In this research, we investigate an SVIR system with distributed delay. We first provide some preliminary results on the proprieties of the Volterra function and the well-posedness of the solution, also the existence of a global compact attractor denoted $\mathcal{D}$. Then we determine the global behavior of the solution in detail, where it is characterized in two different cases in terms of the basic reproduction number ${\mathfrak{R}}_0$. For ${\mathfrak{R}}_0<1$, we employ a proper Lyapunov function to show the global stability, and we claimed that $\mathcal{D}$ is reduced to the disease-free equilibrium using the proprieties of the $\alpha -limit$ and $\omega -limit$ sets. For ${\mathfrak{R}}_0>1$, we prove the uniform persistence and $\mathcal{D}$ is restricted to the set $\left\{E_0\right\}\cup ℭ\cup {\mathcal{D}}_1$, with $ℭ$ is the set of points that connects orbits from $E$ to ${\mathcal{D}}_1$, and ${\mathcal{D}}_1$ attracts all points with initial infection force. For proving that $\mathcal{D}$ just consists of a positive equilibrium we used a Lyapunov approach, where we provided the relationship between the Lyapunov function for the distributed delayed system and the Lyapunov function for the system with the differential system. The results are confirmed using some graphical representations.

在这项研究中，我们研究了具有分布式延迟的SVIR系统。我们首先提供有关Volterra函数的适当性和解的适定性的一些初步结果，以及存在表示为的全局紧致吸引子的一些初步结果。$\mathcal{d}$。然后，我们详细确定解决方案的全局行为，其中在两种不同情况下根据基本复制数来表征该解决方案的全局行为。${\mathfrak{[R}}_0$。为了${\mathfrak{[R}}_0<\mathrm{1个}$，我们采用了适当的Lyapunov函数来显示全局稳定性，因此我们声称 $\mathcal{d}$ 通过使用 $\alpha -升\mathrm{一世}米\mathrm{一世}Ť$ 和 $\omega -升\mathrm{一世}米\mathrm{一世}Ť$套。为了${\mathfrak{[R}}_0>\mathrm{1个}$，我们证明了一致的持久性和 $\mathcal{d}$ 仅限于该集合 $\left\{{Ë}_0\right\}\cup ℭ\cup {\mathcal{d}}_{\mathrm{1个}}$， 和 $ℭ$ 是连接轨道从 $Ë$ 至 ${\mathcal{d}}_{\mathrm{1个}}$， 和 ${\mathcal{d}}_{\mathrm{1个}}$以最初的感染力吸引了所有人群。为了证明$\mathcal{d}$只是由一个正平衡组成，我们使用了Lyapunov方法，在该方法中，我们提供了分布式时滞系统的Lyapunov函数与具有微分系统的系统的Lyapunov函数之间的关系。使用一些图形表示可以确认结果。