Modeling a Tumor Growth with Piecewise Constant Arguments

This study is based on an early brain tumor growth that is modeled as a hybrid system such as (A): dx(t)/dt = x(t){r(1 - alpha x(t) - beta(0)x((sic)t(sic)) - beta(1)x((sic)t - 1(sic))) + gamma(1)x((sic)t(sic)) + gamma(2)x((sic)t - 1(sic))}, where the parameters alpha, beta(0), beta(1), and r denote positive numbers, gamma(1) and gamma(2) are negative numbers and (sic)t(sic) is the integer part of t is an element of [0,infinity). Equation (A) explains a brain tumor growth, where gamma(1) is embedded to show the drug effect on the tumor and gamma(2) is a rate that causes a negative effect by the immune system on the tumor population. Using (A), we have constructed two models of a tumor growth: one is (A) and the other one is a population model at low density by incorporating an Allee function to (A) at time t. To consider the global behavior of (A), we investigate the discrete solutions of (A). Examination of the characterization of the stability shows that increase of the population growth rate decreases the local stability of the positive equilibrium point of (A). The simulations give a detailed description of the behavior of solutions of (A) with and without Allee effect.