In this work we extended our hybrid model [39] with IL-based immunotherapies and Adoptive Cellular Immunotherapies (ACIs), both modeled as piecewise constant or impulsive functions. We performed analytical analysis of the corresponding deterministic model, inspired by earlier work by Panetta and Kirschner [14]. We analyzed our hybrid model via stochastic simulations which seem to suggest results of some interest, which we briefly summarize:

(*i*) by the transitory analysis it turns out that IL-based immunotherapies require very large values of the parameter *p*
_{
E
}, which might substantially reduce the number of patients to whom it may be used as monotherapy;

(*ii*) in IL-based immunotherapies the piece-wise constant delivery seems more effective for tumor eradication than the impulsive one although at the price of very long infusion sessions;

(*iii*) in a daily delivered ACI the piece-wise constant delivery seems more or less equivalent to the impulsive one;

(*iv*) in a ACI the impulsive delivery seems slightly more effective than the daily delivery: less frequent deliveries of larger doses ensure a slightly more rapid eradication than frequent deliveries of smaller doses. Note that the latter type of delivery is called metronomic delivery, and it is of great relevance for other anti-tumor therapies such as anti-angiogenesis therapies and chemotherapies [1, 59, 60]. Furthermore, for those therapies the metronomic delivery is often more effective;

(*v*) in a ACI the weekly impulsive delivery seems slightly more effective than the weekly piecewise constant delivery;

(*vi*) when combined impulsive therapies are considered both the synchronous and the asynchronous delivery seem to be effective and no remarkable differences are observable.

Other more predictable effects were observed such as the synergistic effects of combined therapies, or the dependence of the eradication on the initial values. Of course, these results are strongly linked to the specific model, to its ability in describing the dynamics of real tumors and to the chosen parameters.

As far as the model is concerned, we have previously stressed that maybe the hypothesis that the linear antigenic effect *cT* due to the tumor size should be corrected by assuming a saturating stimulation *cT*/(1+*dT*); here we also add that the assumption that *E*' linearly depend on *E* could be corrected, as there are cases where this dependence might be nonlinear (see [26] and references therein). Note also that, although computationally useful, representing the piece-wise constant delivery of ACI by means of a continuous input *σ*
_{
E
}(*t*) is only an approximation. Indeed, in reality the infusion should be more realistically represented as a series of injections of a group of cells each Δ*t* ≪ 1 time units. The time interval Δ*t* should be modeled as a Poisson random variable.

As far as the parameters are concerned, in order to obtain more general biological inferences an extensive and systematic exploration of the space of parameters is mandatory. Of course this will require the exploitation of intelligent algorithms (e.g. approximated stochastic simulations [61, 62]) to tackle the computational hardness of model analysis.

Finally, here we have only explored the effects of the intrinsic stochasticity on the dynamics of tumor-immune system interplay under therapy. However, it has been shown that without therapy the extrinsic stochasticity may play a significant role in shaping tumor evasion from the immune control [28]. Moreover, it has also been proposed that realistic bounded stochastic fluctuations affecting chemotherapy may deeply influence the outcome of chemotherapies of solid vascularized tumors [63].

Note that the inclusion of realistic extrinsic noise would require minor changes in the proposed hybrid simulation algorithms besides the inclusion of the stochastic nonlinear equations for correlated bounded noises [28, 63]. However, that would require extensive numerical simulations (e.g. a higher number of samples of the stochastic process underlying the hybrid system) when inferring heuristic probability densities of eradication times, for instance.