Decidable categoricity spectra for almost prime models

We study decidable categoricity spectra for almost prime models. For any computable collection $\{D_i\}_{i\in\omega}$, where $D_i$ either is a c.e. set or $D_i=PA$, we construct a sequence of almost prime models $\{\mathcal{M}_i\}_{i\in\omega}$ elementarily embedded in each other, in which case for any $i$ there exists a finite collection of constants such that the model $\mathcal{M}_i$ in the expansion by these constants has degree of decidable categoricity $\deg_T(D_i)$, if $D_i$ is a c.e. set, and has no degree of decidable categoricity if $D_i=PA$. The result obtained extends that of S. S. Goncharov, V. Harizanov, and R. Miller [Sib. Adv. Math., 30, No. 3, 200–212 (2020)].