Scientists claim that children under ten years ago suffer easily childhood diseases due to a weak immune system.
This sentence has several issues that include clarity issues and grammar issues. Let's take this part ;
Scientists claim that children under ten years ago
.... are you talking about "children under ten years" or "children who were born ten years ago" ?
Clarity should have first priority in writing. Make sure your ideas are delivered clearly to the reader.
It is proved that vaccine injection not only improve children's immune system but also help them to develop the physical and intellectual, what is more, apart from children vaccinated, other children who go to nursery and primary school with them could also not be affected by infectious diseases.
This sentence is too very long... The reader does not want to memorize so many details in a sentence to get your idea. So, it is better you write shorter sentences.