Complexity of Index Sets for Several Classes of Models

When we study questions about computable characterization existence for different classes of models the approach suggested by Goncharov and Knight [1] is effective. It consists of obtaining precise estimations of index sets of such classes in corresponding hierarchy.
For the universal numeration of computable models in non-trivial computable language there were found precise estimations of the following index sets of computable models classes: models with Ehrenfeucht theory ($\Pi^1_1$), models with a theory admitting infinitely many countable models ($\Sigma^1_1$).