave of Covid-19, especially in worst hit lower-middle-income countries like India, and the drastic rise of another growing epidemic of Mucormycosis, call for an efficient mathematical tool to model pandemics, analyse their course of outbreak and help in adopting quicker control strategies to converge to an infection-free equilibrium. This review paper on prominent pandemics reveals that their dispersion is chaotic in nature having long-range memory effects and features which the existing integer-order models fail to capture. This paper thus puts forward the use of fractional-order (FO) chaos theory that has memory capacity and hereditary properties, as a potential tool to model the pandemics with more accuracy and closeness to their real physical dynamics. We investigate eight FO models of Bombay plague, Cancer and Covid-19 pandemics through phase portraits, time series, Lyapunov exponents and bifurcation analysis. FO controllers (FOCs) on the concepts of fuzzy logic, adaptive sliding mode and active backstepping control are designed to stabilise chaos. Also, FOCs based on adaptive sliding mode and active backstepping synchronisation are designed to synchronise a chaotic epidemic with a non-chaotic one, to mitigate the unpredictability due to chaos during transmission. It is found that severity and complexity of the models increase as the memory fades, indicating that FO can be used as a crucial parameter to analyse the progression of a pandemic. To sum it up, this paper will help researchers to have an overview of using fractional calculus in modelling pandemics more precisely and also to approximate, choose, stabilise and synchronise the chaos control parameter that will eliminate the extreme sensitivity and irregularity of the models.

Bifurcation and chaos analysis of tumor growth

In this paper, a dynamic model given by three-dimensional ordinary differential equations is studied to determine how the dynamics of tumor growth is controlled by some key parameters. By varying the competition coefficient between healthy host cells and tumor cells, a Hopf bifurcation occurs in this system, leading to the creation of a stable limit cycle. Through numerical analysis of the continuity of this limit cycle, we find that the cascade of period-doubling bifurcations leads to the generation of a chaotic attractor. Moreover, the region of attractors is shown in the parameter space. Numerical simulations, bifurcation diagrams, Lyapunov exponent graph and phase portraits permit to highlight the rich and complex phenomena presented by the model.

Spatiotemporal dynamics of a glioma immune interaction model

We report a mathematical model which depicts the spatiotemporal dynamics of glioma cells, macrophages, cytotoxic-T-lymphocytes, immuno-suppressive cytokine TGF-β and immuno-stimulatory cytokine IFN-γ through a system of five coupled reaction-diffusion equations. We performed local stability analysis of the biologically based mathematical model for the growth of glioma cell population and their environment. The presented stability analysis of the model system demonstrates that the temporally stable positive interior steady state remains stable under the small inhomogeneous spatiotemporal perturbations. The irregular spatiotemporal dynamics of gliomas, macrophages and cytotoxic T-lymphocytes are discussed extensively and some numerical simulations are presented. Performed some numerical simulations in both one and two dimensional spaces. The occurrence of heterogeneous pattern formation of the system has both biological and mathematical implications and the concepts of glioma cell progression and invasion are considered. Simulation of the model shows that by increasing the value of time, the glioma cell population, macrophages and cytotoxic-T-lymphocytes spread throughout the domain.

Various bifurcations in the development of stem cells

Cell development from an undifferentiated stem cell to a differentiated one is essential in forming an organism. In this paper, various bifurcations of a stem cell during this process are studied using a model based on Furusawa and Kaneko's hypothesis. Furusawa and Kaneko's hypothesis tells that the gene expression of stem cells is chaotic. By developing to a differentiated cell, the gene expression in more order, which is the cause of losing pluripotency. In this model, the chaotic dynamics of gene expression in the stem cells become ordered during the developments. Various patterns and bifurcation points can be seen during development. The bifurcation points and their predictions during the process of cell development are studied in this paper. Some well-known critical slowing down indicators are used to show the variations of slowness during the cell's development and predict the bifurcation points. It is vital since the unexpected changes of the state can cause a disaster. All of the indicators have a proper trend by approaching the bifurcation points and faring away.

Mathematical analysis of a cancer model with time-delay in tumor-immune interaction and stimulation processes

In this study, we discuss a cancer model considering discrete time-delay in tumor-immune interaction and stimulation processes. This study aims to analyze and observe the dynamics of the model along with variation of vital parameters and the delay effect on anti-tumor immune responses. We obtain sufficient conditions for the existence of equilibrium points and their stability. Existence of Hopf bifurcation at co-axial equilibrium is investigated. The stability of bifurcating periodic solutions is discussed, and the time length for which the solutions preserve the stability is estimated. Furthermore, we have derived the conditions for the direction of bifurcating periodic solutions. Theoretically, it was observed that the system undergoes different states if we vary the system's parameters. Some numerical simulations are presented to verify the obtained mathematical results.

