Health & Biomedicine
Mammary carcinoma is a common type of cancer that appears mostly in women under a certain age and can be inherited to future generations . Biologist are exploring new treatments that can avoid the development of breast cancer in patients with hight risk of malignant cell mutations by experiment directly in mice. The genetically engineered mice (GEM) have proven extremely useful for studying breast cancer and have become the animal model for human breast cancer . Studies in vivo are useful because represents real dynamics of the tumor cells in presence of immunotherapy for some complex types of cancer, Fomchenko et al. have presented a model of brain tumors that can be applied in preclinical trials, Kwon et al. presents a recent study of a possible model that describe the biological dynamics of the lung cancer by validating the results in mouse, Bianca et al. presents the first nonlinear model of the carcinoma mammary cancer. Mathematical models are tools used in engineering and science to predict functional relationships between certain input and output variables. Mathematical modeling thus provide feedback to biologists on the suitability of experimental data, and they in turn can help improve and refinethe mathematical models. Thus the mathematical modeling provides the opportunity of improving both the understanding and prediction of biological phenomena .
Using the method of localization compact invariant sets (LOCI)  in biological models ,, we define a region that contains the whole complex dynamics of the model where the upper bound represents the maximal concentration of population that can exist and also inside the region we can analyzed equilibrium points or define sufficient conditions of attractivity. In this paper is analyzed at the moment one case of study for the nonlinear tenth dimensional system describing the dynamics between the immune system and mammary carcinoma antigen by Bianca et al., where the variables Xi2R5+;0;i= 1;2;5;8;9. The model is shown below:
The variable X1 represents the number of injected vaccine cells,X2 is the number of P-185 tumor associated antigens,X5the number of interleukin 12,X8is the number of cancer cells andX9the number of activated cytotoxic cells, the parameters are mention in  and all have positive values. In this case of study the vaccine can directly eradicate all the cancer cells and any antigen associate to the tumor due to the combination of cytotoxic cells and immunotherapy interleukin 12. Local stability is made by numerical simulation as well as finding equilibrium points. Has a result of this analysis we found four equilibrium points were three of them are negative equilibrium points. In this work are not consider the negative equilibrium points because they lack of biological sense, which leads to only one equilibrium point that is unique and locally stable. The Lyapunov function is shown below:
Upper bounds for each of the variable stated are presented. The intersection of all the upper bounds provides the localization domain of all compact invariant sets of the model.
For one case of study the model (1) which is a reduced dynamic model of Bianca  we present local stability by numerical simulation and using the Lyapunov function 2 we ensure global asymptotic stability of the equilibrium point. The upper bounds define the region that contains the dynamic of the model including the unique equilibrium point.
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