本篇論文中，我們提出一個具有疫苗接種效力的SIV 傳染病模型，並在
傳播係數β 和治癒率γ 上加入獨立布朗運動，且兩參數之間有相關性的影
響。首先我們先證明了該隨機傳染病模型具有唯一性正解，再來提出了疾
病的滅絕和持續存在的相關條件及其數學證明，並使用量綱分析來探討所
有模型中的參數的物理量。最後運用數值分析的方法，驗證我們所推導出
的定理條件。特別地，我們發現此具有疫苗接種效力的隨機SIV 傳染病模
型，其確定型模型在特定的參數影響下，會產生後向分歧的現象。

This thesis studies a stochastic SIV epidemic model with vaccine efficacy. We add two independent stochastic processes on the transmission coefficient β and the recovery rate γ in the epidemic model to make it more achievable for environmental fluctuations. We first prove that this model has a unique positive solution. Then we establish conditions for the extinction and persistence of the disease. Moreover,
we use the dimensional analysis to find out the physical quantities of all parameters and show that our results RE and RP are dimensionless. Finally, the theoretical results are illustrated by several numerical simulations.

Acknowledgements
摘要i
Abstract ii
1 Introduction 1
2 Mathematical Formulation of Epidemic Model 5
2.1 The Deterministic SIV Epidemic Model with Vaccine Efficacy σ . . . . . . . . 5
2.2 The Stochastic SIV Epidemic Model with Vaccine Efficacy σ . . . . . . . . . . 7
3 Mathematical Analysis of the Stochastic SIV Epidemic Model 10
3.1 Existence of Unique Positive Solution . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Extinction of the Disease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Persistence of the Disease . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Numerical Simulations 31
4.1 Behavior of Backward Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Examples of Extinction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3 Examples of Persistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5 Conclusions 38
References 39
Appendix 41

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