Yes Elie
You could well be in ketosis now - in fact if the sticks say so - you ARE.
Probably as you have been on a diet for a month and already lost the glycogen stores. Basicially that big loss in the first week is losing your glycogen as the glycogen is stored with water and as the body uses it up the water gets peed out which is why you wee a lot at first and lose so much weight the first week.
At least the fact all your glycogen has gone means everything you lose now will be fat - there will be no fluid loss on the scales now.
Congratulations - probably means the first few days won't be hard lol.