A stochastic model incorporating an implicit delay effect for toxoplasmosis: Evaluation of intervention policies for public health

According to the World Health Organization (WHO), toxoplasmosis affects more than 60% of the global population. The prevalence of this infection is particularly high in hot, humid, and low-altitude regions, as such environments favor the survival of oocysts in the ecosystem. In this study, we investigated the transmission dynamics of toxoplasmosis using a stochastic model with an implicit delay effect approach. The host populations were divided into compartments representing susceptible cats $ S\left(t\right) $, infected cats $ {I}_{c}\left(t\right) $, recovered cats $ {V}_{R}\left(t\right) $, susceptible mice $ {S}_{m}\left(t\right) $, infected mice $ {I}_{m}\left(t\right) $, and the number of oocysts in the environment $ O\left(t\right) $. In the delayed deterministic model, fundamental mathematical properties such as positivity, boundedness, existence, and uniqueness of solutions were established. Furthermore, the local and global stability of the steady states were analyzed using second-order stability conditions. In the stochastic delayed formulation, we investigated the positivity, boundedness, extinction, and persistence of the infection under random environmental fluctuations. To address the nonlinear complexity of the proposed system, several computational methods were employed, including the Euler–Maruyama, stochastic Euler, stochastic Runge–Kutta, and the stochastic non-standard finite difference (SNSFD) schemes. A comparative numerical analysis demonstrated that the SNSFD scheme preserves the qualitative features of the continuous model and remains stable under large time steps, confirming its suitability for modeling biologically realistic epidemic dynamics.