Aspirin won't cure diseases but it can help prevent them from establishing on your tomato plants and it can greatly slow the progression of the diseases down when established. Aspirin is a preventative measure to help reduce tomato plant disease.
The process is preventative so it is best to start using it before diseases arrive. The aspirin spray essentially tricks the tomato plant.
The salicylic acid in aspirin mimics a natural hormone that tricks the tomato into initiating the SAR response or the Systemic Acquired Resistance response... which means you are turning on all your tomatoes defenses without a true attack occurring. I believe this method of management makes it harder for disease to establish themselves on tomato leaves. I use it yearly.
There is a lot of research on this topic. The video explains how to use aspirin in detail, including the mixture and spraying method.
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