Mathematical Science

Permanent URI for this collection

https://hdl.handle.net/10530/903

Browse

Browsing Mathematical Science by Author "Matadi, M.B."

Now showing 1 - 1 of 1
  • Thumbnail Image
    Item
    Mathematical modeling for optimal control of breast cancer
    (University of Zululand, 2019) Oke, Segun Isaac; Matadi, M.B.; Xulu, S.S.
    Breast cancer, which often occurs in the inner lining of milk ducts, is the deadliest and most common form of invasive cancer among females according to a 2017 report of the World Health Organization. The purpose of this study was to develop a four compartmental mathematical model using a system of nonlinear Ordinary Di erential Equations (ODEs) which investigates the impact of anti-cancer drugs, ketogenic-diets and immune boosters on the dynamics of breast cancer. The study focused on the dynamical interaction of normal and tumor cells as well as the invasion of tumor cells during the metastasis stage of breast cancer. The systems of ODEs were analytically solved for the equilibria. Using the next generation matrix method, a threshold quantity called the treatment in- duced invasion reproduction number (R i ) was computed. Center manifold theory was used to investigate the possibility of the bifurcation analysis of R i being greater than unity. Using a suitable Lyapunov functions, the global stability of the tumor-free equilibrium was achieved in conjuction with LaSalle's invariance principle. Uncertainty and sensitivity analyses were performed on R i using Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coe cient (PRCC). R i was used as the response function while investigating the most signi cant parameters (such as: 1, 2, 1, d, and 1 ) that a ects disease progression and cell invasion. Optimal control theory was applied using the Pontryagins' Maximum Principle to investigate optimal strategies for controlling and eliminating tumor cells using time dependent controls such as u1(t) (anti-cancer drugs) and u2(t) (ketogenic diets). Numerical simulation results using a set of parameter values were provided to validate the analytical results. It was found that the tumor-free equilibrium points for ix breast cancer was locally asymptotically stable when the associated invasion reproduction number was less than unity and that it was otherwise unstable. The tumor-free equilibrium was found to be globally asymptotically stable if (Ri) < 1 . Sensitivity analysis showed that the natural death rate of normal cells has the most positive sensitivity index. However, increasing the death rate as a control measure is unreasonable biologically. The level of ketogenic diet rate was found to be most negatively sensitive to Ri. Therefore, the formulated model showed that reduction of the invasion reproduction number (R i ) below unity can be achieved by maintaining the level of ketogenic diet and by reducing tumor progression rate. It was shown from this study that the breast cancer model exhibited backward bifurcation with bifurcation parameter 1 which implies that the reduction R i below unity alone is not su cient to eradicate tumor cells from the body system while in the case of forward bifurcation, the reduction of R i above unity is su cient to eradicate tumor cells from the body system . The incremental cost-e ectiveness analysis of control strategies adapted in treating breast cancer has shown that the integration of ketogenic diet and anti-cancer drugs as intervention strategy is the most cost-e ective in ghting tumor cells. x
University of Zululand