傳染病的防治是公共衛生領域中，相當重要的議題，隔離與疫苗接種
是常見的防治方法。在本研究中，我們建構了一個具有隨機性的 SIQV 模
型，此模型是基於傳統的 SIS 模型，並增加隔離與接種疫苗的人群，以呈
現傳染病在這個防治措施下的感染人數變化情形。由於防治措施的效果受
到人際交流的影響，我們也將人際網路納入模型中，由數學工具推導得知，
當理論閾值 R0 < 1 時，疾病幾乎必然地滅絕; R0 > 1 且疾病死亡率 α = 0
時，病毒會幾乎必然地持續存在。在數值計算中，我們利用各種參數、分
群個數與閾值 R0 的不同，驗證理論的結果。另外，我們也利用數值模擬疾
病死亡率的大小，探討疾病存在的可能情況。特別地，我們在模型加了一
個特別的參數，提高防疫意識，各群會從可感染的人直接到被感染的人，
加入此參數可以更符合現實社會。

The prevention and control of infectious diseases are important issues in the field
of public health. Quarantine and vaccination are common methods of prevention
and control. In this thesis, we constructed a stochastic SIQV model based on the
traditional SIS model, incorporating the populations of individuals undergoing
quarantine and vaccination to investigate the changes in the number of infected
individuals under these preventive measures. Since the effectiveness of these measures is influenced by interpersonal interactions, we also included a social network
in the model. The theoretical results indicate that when the threshold value R0 < 1,
the disease will become extinct almost surely. Moreover, the disease will persist
almost surely if R0 > 1 and the infection-induced death rates α = 0. In numerical
calculations, we study the various values of parameters, the classification of the individuals, and the threshold value R0 to verify our theoretical results. In addition,
we also use numerical simulations to discuss the possibility of the existence of the
disease when the infection-induced death rate is positive. We have added a special parameter to the model to enhance awareness of epidemic prevention. People
will transition directly from being susceptible to being infected when exposed to
infected individuals. By incorporating this parameter, the model can better reflect
the reality of society.

Acknowledgements
摘要 i
1 Introduction 1
2 Formulation for Epidemic Model 3
2.1 The Deterministic Epidemic Model 3
2.2 The Stochastic Epidemic Model 6
3 The Stochastic Mathematic Model 8
3.1 Preliminary 8
3.2 Existence of Positive Solution of the Disease 9
3.3 Extinction of the Disease 13
3.4 Persistence of the Disease 15
4 Numerical Simulations 24
4.1 Behavior of Stochastic Equation 24
4.2 Example of Extinction 24
4.3 Example of Persistance 26
4.4 Discuss the Disease with Different Value of α and k 29
5 Conclusion 36
References 37
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