Mathematical Biosciences and Engineering, 2013, 10(3): 787-802. doi: 10.3934/mbe.2013.10.787

Primary: 49K15, 92C50; Secondary: 93C95.

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On optimal chemotherapy with a strongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth

1. Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Illinois, 62026-1653
2. Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Mo 63130

In this paper, a mathematical model for chemotherapy that takestumor immune-system interactions into account is considered for astrongly targeted agent. We use a classical model originallyformulated by Stepanova, but replace exponential tumor growth with ageneralised logistic growth model function depending on a parameter$\nu$. This growth function interpolates between a Gompertzian model(in the limit $\nu\rightarrow0$) and an exponential model (in thelimit $\nu\rightarrow\infty$). The dynamics is multi-stable andequilibria and their stability will be investigated depending on theparameter $\nu$. Except for small values of $\nu$, the system hasboth an asymptotically stable microscopic (benign) equilibrium pointand an asymptotically stable macroscopic (malignant) equilibriumpoint. The corresponding regions of attraction are separated by thestable manifold of a saddle. The optimal control problem of movingan initial condition that lies in the malignant region into thebenign region is formulated and the structure of optimal singularcontrols is determined.
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