Stochastic Hybrid Automata with delayed transitions to model biochemical systems with delays

To study the effects of a delayed immune-response on the growth of an immunogenic neoplasm we introduce Stochastic Hybrid Automata with delayed transitions as a representation of hybrid biochemical systems with delays. These transitions abstractly model unknown dynamics for which a constant duration can be estimated, i.e. a delay. These automata are inspired by standard Stochastic Hybrid Automata, and their semantics is given in terms of Piecewise Deterministic Markov Processes. The approach is general and can be applied to systems where (i) components at low concentrations are modeled discretely (so to retain their intrinsic stochastic fluctuations), (ii) abundant component, e.g., chemical signals, are well approximated by mean-field equations (so to simulate them efficiently) and (iii) missing components are abstracted with delays. Via simulations we show in our application that interesting delay-induced phenomena arise, whose quantification is possible in this new quantitative framework.