Analyzing the Dynamics of Breast Cancer Progression with Fractional Modeling and Simulation: Effects of Estrogen and Unhealthy Diets

A number of studies in the past have shown that an unhealthy diet can lead to tumors progressing more rapidly, leading to a higher mortality rate due to their progression. In addition, it has been observed that hormonal imbalances within the human body are a strong factor that contributes to the proliferation rate of different types of cancer. Among the various hormones which make up the class of estrogens, this one plays a very important role in the proper development of a woman’s sexual and reproductive life. In addition, studies have been conducted that have demonstrated that estrogen and testosterone levels have a direct relationship with a woman’s risk of developing breast cancer. Exposure to these hormones over a long period of time and/or at high levels has been linked to an increased risk of breast cancer in women. In this study, estrogen levels in females are integrated with unhealthy diet effects to study how these levels affect cancer progression in females. The levels of estrogen are fixed to explore the other cell densities. It has been concluded that an unhealthy diet can significantly impact a patient’s immune system if they consume a lot of estrogens. A mathematical model is presented in this paper to analyze how breast cancer affects the dynamics of breast tissue under the influence of estrogen levels and an unhealthy diet. This paper presents the results of a study that uses fractional Atangana–Baleanu (AB), Caputo–Fabrizio (CF) and Caputo operators to analyze the differential equations for breast cancer. To investigate the uniqueness and the existence of solutions, we use the fixed-point postulate in our study. This research was carried out using the Adams–Bashforth and Toufik–Atangana techniques for solving the generalized nonlinear system, which contains the Atangana–Baleanu, Caputo–Fabrizio and Caputo noninteger operators. Using the noninteger breast cancer model as a reference, numerical simulations have been conducted to demonstrate the applicability and effectiveness of these methods.