Kawasaki disease (KD) is an acute, self-limited pediatric vasculitis of unknown etiology and is one of the leading causes of acquired coronary artery complications in children. Endothelial dysfunction, vascular endothelial growth factor (VEGF) activity, adhesion molecule/chemokine activation, and inflammatory cytokine responses play important roles in its pathogenesis. This paper presents a delay differential equation model with stochastic perturbations to study lesion-level inflammatory mechanisms involved in Kawasaki disease pathogenesis. The model describes interactions among healthy endothelial cells, vascular endothelial growth factor (VEGF), adhesion molecules/chemokines, and inflammatory cytokine activity. Mathematically, endothelial-cell injury promotes VEGF production, VEGF contributes to adhesion molecule and chemokine activation, and the combined adhesion molecule/chemokine activity stimulates inflammatory cytokine production after a time delay. The variables are interpreted as aggregated biological activities, not as individual molecular species. The model is not designed to represent the acute, subacute, and convalescent clinical phases of Kawasaki disease separately, and coronary artery inflammation is not included as an independent state variable. Instead, endothelial dysfunction and inflammatory cytokine activity are used as indirect mechanistic indicators of vascular inflammatory progression. The model is shown to preserve positivity and boundedness under suitable dissipativity assumptions. Equilibrium points and an inflammatory feedback threshold quantity are discussed, and local stability is analyzed through the characteristic equations of the delayed system. Reported incidence data from 2020–2025 are used only as qualitative motivation for considering variability and delayed biological responses. A stochastic extension is then formulated to represent random biological and environmental fluctuations, and a stochastic nonstandard finite difference scheme is proposed to preserve positivity and boundedness in numerical simulations. The results provide a mathematical framework for studying delayed stochastic inflammatory interactions in Kawasaki disease, while highlighting that explicit modeling of clinical phases and coronary artery involvement remains an important direction for future work.

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.