Dynamics of vertically and horizontally transmitted parasites: Continuous vs Discrete Models

In this paper we analyze a continuous-time epidemic model and its discrete counterpart, where infection spreads both horizontally and vertically. We consider three cases: model with horizontal and imperfect vertical transmissions, model with horizontal and perfect vertical transmissions, and model with perfect vertical and no horizontal transmissions. Stability of different equilibrium points of both the continuous and discrete systems in all cases are determined. It is shown that the stability criteria are identical for continuous and discrete systems. The dynamics of the discrete system have also shown to be independent of the step-size. Numerical computations are presented to illustrate analytical results of both the systems and their subsystems.