This paper presents a stochastic vector-borne epidemic model with direct transmission and media coverage. It proves the existence and uniqueness of positive solutions through the construction of a suitable Lyapunov function. Immediately after that, we study the transmission mechanism of vector-borne diseases and give threshold conditions for disease extinction and persistence; in addition we show that the model has a stationary distribution that is determined by a threshold value, i.e., the existence of a stationary distribution is unique under specific conditions. Finally, a stochastic model that describes the dynamics of vector-borne diseases has been numerically simulated to illustrate our mathematical findings.

The dynamic analysis of epidemic models plays a crucial role in understanding disease transmission mechanisms and prevention strategies. Building upon the research of Tan et al. ^[1], this paper investigates novel dynamic properties of solutions and $\alpha$-paths of an uncertain SIRS model. First, we prove the existence, uniqueness, and stability of a solution for the SIRS model. Using the Yao-Chen formula, we derive $\alpha$-paths of the model and prove that both the disease-free and endemic equilibria are globally asymptotically stable, thereby refining existing research. When the threshold ${\mathcal{R}}_0^u\le 1$, the disease-free equilibrium is globally asymptotically stable, while the endemic equilibrium is globally asymptotically stable if ${\mathcal{R}}_0^u>1$. Finally, numerical simulations demonstrate that increasing the recovery rate and decreasing the disease-induced mortality rate can effectively reduce the disease spread. The results show that the disease transmission rate and the intensity of the Liu process are essential to prevent the disease spread.

The paper aims to study the formulas of the confounding index pattern between main effects and other-order effects in two-level regular designs. The relationship between the alias matrix and aliased effect-number pattern is investigated with the index pattern. Through the aliased effect-number pattern, we provide the formula of the important elements in the confounding index pattern. The optimal designs are listed and compared under different conditions in tables.

Asymptotic methods have usually poor performance when sample size is small. For small samples in stratified bilateral data, the paper proposes exact tests of relative risk ratio under Donner's model. Based on score test, two exact methods are introduced in small samples such as E and M methods. Simulations show E method performs better in terms of empirical type I error rate and power.