In this paper, we construct a generalized epidemiological mathematical model to study toxoplasmosis dynamics, taking into consideration both cat and mouse populations. The model incorporates generalized proportions for the congenital transmission in the mouse and cat populations, along with the oocysts available in the environment. We focus on determining the conditions under which toxoplasmosis can be eradicated. We conduct a stability analysis in order to reveal the dynamics of toxoplasmosis in the cat and mouse populations; moreover, we compute the basic reproduction number R0, which is crucial for the long-term behavior of the toxoplasmosis disease in these populations as well as the steady states related to both populations. We find that vertical transmission in the cat population is essential, and affects the basic reproduction number R0. If full vertical transmission is considered in the mouse population and R0<1, we find that all solutions converge to the limit set comprised by the infinitely many toxoplasmosis-free-cat steady states, meaning that toxoplasmosis would vanish from the cat population regardless of the initial conditions. On the other hand, if R0>1, then there is only one toxoplasmosis-endemic steady state. When full vertical transmission is not considered in the mouse population, then a unique toxoplasmosis-free equilibrium exists and toxoplasmosis can be eradicated for both the cat and mouse populations. This has important public health implications. Numerical simulations are carried out to reinforce our theoretical stability analysis and observe the repercussion of some parameters on the dynamics.