To stay is to keep, and physical and mental health is to be healthy. Maintaining physical and mental health is expressed in English as maintaining physical and mental health.
Related example: The title of women's magazine is "Stress leads to mistakes!" The article says, "If you want to keep healthy, avoid stress events" is the title of some women's magazines. The article points out that if you want to maintain physical and mental health, you should avoid doing things that make you too stressed.