//$date_from_db - дата из БД

if ($date_from_db > time()){
    $start_date = date('d.m.Y', strtotime('-5 days'));
    $end_date = date('d.m.Y', strtotime('+5 days'));

    $date_from_db = time();
    
    // Тут сохранить все три значения в базу
}

$now = new DateTime('now');
$date = new DateTime('2019-05-01');
$diff = $now->diff($date);
$interval = $diff->format('%m');

<?php


use Ddeboer\Imap\Search\Flag\Unseen;
use Ddeboer\Imap\SearchExpression;
use Ddeboer\Imap\Server;


$server = (new Server('imap.example.com'))->authenticate('user', 'pass');

$mailbox = $server->getMailbox('INBOX');
$sr = new SearchExpression();
$sr->addCondition(new Unseen());
$messages = $mailbox->getMessages($sr);

foreach ( $messages as $msg) {

    try {
        if($html = $msg->getBodyHtml()){
            $options = array(
                'ignore_errors' => true,
            );
            $text = \Soundasleep\Html2Text::convert($html, $options);
        } else {
            $text = $msg->getBodyText();
        }

        foreach ( $msg->getAttachments() as $attachment) {
            $path_to_save = '/files/' . $attachment->getFilename();
            file_put_contents($path_to_save,
                              $attachment->getDecodedContent());
        }
    } catch (Exception $e) {
        error_log($e);
    } finally {
        $msg->markAsSeen();
    }
}