SELECT `people_id`, SUM(DATEDIFF(end_date, `start_date`)) AS wd FROM test_days GROUP BY people_id

SELECT SUM(days) total 
FROM 
(
	SELECT datediff(`end_date`, `start_date`) days FROM test_days
) AS get_days

CREATE TABLE `test_days` (
  `people_id` varchar(100) DEFAULT NULL,
  `start_date` date DEFAULT NULL,
  `end_date` date DEFAULT NULL
) ENGINE=InnoDB DEFAULT CHARSET=utf8;


INSERT INTO `test_days` (`people_id`, `start_date`, `end_date`) VALUES
('Петров', '2018-06-05', '2018-06-09'),
('Петров', '2018-05-01', '2018-05-19'),
('Петров', '2018-05-06', '2018-05-19'),
('Петров', '2018-05-03', '2018-05-23');

create table seqnum(X int not null);
-- Первые 8 записей
insert into seqnum values(0),(1),(2),(3),(4),(5),(6),(7);
-- И еще 512
insert into seqnum
select s1.x*64+s2.x*8+s3.x+8
  from seqnum s1, seqnum s2, seqnum s3;

select d.people_id, count(distinct d.start_date + interval s.x day) days
  from test_days d, seqnum s
 where s.x<=DATEDIFF(end_date, start_date)
 group by d.people_id

CREATE DEFINER=`root`@`%` FUNCTION `p2` (`name` VARCHAR(255)) RETURNS INT(10) BEGIN

        DECLARE d1 date;
        DECLARE d2 date;
        DECLARE prev_d1 date;
        DECLARE prev_d2 date;
        DECLARE done INT DEFAULT 0;
        DECLARE summ_days INT DEFAULT 0;
        DEClARE cur CURSOR FOR 
            SELECT start_date, end_date FROM test_days WHERE people_id = name         ORDER BY start_date ;
        DECLARE CONTINUE HANDLER FOR SQLSTATE '02000' SET done = 1;
        OPEN cur;
        SET summ_days:=0;
        FETCH cur INTO prev_d1,prev_d2;
        REPEAT
            FETCH cur INTO d1,d2;
                IF NOT done THEN
                    IF d2 >= prev_d2 AND d1 <= prev_d2 THEN

                        SET summ_days:= summ_days + DATEDIFF(d2, prev_d1);
                        SET prev_d2:= d2;
                       ELSEIF d1 > prev_d2 THEN
                        IF summ_days = 0 THEN
                            SET summ_days:= summ_days + DATEDIFF(d2, d1) + DATEDIFF(prev_d2, prev_d1);
                            SET prev_d1:= d1;
                            SET prev_d2:= d2;
                        ELSE
                            SET summ_days:= summ_days + DATEDIFF(d2, d1);
                            SET prev_d1:= d1;
                            SET prev_d2:= d2;
                        END IF;
                    END IF;
                END IF;


        UNTIL done  END REPEAT;

       CLOSE cur;

       RETURN summ_days;

    END$$

    DELIMITER ;

SELECT `people_id`, p2(`people_id`) FROM test_days GROUP BY `people_id`