Sensitivity and uncertainty analysis of a simplified Kirschner-Panetta model for immunotherapy of tumor-immune interaction

In this study, we have simplified the Kirschner-Panetta model on the interaction of tumor cells and effector cells by considering linear growth term as opposed to logistic growth term used by Kirschner and Panetta. We have done a comprehensive mathematical analysis and established the existence of positive equilibrium. In addition, a fixed point bifurcation is investigated using the rate of spread of tumor as a varying parameter, suggesting that backward bifurcation can occur under reasonable choice of parameters. However, bistable dynamics is unlikely to happen in this case, which implies that the strategies that could reduce the rate of spread of tumor are the most influential to cancer treatment. Through mathematical deduction and numerical simulation, an elaborate uncertainty and sensitivity analysis of the rate of spread of tumor Rs$R_{s}$ is performed. The distribution of Rs$R_{s}$ is derived, and the sensitivity of the magnitude of Rs$R_{s}$ to the uncertainty in estimating values of input parameters is assessed. The results indicate that the external source of effector cells and its death rate are influential in the rate of spread of tumor.