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In this paper, a mathematical model of breast cancer governed by a system of ordinary differential equations in the presence of chemotherapy treatment and ketogenic diet is discussed. Several comprehensive mathematical analysis was carried out using varieties of analytical methods to study the stability of the breast cancer model. Also, sufficient conditions on parameter values to ensure cancer persistence in the absence of anti-cancer drugs ketogenic diet and cancer emission when anti-cancer drugs, immune-booster, ketogenic diet are included were established. Furthermore, optimal control theory is applied to find out the optimal drug adjustment as an input control of the system therapies to minimize the number of cancerous cells by considering different controlled combinations of administering the chemotherapy agent and ketogenic diet using the popular Pontryagin's Maximum Principle. Numerical simulations were presented to validate our theoretical results.

The energy distribution in the Kerr-Newman space-time is computed using the Møller energy-momentum complex. This agrees with the Komar mass for this space-time obtained by Cohen and de Felice. These results support the Cooperstock hypothesis. * E-mail: ssxulu@pan.uzulu.ac.za 1511 Mod. Phys. Lett. A 2000.15:1511-1517. Downloaded from www.worldscientific.com by UNIVERSITY OF CALIFORNIA @ DAVIS on 02/08/15. For personal use only.

We use the energy-momentum complexes of Landau and Lifshitz and Papapetrou to obtain the energy distribution in Melvin's magnetic universe. For this space-time we find that these definitions of energy give the same and convincing results. The energy distribution obtained here is the same as we obtained earlier for the same space-time using the energy-momentum complex of Einstein. These results uphold the usefulness of the energy-momentum complexes.