Abstract An interesting point in studying the oscillatory behavior of solutions delay differential equations is abbreviation conditions that ensure oscillation all solutions, especially when noncanonical case. Therefore, this study aims to reduce fourth-order with a operator. Moreover, approach used gives more accurate results applied some special cases, as we explained examples.

In this paper, a generalized mathematical model of spread of infectious disease as SIRS epidemic model is considered as a nonlinear system of differential equation. We prove that for positive initial conditions the resulting equivalence system has positive solution and under some hypothesis, this system with initial positive condition, has a positive $T$-periodic solution which is globally asym...