Application of Operators with Shift in the Modeling of Renewable Systems with Elements in Four States

In our recent works, we developed models for renewable systems with elements in three states. For the sake of clarity, states were interpreted as being healthy, being sick, and having immunity acquired after recovery. The states differ significantly in their processes of mortality, reproduction, and mutual influence. In this work, along with the three states listed, a fourth state is introduced, which we will interpret as the state of the elements of the system that have undergone vaccination. As a consequence of this, balance relations become more complicated: unknown functions with shifts and iterations from these shifts appear. A need arises to develop the applied mathematical methods. The content of the article is an expansion and development of our works on modeling systems with elements in several states based on the application of new results on solving functional equations with shifts.