On Mathematical Models Based on Delay Differential Equations in Epidemiology

This paper examines solutions to mathematical models based on functional-differential equations, which have applications in immunology. This new approach allows us to study discontinuous solutions that more accurately depict real-world phenomena. It also enables us to exploit the information contained in the initial function. We discuss immunology models by generalizing existing impulsive delay differential equation models to the proposed form. The new phase space introduced here enables a unified approach to continuous and impulsive solutions that were previously studied, as well as the development of new properties that depend on the initial function. To illustrate our work, we present extensions of current immunological models and demonstrate some applications in fields beyond immunology. This paper focuses on establishing the theoretical basis for modifying models based on delayed differential equations, which are not limited to immunology. It also provides some examples.