Stability analysis of a fractional-order cancer model with chaotic dynamics

In this paper, we develop a three-dimensional fractional-order cancer model. The proposed model involves the interaction among tumor cells, healthy tissue cells and activated effector cells. The detailed analysis of the equilibrium points is studied. Also, the existence and uniqueness of the solution are investigated. The fractional derivative is considered in the Caputo sense. Numerical simulations are performed to illustrate the effectiveness of the obtained theoretical results. The outcome of the study reveals that the order of the fractional derivative has a significant effect on the dynamic process. Further, the calculated Lyapunov exponents give the existence of chaotic behavior of the proposed model. Also, it is observed from the obtained results that decrease in fractional-order [Formula: see text] increases the chaotic behavior of the model.

A review of dynamical systems approaches for the detection of chaotic attractors in cancer networks

Cancers are complex dynamical systems. They remain the leading cause of disease-related pediatric mortality in North America. To overcome this burden, we must decipher the state-space attractor dynamics of gene expression patterns and protein oscillations orchestrated by cancer stemness networks. The review provides an overview of dynamical systems theory to steer cancer research in pattern science. While most of our current tools in network medicine rely on statistical correlation methods, causality inference remains primitively developed. As such, a survey of attractor reconstruction methods and machine algorithms for the detection of causal structures applicable in experimentally derived time series cancer datasets is presented. A toolbox of complex systems approaches are discussed for reconstructing the signaling state space of cancer networks, interpreting causal relationships in their time series gene expression patterns, and assisting clinical decision making in computational oncology. As a proof of concept, the applicability of some algorithms are demonstrated on pediatric brain cancer datasets and the requirement of their time series analysis is highlighted.

Exploring dynamical complexity in a time-delayed tumor-immune model

The analysis of dynamical complexity in nonlinear phenomena is an effective tool to quantify the level of their structural disorder. In particular, a mathematical model of tumor-immune interactions can provide insight into cancer biology. Here, we present and explore the aspects of dynamical complexity, exhibited by a time-delayed tumor-immune model that describes the proliferation and survival of tumor cells under immune surveillance, governed by activated immune-effector cells, host cells, and concentrated interleukin-2. We show that by employing bifurcation analyses in different parametric regimes and the 0-1 test for chaoticity, the onset of chaos in the system can be predicted and also manifested by the emergence of multi-periodicity. This is further verified by studying one- and two-parameter bifurcation diagrams for different dynamical regimes of the system. Furthermore, we quantify the asymptotic behavior of the system by means of weighted recurrence entropy. This helps us to identify a resemblance between its dynamics and emergence of complexity. We find that the complexity in the model might indicate the phenomena of long-term cancer relapse, which provides evidence that incorporating time-delay in the effect of interleukin in the tumor model enhances remarkably the dynamical complexity of the tumor-immune interplay.

Cancer: A turbulence problem

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Application of nonlinear dynamics to cancer ecosystems.

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Chemical turbulence and strange attractor models in tumor growth, invasion and pattern formation are investigated.

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Computational algorithms for detecting such structures are proposed.

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Complex systems applications to cancer dynamics.

Cancers are complex, adaptive ecosystems. They remain the leading cause of disease-related death among children in North America. As we approach computational oncology and Deep Learning Healthcare, our mathematical models of cancer dynamics must be revised. Recent findings support the perspective that cancer-microenvironment interactions may consist of chaotic gene expressions and turbulent protein flows during pattern formation. As such, cancer pattern formation, protein-folding and metastatic invasion are discussed herein as processes driven by chemical turbulence within the framework of complex systems theory. To conclude, cancer stem cells are presented as strange attractors of the Waddington landscape.

