Custom Post Status 404 URL

I have a function that expires posts after a certain amount of time. It also sets a custom post status of 'expired'. I would like it to function like a 'draft' post in that the URL is no longer accessible to the public. At the moment, once the post is set to 'expired', the full permalink is still visible. Here is my code:

function hrl_custom_status_creation(){
    register_post_status( 'expired', array(
        'label'                     => _x( 'Expired', 'post_type_listings' ),
        'label_count'               => _n_noop( 'Expired <span class="count">(%s)</span>', 'Expired <span class="count">(%s)</span>'),
        'public'                    => true,
        'exclude_from_search'       => false,
        'show_in_admin_all_list'    => true,
        'show_in_admin_status_list' => true,
        'protected' => true,
        '_builtin'  => true
    ));
}
add_action( 'init', 'hrl_custom_status_creation' );