我們提出了一個考慮疫苗接種和相關性影響的隨機SIS傳染病模型，更準確地說，我們利用兩個獨立布朗運動在傳播係數和治癒率上引入相關性影響。本篇論文中，我們首先證明了該模型具有唯一性正解，接著提出了保證疾病滅絕和持續存在的特定充分條件。特別地，這些噪聲的強度和相關性參數都對控制疾病的行為起到了重要的作用；此外，我們使用量綱分析來探討所有模型參數和閾值的物理量，達到符合理論結果一致性的驗證。最後，我們討論了相關性參數ρ對於模型的影響，並且給出了一些數值模擬來支持我們的理論結果。

A stochastic SIS epidemic model is proposed to take vaccination and correlation effects into account. More precisely, the correlation effects are involved in the transmission coefficient and the recovery rate by two independent Brownian motions. In this thesis, we presented the specific sufficient conditions to guarantee the extinction and persistence of the disease. In particular, the intensities of these noises and the correlation coefficient both play important roles to control the behaviors of the disease. Based on the dimensional analysis, the physical quantities of all parameters and the threshold values are characterized to consist with our results. Finally, we discuss the effects of the correlation coefficient ρ and give some numerical simulations to support our theoretical results.

Abstract----------------------------------------------i
Acknowledgments-------------------------------------iii
1 Introduction----------------------------------------1
2 Preliminaries---------------------------------------6
3 Mathematical Formulation for Epidemic Model--------10
3.1 Existence of Unique Positive Solution------------10
3.2 Extinction of the Disease------------------------14
3.3 Persistence of the Disease-----------------------19
3.4 Dimensional Analysis-----------------------------22
4 Numerical Simulations------------------------------25
4.1 Examples of Extinction---------------------------25
4.2 Examples of Persistence--------------------------29
4.3 The Effects of the Parameter ρ-------------------33
5 Conclusions----------------------------------------39
References-------------------------------------------40
Appendix---------------------------------------------42

[1] R. M. Anderson and R. M. May, Population biology of infectious diseases: Part I., Nature, 280 (1979), 361-367.
[2] F. Brauer, L. J. S. Allen, P. Van den Driessche and J. Wu, Mathematical Epidemiology, Lecture Notes in Math. 1945, Math. Biosci. Subser., Springer-Verlag, Berlin, 2008.
[3] F. Brauer, Compartmental models in epidemiology, Mathematical Epidemiology, 1945 (2008), 19–79.
[4] V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Mathematical Biosciences, 42 (1978), 43–61.
[5] S. Cai, Y. Cai and X. Mao, A stochastic diff erential equation SIS epidemic model with two independent Brownian motions, Journal of Mathematical Analysis and Applications, 474 (2019), 1536–1550.
[6] S. Cai, Y. Cai, X. Mao, A stochastic diff erential equation SIS epidemic model with two correlated Brownian motions, Nonlinear Dynamics, 97 (2019), 2175–2187.
[7] A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic diff erential equation SIS epidemic model, SIAM Journal on Applied Mathematics, 71(2011), 876–902.
[8] H. W. Hethcote and J. A. Yorke, Gonorrhea Transmission Dynamics and Control, Lecture Notes in Biomath. 56, Springer-Verlag, Berlin, 1984.
[9] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic diff erential equations, SIAM Review, 43 (2001), 525–546.
[10] W. O. Kermack and A. G. McKendrick, A contributions to the mathematical theory of epidemics, part I, Proc. R. Soc. Lond. A, 115 (1927), 700-721.
[11] W. O. Kermack and A. G. McKendrick, A contributions to the mathematical theory of epidemics, part II, Proc. R. Soc. Lond. A, 138 (1932), 55-83.
[12] J. Li and Z. Ma, Qualitative analyses of SIS epidemic model with vaccination and varying total population size, Mathematical and Computer Modelling, 35 (2002), 1235–1243.
[13] J. Li and Z. Ma, Global analysis of SIS epidemic models with variable total population size, Mathematical and Computer Modelling, 39 (2004), 1231–1242.
[14] M. Martcheva, An Introduction to Mathematical Epidemiology, Springer, New York, 2015.
[15] X. Mao, Stochastic Diff erential Equations and Applications, 2nd ed., Horwood, Chichester, UK, 2008.
[16] 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, 392 (2013), 4916-4927.