Optimal solution of the fractional order breast cancer competition model

A fractional-order mathematical model for lung cancer incorporating integrated therapeutic approaches

This paper addresses the dynamics of lung cancer by employing a fractional-order mathematical model that investigates the combined therapy of surgery and immunotherapy. The significance of this study lies in its exploration of the effects of surgery and immunotherapy on tumor growth rate and the immune response to cancer cells. To optimize the treatment dosage based on tumor response, a feedback control system is designed using control theory, and Pontryagin's Maximum Principle is utilized to derive the necessary conditions for optimality. The results reveal that the reproduction number [Formula: see text] is 2.6, indicating that a lung cancer cell would generate 2.6 new cancer cells during its lifetime. The reproduction coefficient [Formula: see text] is 0.22, signifying that cancer cells divide at a rate that is 0.22 times that of normal cells. The simulations demonstrate that the combined therapy approach yields significantly improved patient outcomes compared to either treatment alone. Furthermore, the analysis highlights the sensitivity of the steady-state solution to variations in [Formula: see text] (the rate of division of cancer stem cells) and [Formula: see text] (the rate of differentiation of cancer stem cells into progenitor cells). This research offers clinicians a valuable tool for developing personalized treatment plans for lung cancer patients, incorporating individual patient factors and tumor characteristics. The novelty of this work lies in its integration of surgery, immunotherapy, and control theory, extending beyond previous efforts in the literature.

Control and Anticontrol of chaos in Fractional-order models of Diabetes, HIV, Dengue, Migraine, Parkinson's and Ebola Virus diseases

This work proposes new fractional-order (FO) models of six chaotic diseases whose fractional dynamics have not been studied so far in literature. Secondly, design and analysis of suitable controllers to control chaos where present, and that of anticontrollers to generate chaos where absent, for these newly proposed FO models of diseases, are put forward. The proposed controllers and anticontrollers address the problem of the health hazards arising from the dysfunctionalities due to the impact of chaos in these biological models. Controllers to supress chaos in four diseases, namely, FO Diabetes Mellitus, FO Human Immunodeficiency Virus (HIV), FO Ebola Virus and FO Dengue models are designed by Back-stepping, Adaptive Feedback and Sliding Mode Control strategies, whereas anticontrollers to introduce chaos in diseases, namely, FO Parkinson's illness and FO Migraine models, are carried out by Linear State Feedback, Single State Sinusoidal Feedback and Sliding Mode Anticontrol strategies. The equilibrium points, eigenvalues and Lyapunov Exponents of the FO disease models are evaluated and indicate the significance of chaos in them and necessitate upon the requirement of controllers and anticontrollers accordingly. The simulation results in terms of bifurcation diagrams, time series plots and phase portraits confirm the successful accomplishment of the control objectives.