Dynamical Behavior of a Cancer Growth Model with Chemotherapy and Boosting of the Immune System

Abstract

In this study, we set up and analyze a cancer growth model that integrates a chemotherapy drug with the impact of vitamins in boosting and strengthening the immune system. The aim of this study is to determine the minimal amount of treatment required to eliminate cancer, which will help to reduce harm to patients. It is assumed that vitamins come from organic foods and beverages. The chemotherapy drug is added to delay and eliminate tumor cell growth and division. To that end, we suggest the tumor-immune model, composed of the interaction of tumor and immune cells, which is composed of two ordinary differential equations. The model’s fundamental mathematical properties, such as positivity, boundedness, and equilibrium existence, are examined. The equilibrium points’ asymptotic stability is analyzed using linear stability. Then, global stability and persistence are investigated using the Lyapunov strategy. The occurrence of bifurcations of the model, such as of trans-critical or Hopf type, is also explored. Numerical simulations are used to verify the theoretical analysis. The Runge–Kutta method of fourth order is used in the simulation of the model. The analytical study and simulation findings show that the immune system is boosted by regular vitamin consumption, inhibiting the growth of tumor cells. Further, the chemotherapy drug contributes to the control of tumor cell progression. Vitamin intake and chemotherapy are treated both individually and in combination, and in all situations, the minimal level required to eliminate the cancer is determined.

Stability analysis of the depletion of dissolved oxygen for the Phytoplankton-Zooplankton model in an aquatic environment

“…Hence, condition (12) gives ( ) ( ) ( ) ( ) ( ) ( ) That means the Hop bifurcation has occurred at then the proof is completed. From Theorem 3, the stability condition of the stable limit cycle in is presented using the coefficient of curvature of the limit cycle.…”

“…There are many studies on this important issue, for instance, Mondal et al [8] studied the coupled plankton-oxygen dynamics in the ocean that are affected by a low oxygen production rate which can result in oxygen depletion and species extinction. Furthermore, the primary objective of the study of theoretical ecology is to identify the various dynamical mechanisms underlying interactions between prey and predator [9][10][11][12][13][14][15]. The relationship between phytoplankton and zooplankton is an example of a predator-prey interaction that reveals numerous aspects of marine ecology.…”

“…Hence, is a Lyapunov function. Therefore, CEP is globally asymptotically stable in if , and are controlled by condition (12).…”

Personalized Plasma Medicine for Cancer: Transforming Treatment Strategies with Mathematical Modeling and Machine Learning Approaches

“…It complements cancer research by providing valuable quantitative predictions to unravel the complexity of cancer and open new avenues for developing effective treatments [25]. Numerous mathematical models, often rooted in ordinary differential equations (ODEs), have explored various aspects of cancer research, including treatment strategies [26][27][28][29][30][31][32][33][34], immune responses [35][36][37][38][39][40][41], treatment sensitivity and resistance [42], the effects of treatment combinations [43][44][45][46][47][48][49], habitat dynamics [50], tumor heterogeneity [51,52], and treatment optimization [53]. Their collective contribution lies in furnishing predictions for optimal dosing, treatment regimens, and scheduling, all while minimizing adverse side effects and maximizing therapeutic gains.…”

The role of noise in the tumour-immune interactions reveals memoryless shifting response

“…Many mathematical models have been introduced to investigate the development of the tumour in the host [3][4][5][6][7][8]. These models, depending on the level of their complexity, can predict the tumour cells proliferation.…”