On the modeling of the interaction between tumor growth and the immune system using some new fractional and fractional-fractal operators

Humans are always exposed to the threat of infectious diseases. It has been proven that there is a direct link between the strength or weakness of the immune system and the spread of infectious diseases such as tuberculosis, hepatitis, AIDS, and Covid-19 as soon as the immune system has no the power to fight infections and infectious diseases. Moreover, it has been proven that mathematical modeling is a great tool to accurately describe complex biological phenomena. In the recent literature, we can easily find that these effective tools provide important contributions to our understanding and analysis of such problems such as tumor growth. This is indeed one of the main reasons for the need to study computational models of how the immune system interacts with other factors involved. To this end, in this paper, we present some new approximate solutions to a computational formulation that models the interaction between tumor growth and the immune system with several fractional and fractal operators. The operators used in this model are the Liouville-Caputo, Caputo-Fabrizio, and Atangana-Baleanu-Caputo in both fractional and fractal-fractional senses. The existence and uniqueness of the solution in each of these cases is also verified. To complete our analysis, we include numerous numerical simulations to show the behavior of tumors. These diagrams help us explain mathematical results and better describe related biological concepts. In many cases the approximate results obtained have a chaotic structure, which justifies the complexity of unpredictable and uncontrollable behavior of cancerous tumors. As a result, the newly implemented operators certainly open new research windows in further computational models arising in the modeling of different diseases. It is confirmed that similar problems in the field can be also be modeled by the approaches employed in this paper.

The Effect of a Non-Local Fractional Operator in an Asymmetrical Glucose-Insulin Regulatory System: Analysis, Synchronization and Electronic Implementation

For studying biological conditions with higher precision, the memory characteristics defined by the fractional-order versions of living dynamical systems have been pointed out as a meaningful approach. Therefore, we analyze the dynamics of a glucose-insulin regulatory system by applying a non-local fractional operator in order to represent the memory of the underlying system, and whose state-variables define the population densities of insulin, glucose, and β-cells, respectively. We focus mainly on four parameters that are associated with different disorders (type 1 and type 2 diabetes mellitus, hypoglycemia, and hyperinsulinemia) to determine their observation ranges as a relation to the fractional-order. Like many preceding works in biosystems, the resulting analysis showed chaotic behaviors related to the fractional-order and system parameters. Subsequently, we propose an active control scheme for forcing the chaotic regime (an illness) to follow a periodic oscillatory state, i.e., a disorder-free equilibrium. Finally, we also present the electronic realization of the fractional glucose-insulin regulatory model to prove the conceptual findings.

A New ODE-Based Model for Tumor Cells and Immune System Competition

Changes in diet are heavily associated with high mortality rates in several types of cancer. In this paper, a new mathematical model of tumor cells growth is established to dynamically demonstrate the effects of abnormal cell progression on the cells affected by the tumor in terms of the immune system’s functionality and normal cells’ dynamic growth. This model is called the normal-tumor-immune-unhealthy diet model (NTIUNHDM) and governed by a system of ordinary differential equations. In the NTIUNHDM, there are three main populations normal cells, tumor cell and immune cells. The model is discussed analytically and numerically by utilizing a fourth-order Runge–Kutta method. The dynamic behavior of the NTIUNHDM is discussed by analyzing the stability of the system at various equilibrium points and the Mathematica software is used to simulate the model. From analysis and simulation of the NTIUNHDM, it can be deduced that instability of the response stage, due to a weak immune system, is classified as one of the main reasons for the coexistence of abnormal cells and normal cells. Additionally, it is obvious that the NTIUNHDM has only one stable case when abnormal cells begin progressing into early stages of tumor cells such that the immune cells are generated once. Thus, early boosting of the immune system might contribute to reducing the risk of cancer.

Mathematical analysis of a time-delayed model on brucellosis transmission with disease testing information

Testing–culling is one of the important prevention and control measures considered in the study of animal infectious diseases. However, the process of finding infected animals (animal testing) is still not well studied through the kinetic model. In this paper, based on the characteristics of animal testing, a time-delayed model on brucellosis transmission is established. Under the general hypothesis of biological significance, the existence and stability of equilibria are first investigated. The results find that the global stability of equilibria depends on the basic reproduction number [Formula: see text] without the information delay: if [Formula: see text], the disease dies out; if [Formula: see text], the endemic equilibrium exists and the disease persists. Next, the impact of information delay on the dynamics of the model is analyzed and Hopf bifurcation is found in the established model when the information delay is greater than a critical value. Finally, the theoretical results are then further explained through numerical analysis and the significance of these results for the development of risk management measures is elaborated.

Investigation of Early Warning Indexes in a Three-Dimensional Chaotic System with Zero Eigenvalues

A rare three-dimensional chaotic system with all eigenvalues equal to zero is proposed, and its dynamical properties are investigated. The chaotic system has one equilibrium point at the origin. Numerical analysis shows that the equilibrium point is unstable. Bifurcation analysis of the system shows various dynamics in a period-doubling route to chaos. We highlight that from the evaluation of the entropy, bifurcation points can be predicted by identifying early warning signals. In this manner, bifurcation points of the system are analyzed using Shannon and Kolmogorov-Sinai entropy. The results are compared with Lyapunov exponents.

