Motivated by the result of invariance of regular-boundary open sets in [P. Cannarsa, G. D. Prato, and H. Frankowska, Indiana Univ. Math. J., 59 (2010), pp. 53-78] and multi-stability issues in gene networks, our paper focuses on three closely related aims. First, we give a necessary local Lipschitz-like condition in order to expect invariance of open sets (for deterministic systems). Comments on optimality are provided via examples. Second, we provide a border avoidance (near-viability) counterpart of [P. Cannarsa, G. D. Prato, and H. Frankowska, Indiana Univ. Math. J., 59 (2010), pp. 53-78] for controlled Brownian diffusions and piecewise deterministic switched Markov processes (PDsMP). We equally discuss to what extent Lipschitz-continuity of the driving coefficients is needed. Finally, by applying the theoretical result on PDsMP to Hasty's model of bacteriophage [J. Hasty, J. Pradines, M. Dolnik, and J. Collins, Proc. Natl. Acad. Sci. USA, 97 (2000), pp. 2075-2080; A. Crudu, A. Debusche, and O. Radulescu, BMC Syst. Biol., 3 (2009), 89], we show the necessity of explicit modeling for the environmental cue triggering lysis.