%0 Journal Article
%T 一类具有无症状感染及接种疫苗的随机SAIRS传染病模型的平稳分布<br>The Stationary Distribution of a Stochastic SAIRS Infectious Disease Model with Asymptomatic Infections and Vaccination
%A 高琴
%J Pure Mathematics
%P 17-28
%@ 2160-7605
%D 2025
%I Hans Publishing
%R 10.12677/pm.2025.156185
%X 本文综合研究了一类无症状感染&#65292;有症状感染&#65292;以及接种疫苗的随机SAIRS传染病模型的平稳分布&#65292;首先证明了模型正解的存在唯一性。然后&#65292;利用构造Lyapunov函数的方法建立了参数<math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msubsup>
   <mi>&#x211B;</mi>
   <mn>0</mn>
   <mi>s</mi>
  </msubsup>
  </mrow>
</math>

&#65292;并且证明了当<math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msubsup>
   <mi>&#x211B;</mi>
   <mn>0</mn>
   <mi>s</mi>
  </msubsup>
  <mo>&#x003E;</mo><mn>1</mn></mrow>
</math>

时&#65292;模型的解在<math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msubsup>
   <mi>&#x211D;</mi>
   <mo>+</mo>
   <mn>3</mn>
  </msubsup>
  </mrow>
</math>

上存在一个唯一的平稳分布。最后&#65292;对本文主要研究内容进行了总结&#65292;发现<math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msubsup>
   <mi>&#x211B;</mi>
   <mn>0</mn>
   <mi>s</mi>
  </msubsup>
  </mrow>
</math>

受到白噪声的影响&#65292;并且<math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msubsup>
   <mi>&#x211B;</mi>
   <mn>0</mn>
   <mi>s</mi>
  </msubsup>
  </mrow>
</math>

小于等于确定型SAIRS模型的基本再生数<math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msub>
   <mi>&#x211B;</mi>
   <mn>0</mn>
  </msub>
  </mrow>
</math>

。<br>This article comprehensively studies the stationary distribution of a stochastic SAIRS infectious disease model with asymptomatic infection, symptomatic infection, and vaccination. Firstly, we prove the existence and uniqueness of the positive solution of the model. Then, we established the parameters <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msubsup>
   <mi>&#x211B;</mi>
   <mn>0</mn>
   <mi>s</mi>
  </msubsup>
  </mrow>
</math>

 by using the method of constructing Lyapunov function, and proven that when <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msubsup>
   <mi>&#x211B;</mi>
   <mn>0</mn>
   <mi>s</mi>
  </msubsup>
  <mo>&#x003E;</mo><mn>1</mn></mrow>
</math>

, the solution of the model has a unique stationary distribution in <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msubsup>
   <mi>&#x211D;</mi>
   <mo>+</mo>
   <mn>3</mn>
  </msubsup>
  </mrow>
</math>

. Finally, we summarize the main results of this article and find that <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msubsup>
   <mi>&#x211B;</mi>
   <mn>0</mn>
   <mi>s</mi>
  </msubsup>
  </mrow>
</math>

 is affected by white noise. In addition, <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msubsup>
%K 随机SAIRS传染病模型&#65292
%K 解的存在唯一性&#65292
%K 平稳分布<br>Stochastic SAIRS Infectious Disease Model
%K The Existence and Uniqueness of Solutions
%K Stationary Distribution
%U http://www.hanspub.org/journal/PaperInformation.aspx?PaperID=117596