Omg I already have aphids.

[ChiliPeppa]

Went out today to find most of my 60 tomato plants with little black specks all over them. I assumed they were aphids. I've been noticing a fair amount of ladybugs all over the garden but nowhere near the amount able to cope with the aphids. I mixed up some Dr Bronners and Neem oil and sprayed all of them. Will this do the trick? How often should I spray with this? We do have ants of various types all over the farm. I have no idea which are the ones that cultivate aphids. I will not use 'poison'. The weather is still somewhat cool and yet these nasty critters arrived. The plants are already a little bit stressed from no less than five wind storms thus far this month. Any ideas/suggestions appreciated. I really want tomato success this year.