  In the paper we investigate a mathematical model describing the growth of
tumor in the presence of immune response of a host organism. The dynamics of
tumor and immune cells is based on the generic Michaelis-Menten kinetics
depicting interaction and competition between the tumor and the immune system.
The appropriate phenomenological equation modeling cell-mediated immune
surveillance against cancer is of the predator-prey form and exhibits
bistability within a given choice of the immune response-related parameters.
Under the influence of weak external fluctuations, the model may be analyzed in
terms of a stochastic differential equation bearing the form of an overdamped
Langevin-like dynamics in the external quasi-potential represented by a double
well. We analyze properties of the system within the range of parameters for
which the potential wells are of the same depth and when the additional
perturbation, modeling a periodic treatment, is insufficient to overcome the
barrier height and to cause cancer extinction. In this case the presence of a
small amount of noise can positively enhance the treatment, driving the system
to a state of tumor extinction. On the other hand, however, the same noise can
give rise to return effects up to a stochastic resonance behavior. This
observation provides a quantitative analysis of mechanisms responsible for
optimization of periodic tumor therapy in the presence of spontaneous external
noise. Studying the behavior of the extinction time as a function of the
treatment frequency, we have also found the typical resonant activation effect:
For a certain frequency of the treatment, there exists a minimum extinction
time.
