Effects of delay in a biological environment subject to tumor dynamics

In the present work, we perform a study investigating a generic model of tumor growth with a delay distribution in the proliferation of tumor-stimulating effectors using combinations of analytical and numerical methods. We examine two borderline cases of the distribution: the first limit case is the Dirac distribution, leading to a model with constant delay and the second limit case is the exponential distribution leading to a model with an additional equation. The main objective is to assess the effect of delays in the response of the immune system on the dynamic stability of interaction between tumor, immune and host cells. Analytical and numerical investigations reveal that in the absence of delay, the stationary states of the two models can be stable or unstable for all the parameters used. In the case of constant delay, the analysis focuses on the stability switch with increasing delay. We show using the generalized Sturm criterion that the space of the parameters of concern is divided into four regions determined by a sequence of discrimination and the Routh-Hurwitz conditions: the system can undergo no stability switch and remain unstable regardless of the delay or undergo exactly a stability switch causing the coexisting equilibrium to pass from stable to unstable when the parameters are chosen in a well-defined region. This shows that the delay plays the role of destabilizer and not of stabilizer. We also show in this case that the destabilization of the system by the delay induces a chaotic behavior in the a priori non-chaotic system in the absence of delay. In the case of exponential distribution, we show that the delay induces certain phenomena such as the Hopf bifurcation, the doubling of periods, the intermittence by saddle-node bifurcation and chaos. We show the importance of characterizing the delay-induced chaos and dynamic states of the system by examining the maximum tumor size for each dynamic state. In both cases of study, it is observed that small delays guarantee stability at the stable equilibrium level, but delays greater than a critical value can produce periodic solutions by Hopf bifurcation and larger delays can even lead to chaotic attractors. The implications of these results are discussed. We examined the other scenarios by showing the influence of the probability density parameters on the behavior of the solutions as well as the dynamics of the model. It is shown that, the region of stability for distributed delays is relatively larger than that of the presence of any discrete delay.