Can a model be called from a UI button using ArcObjects code?

Answer

Note:
Please note that VBA will probably raise a syntax error for the reference to IArcToolboxExtension. To fix this, add a reference to the ESRI Geoprocessing UI library. Do this using the Tools > References menu item.

Code:
Private Sub UIButtonControl1_Click() 
'Ensure that there are references set to the 
'ESRI GeoprocessingUI Object Library and the
'ESRI Geoprocessing Object Library.
'Go to Tools > References to do this

Dim pUID As New UID 
pUID = "esriGeoprocessingUI.ArcToolboxExtension" 

Dim pATBExt As IArcToolboxExtension 
Set pATBExt = Application.FindExtensionByCLSID(pUID) 

Dim pAtb As IArcToolbox 
Set pAtb = pATBExt.ArcToolbox 

Dim pTool As IGPTool 

'* Use this line for custom tools or models
'* ========================================
' Set pTool = pAtb.GetToolbyNameString("MyModel") 

'* Use this line for existing tools
'* ================================
Set pTool = pAtb.GetToolbyNameString("buffer_analysis") 

Dim pCommand As IGPToolCommandHelper 
Set pCommand = New GPToolCommandHelper 
pCommand.SetTool pTool 
pCommand.Invoke Nothing 

End Sub

Code:
Set pTool = pAtb.GetToolbyNameString("MyModel")

Code:
Set pTool = pAtb.GetToolbyNameString("buffer_analysis")