Chaos in a nonautonomous eco-epidemiological model with delay

In this paper, we propose and analyze a nonautonomous predator-prey model with disease in prey, and a discrete time delay for the incubation period in disease transmission. Employing the theory of differential inequalities, we find sufficient conditions for the permanence of the system. Further, we use Lyapunov's functional method to obtain sufficient conditions for global asymptotic stability of the system. We observe that the permanence of the system is unaffected due to presence of incubation delay. However, incubation delay affects the global stability of the positive periodic solution of the system. To reinforce the analytical results and to get more insight into the system's behavior, we perform some numerical simulations of the autonomous and nonautonomous systems with and without time delay. We observe that for the gradual increase in the magnitude of incubation delay, the autonomous system develops limit cycle oscillation through a Hopf-bifurcation while the corresponding nonautonomous system shows chaotic dynamics through quasi-periodic oscillations. We apply basic tools of non-linear dynamics such as Poincaré section and maximum Lyapunov exponent to confirm the chaotic behavior of the system.

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. Mathematical 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 tumor 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 distribution of eigenvalues, 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 simulations demonstrate that the delayed tumor model exhibits regular and irregular periodic oscillations or chaotic behaviors, which indicate the scenario of long-term tumor relapse.

Dynamical analysis of tumor-immune-help T cells system

In this paper, we construct a mathematical model to investigate the interaction between the tumor cells, the immune cells and the helper T cells (HTCs). We perform mathematical analysis to reveal the stability of the equilibria of the model. In our model, the HTCs are stimulated by the identification of the presence of tumor antigens. Our investigation implies that the presence of tumor antigens may inhibit the existence of high steady state of tumor cells, which leads to the elimination of the bistable behavior of the tumor-immune system, i.e. the equilibrium corresponding to the high steady state of tumor cells is destabilized. Choosing immune intensity [Formula: see text] as bifurcation parameter, there exists saddle-node bifurcation. Besides, there exists a critical value [Formula: see text], at which a Hopf bifurcation occurs. The stability and direction of Hopf bifurcation are discussed.

A STRATEGY OF OPTIMAL EFFICACY OF T11 TARGET STRUCTURE IN THE TREATMENT OF BRAIN TUMOR

We report a mathematical model depicting gliomas and immune system interactions by considering the role of immunotherapeutic drug T11 target structure (T11TS). The mathematical model comprises a system of coupled nonlinear ordinary differential equations involving glioma cells, macrophages, activated cytotoxic T-lymphocytes (CTLs), immunosuppressive cytokine transforming growth factor-[Formula: see text] (TGF-[Formula: see text]), immunostimulatory cytokine interferon-[Formula: see text] (IFN-[Formula: see text]) and the concentrations of immunotherapeutic agent T11TS. For the better understanding of the circumstances under which the gliomas can be eradicated from a patient, we use optimal control strategy. We design the objective functional by considering the biomedical goal, which minimizes the glioma burden and maximizes the macrophages and activated CTLs. The existence and the characterization for the optimal control are established. The uniqueness of the quadratic optimal control problem is also analyzed. We demonstrate numerically that the optimal treatment strategies using T11TS reduce the glioma burden and increase the cell count of activated CTLs and macrophages.

Bistable Firing Pattern in a Neural Network Model

Excessively high, neural synchronization has been associated with epileptic seizures, one of the most common brain diseases worldwide. A better understanding of neural synchronization mechanisms can thus help control or even treat epilepsy. In this paper, we study neural synchronization in a random network where nodes are neurons with excitatory and inhibitory synapses, and neural activity for each node is provided by the adaptive exponential integrate-and-fire model. In this framework, we verify that the decrease in the influence of inhibition can generate synchronization originating from a pattern of desynchronized spikes. The transition from desynchronous spikes to synchronous bursts of activity, induced by varying the synaptic coupling, emerges in a hysteresis loop due to bistability where abnormal (excessively high synchronous) regimes exist. We verify that, for parameters in the bistability regime, a square current pulse can trigger excessively high (abnormal) synchronization, a process that can reproduce features of epileptic seizures. Then, we show that it is possible to suppress such abnormal synchronization by applying a small-amplitude external current on > 10% of the neurons in the network. Our results demonstrate that external electrical stimulation not only can trigger synchronous behavior, but more importantly, it can be used as a means to reduce abnormal synchronization and thus, control or treat effectively epileptic seizures.