Abstract
<abstract><p>This research provides an improved theoretical framework of the Kermack-McKendrick system. By considering the general interference function and the polynomial perturbation, we give the sharp threshold between two situations: the disappearance of the illness and the ergodicity of the higher-order perturbed system. Obviously, the ergodic characteristic indicates the continuation of the infection in the population over time. Our study upgrades and enhances the work of Zhou et al. (2021) and suggests a new path of research that will serve as a basis for future investigations. As an illustrative application, we discuss some special cases of the polynomial perturbation to examine the precision of our outcomes. We deduce that higher order fluctuations positively affect the illness extinction time and lead to its rapid disappearance.</p></abstract>
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