M.R.
You shouldn't jog with him until he is 6-9 months old, but ask your child's pediatrician as the recommendation vary widely.
At 4.5 months you can probably put him in the stroller without the carseat if he is doing lots of tummy time and holding his head up.
I have a BOB Duallie and LOVE jogging with my kids. I'm sure you will too!