Magento 1.9 how to run product alert notifications after updating stock item

I am updating products stock like this:

$_product = Mage::getModel('catalog/product')->loadByAttribute('sku',  $sku);
Mage::app()->setCurrentStore(Mage_Core_Model_App::ADMIN_STORE_ID);
if ($_product) {
    $stockItem = Mage::getModel('cataloginventory/stock_item')->loadByProduct($_product);
    if (!$stockItem->getId()) {
        echo "No stock item for $sku".PHP_EOL;
    } else { // if there is, update it
        $stockItem->setQty($qty);
        $stockItem->setManageStock(true);
    }
    echo "$sku UPDATED to $qty".PHP_EOL;
    $stockItem->save();

} else {
    echo "$sku does not exist".PHP_EOL;
}

This works fine, but I need the alert emails to be sent (back in stock notifications), but with this code they do not run. I have also tried adding the below:

$_product->setData("_hasDataChanges", true);
$_product->save();

To try to force the catalog_product_save_after event to fire. But it doesn’t
Any ideas?