Positive model structures for abstract symmetric spectra
  We give a general method of constructing positive stable model structures for symmetric spectra over an abstract simplicial symmetric monoidal model category. The method is based on systematic localization, in Hirschhorn's sense, of a ceratin positive projective model structure on spectra, where positivity basically means the truncation of the zero slice. The localization above is by the set of stabilizing morphisms, or their truncated version. 
Sergey Gorchinskiy, Vladimir Guletskii
math.AT