What is the name and the mathematical model of this effect Consider the case when some economic variable affects some another (for example, interest rate affects price index) because it involves human decisions. The effect is not immediate and would take some time to begin. I am searching for the name of this delayed effect and the formula or mathematical or statistical model behind.

I am not economist, I just thought I would find an answer in economics realm
I know that this is called price “stickiness”. Price “flexibility” is when something such as interest rates would bring prices to equilibrium quickly. Price stickiness in this case could have a delayed effect as you mentioned. We could also call this “elasticity”. For simple models rise over run can measure price elasticity. For example, a 5% change in interest rates, and it’s corresponding impact on prices could be measured by slope (rise/run).