Summary: We prove the existence of \(C_0\)-semigroups related to some birth-and-death type infinite systems of ODEs with possibly unbounded coefficients, in the scale of spaces \(l^p\), \(1 \leq p < \infty\). For some particular cases, we also provide a characterization of the spectra of their generators. For the proof of the generation theorem in the case \(p > 1\), we extend P.R.Chernoff’s perturbation result on relatively bounded perturbations of generators [Proc.Am.Math.Soc.33, 72–74 (1972; Zbl 0265.47015)]. The results presented here have been used in [J.Banasiak, M.Lachowicz and M.Moszyński, Math.Biosci.206, No.2, 200–215 (2007; Zbl 1118.92022)] and they play important role for analysing chaoticity of dynamical systems considered there. As a by-product of our approach, we obtain a result related to the classical Shubin theorem [M.A.Shubin, Izv.Akad.Nauk SSSR, Ser.Mat.49, No.3, 652–671 (1985) (Russian); English translation in Math.USSR, Izv.26, No.3, 605–622 (1986; Zbl 0574.39006)]. We show that this theorem, saying that for a class of bounded infinite matrices the spectrum of the corresponding maximal operator in \(l^p\) is independent on \(p \in [1,\infty)\), cannot be extended to unbounded matrices.