The　reduction　of　the　seismic　effects　on　new　and　existing　structures　is　a　relevant　topic　of　the　structural　mechanics　applied　to　the　civil　engineering.　Usually,　the　conceptual　aspects　related　to　a　new　approach　are　studied　by　means　of　low-dimensional　mechanical　models　able　to　capture　the　main　dynamic　aspects　of　the　method.　The　present　paper　can　be　framed　in　this　context.　Specifically,　the　paper　investigates　the　possibility　of　reducing　the　seismic　response　of　a　frame　structure　by　using　a　vibrating　mass　connected　to　an　inerter　device,　which　interacts　through　the　soil　to　protect　the　structure.　The　problem　is　studied　by　using　existing　soil-structure　interaction　(SSI)　and　structure-soil-structure　interaction　(SSSI)　models,　which　describe　the　actions　between　the　structure　and　the　soil,　and　among　adjacent　structures　through　linear　visco-elastic　devices.　A　seven-degrees-of-freedom　mechanical　model　is　used　to　describe　the　problem,　where　a　general　multi-story　frame　structure　is　mathematically　described　by　means　of　an　equivalent　2-degrees-of-freedom　system.　The　external　vibrating　mass　is　coupled　with　the　inerter　device　to　increase　its　inertia　without　using　high　real　mass.　The　aim　of　the　paper　is　to　point　out　the　role　of　the　many　parameters　that　characterize　the　interaction　system.　Particular　attention　is　devoted　to　the　mechanical　characteristics　of　the　soil,　in　order　to　know　the　effectiveness　of　the　SSSI　system　as　a　function　of　the　characteristics　of　the　soil.　Results　show　that　the　vibrating　mass　equipped　with　the　inerter　device　is　almost　always　beneficial　for　the　frame　structure　to　be　protected.　However,　sufficient　good　performances　justifying　the　costs　of　this　method　can　be　reached　only　in　limited　ranges　of　the　characterizing　parameters.