A Hybrid System of Horizontal and Horizontal–Vertical Transmissions via the Concept of Piecewise Differential Equations

ABSTRACT This study examines the practical applications of piecewise differential equations, a powerful mathematical tool for modeling processes that occur over different time intervals, with a specific emphasis on horizontal–vertical models. This innovative approach seeks to offer a new insight into HIV infection models, focusing on both horizontal and vertical transmission routes when an infected newborn is introduced into a population. Alongside the global existence theory for piecewise models, we investigate the conditions under which solutions exist and are unique. We examine the scenarios where these new models involve fractional derivatives and analyze various patterns that emerge from their use. To facilitate this, we present the derivation of the parameterized method to solve piecewise differential equations for vertical and horizontal transmissions. Also, our study includes conducting numerical simulations to visually illustrate the different behaviors exhibited by these two models across various scenarios.