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.

[1] W. O. Kermack and A. G. McKendrick, “A contribution to the mathematical theory of epidemics,” Part I, Proc. R. Soc. Lond. A, vol. 115, pp. 700–721, 1927.
[2] W. O. Kermack and A. G. McKendrick, “A contribution to the mathematical theory of epidemics,” Part II, Proc. R. Soc. Lond. A, vol. 138, pp. 55–83, 1932.
[3] D. Greenhalgh, “Some results for an SEIR epidemic model with density dependence in the death rate,” IMA Journal of Mathematics Applied in Medicine and Biology, vol. 9, pp. 67–106, 1992.
[4] H.W. Hethcote and J. A. Yorke, “Gonorrhea transmission dynamics and control,” Lecture Notes in Biomath. 56, Springer Verlag, Berlin, 1984.
[5] J. Li and Z. Ma, “Qualitative analyses of SIS epidemic model with vaccination and varying total population size,” Mathematical and Computer Modelling, vol. 35, p. 1235–1243, 2002.
[6] J. Li and Z. Ma, “Global analysis of SIS epidemic models with variable total population size,” Math. Comput. Modeling, vol. 39, p. 1231–1242, 2004.
[7] Y. Zhao, D. Jiang, and D. O＇Regan, “The extinction and persistence of the stochastic SIS epidemic model with vaccination,” Physica A: Statistical Mechanics and its Applications,
vol. 392, pp. 4916–4927, 2013.
[8] Y. Zhao and D. Jiang, “The threshold of a stochastic SIS epidemic model with vaccination,” Applied Mathematics and Computation, vol. 243, p. 718–727, 2014.
[9] Y. Zhou, S. Yuan, and D. Zhao, “Threshold behavior of a stochastic SIS model with lévy jumps,” Applied Mathematics and Computation, vol. 275, p. 255–267, 2016.
[10] C. Chen and Y. Kang, “Dynamics of a stochastic multi-strain SIS epidemic model driven by lévy noise,” Commun Nonlinear Sci Numer Simulat, vol. 42, p. 379–395, 2017.
[11] Y. Guo, “Stochastic regime switching SIS epidemic model with vaccination driven by lévy noise,” Advances in Difference Equations, 2017.
[12] J. Arino, C. McCluskey, and P. V. den Driessche, “Global results for an epidemic model with vaccination that exhibits backward bifurcation,” SIAM J. Appl. Math., vol. 64, p. 260–276, 2003.
[13] P. V. den Driessche and J.Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Math. Biosci., vol. 180, p. 29–48, 2002.
[14] P. van den Driessche and K. P. Hadeler, “Backward bifurcation in epidemic control,” Mathematical biosciences, vol. 146, pp. 15–35, 1997.
[15] E. Tornatore, S. Buccellato, and P. Vetro, “Stability of a stochastic SIR system,” Physica A, vol. 354, pp. 111–126, 2005.
[16] E. Tornatore, S. Buccellato, and P. Vetro, “On a stochastic disease model with vaccination,” Rend. Circ. Mat. Palermo, vol. 55, p. 223–240, 2006.
[17] D. G. A. Gray, L. Hu, X. Mao, and J. Pan, “A stochastic differential equation SIS epidemic model,” SIAM Journal on Applied Mathematics, vol. 71, pp. 876–902, 2011.
[18] YongliCai, YunKang, MalayBanerjee, and WeimingWanga, “A stochastic SIRS epidemic model with infectious force under intervention strategies,” J.Differential Equations,
vol. 259, p. 7463–7502, 2015.
[19] J. W. P. van den Driessche, “A simple SIS epidemic model with a backward bifurcation,” Mathematical Biology, vol. 40, p. 525–540, 2000.
[20] R. Anguelov, S. M. Garba, and S. Usaini, “Backward bifurcation analysis of epidemiological model with partial immunity,” Computers and Mathematics with Applications, vol. 68, p. 931–940, 2014.
[21] P. van den Driessche, “Stochastic epidemic models with a backward bifurcation,” Mathematical biosciences and engineering, pp. 445–458, 2006.
[22] C. Chen and Y. Kang, “The asymptotic behavior of a stochastic vaccination model with backward bifurcation,” Applied Mathematical Modelling, vol. 40, p. 6051–6068, 2016.
[23] G. Hu, M. Liu, and K.Wang, “The asymptotic behaviours of an epidemic model with two correlated stochastic perturbations,” Applied Mathematics and Computation, vol. 218, p. 10520–10532, 2012.
[24] S. Cai, Y. Cai, and X. Mao, “A stochastic differential equation SIS epidemic model with two correlated brownian motions,” Nonlinear Dyn, vol. 97, p. 2175–2187, 2019.
[25] D. J. Higham, “An algorithmic introduction to numerical simulation of stochastic differential equations,” SIAM Review, vol. 43, pp. 525–546, 2001.