class HomePage extends Page {
...

public function doPublish() {
$live = Versioned::get_by_stage("QuadArea", "Live", "\"QuadArea\".\"ParentID\" = {$this->ID}");

if($live) {
foreach($live as $area) {
$area->doDeleteFromStage('Live');
}
}

$areas = Versioned::get_by_stage("QuadArea", "Stage", "\"QuadArea\".\"ParentID\" = {$this->ID}");
// publish the draft pages
if($areas) {
foreach($this->QuadAreas() as $area) {
$area->doPublish('Stage', 'Live');
}
}

parent::doPublish();
}
}

class QuadArea extends DataObject {
...

static $extensions = array(
"Versioned('Stage', 'Live')"
);

public function doPublish($fromStage, $toStage, $createNewVersion = false) {
$this->publish($fromStage, $toStage, $createNewVersion);
}

...
}