An investigation on a discrete-time infectious disease model that incorporating vertical transmission is presented in this paper. Departing from prior research centered on continuous-time frameworks, our study adopts a discrete-time formulation to better capture the complex epidemiological dynamics. We establish a model and conduct a bifurcation analysis of its equilibrium points. In particular, sufficient conditions for the local stability and the emergence of Neimark–Sacker and flip bifurcations are rigorously derived and analytically verified. As anticipated, variations in the bifurcation parameter give rise to distinct periodic regimes in the system response. To mitigate the instabilities and chaotic behaviors resulting from these bifurcations, we propose and validate two control strategies, which are Hybrid Control Method and State Feedback Control. Numerical simulations futher substantiated the analytical results, demonstrating that appropriate parameter adjustments can shift the system behavior from chaotic attractors and limit cycles toward stable equilibria. Our results show that by dynamically adjusting the intensity of prevention and control measures to mitigate unstable factors such as vertical transmission and high infection rates, or reducing the frequency of system updates to slow down the growth of infections, the epidemic can be transitioned from repeated outbreaks to a stable and manageable state.