Optimal control of the dynamics of a tumor growth model with Hollings’ type-II functional response

Abstract

In this paper, a simple predator–prey type model for interaction of immune system (consisting of resting and hunting cells) with tumor cells using Hollings’ type-II functional response is considered. It is found that the model system has no positive equilibrium if the half-saturation population of resting cells is greater than a critical value. In case of violation of this condition, model system mostly has unique positive equilibrium but when the half-saturation population of cancer cells is smaller than one, it may have three positive equilibriums. Local stability of the feasible equilibriums is studied. These results suggest that a positive equilibrium can be locally asymptotically stable or unstable based on the system parameter values. In case of unstable positive equilibrium (equilibriums), long-time sustenance of immune system with tumor in terms of limit cycle through Hopf bifurcation is observed with half-saturation population of resting cells as bifurcation parameter. The necessary optimal control inputs that make the unstable equilibriums of the model system asymptotically stable and minimize the required performance measure are given as nonlinear functions of the system densities.