本研究探討了一個包含疫苗效力的隨機流行傳染病模型。我們
使用最小平方法來擬合這個模型，通過最小化殘差平方和進行回歸
分析，以尋找模型參數的最佳估計。我們採用Euler-Maruyama方法
對模型進行離散化處理，藉由數學理論推導參數的點估計值。並使
用MATLAB™進行數據模擬，並通過結果驗證我們的理論計算。很
重要的是，我們發現噪聲強度和相關參數都和疾病的行為息息相
關，運用數值計算的方法，給出預測理論。並討論在不同的參數調
整下，傳染病傳播的消長現象。

This study examines a stochastic epidemic infectious disease model
incorporating vaccine efficacy. We used the least square method to fit this
model by minimizing the residual sum of squares for regression analysis to
find the best estimates of the model parameters. The model was discretized using the Euler-Maruyama method to derive point estimates of the parameters using mathematical theory. We also used MATLAB™ to simulate the
data and validate our theoretical calculations with the results. Importantly,
we found that noise intensity and related parameters are closely related
to disease behavior, and numerical calculations are used to give predictive theories. We also discuss the growth and decline of infectious disease
transmission under different parameter adjustments.

Acknowledgements iii
摘要 v
Abstract vii
1 Introduction 1
2 Preliminaries 5
3 Least Square Estimation 9
3.1 Regression model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Point Estimators for β and (μ + γ + α) . . . . . . . . . . . . . . . . . 12
3.3 Point estimators for q, p and (μ + ε) . . . . . . . . . . . . . . . . . . . 14
3.4 Point estimator for γ, μ, and ε . . . . . . . . . . . . . . . . . . . . . . 18
3.5 Evaluate the variance of estimated parameters . . . . . . . . . . . . . . 25
4 Numerical Simulations 29
4.1 Examples of Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 The Effects of the Parameter σ . . . . . . . . . . . . . . . . . . . . . . 36
4.3 The Effects of the other Parameter . . . . . . . . . . . . . . . . . . . . 37
5 Conclusions 39
References 41

[1] W. O. Kermack, A. G. McKendrick, and G. T. Walker, “A contribution to the
mathematical theory of epidemics,” Proceedings of the Royal Society of London.
Series A, Containing Papers of a Mathematical and Physical Character, vol. 115,
no. 772, pp. 700–721, 1927.
[2] W. O. Kermack, A. G. McKendrick, and G. T. Walker, “Contributions to the math-
ematical theory of epidemics. ii. —the problem of endemicity,” Proceedings of
the Royal Society of London. Series A, Containing Papers of a Mathematical and
Physical Character, vol. 138, no. 834, pp. 55–83, 1932.
[3] D. GREENHALGH, “Some results for an seir epidemic model with density de-
pendence in the death rate,” Mathematical Medicine and Biology: a Journal of the
IMA, vol. 9, no. 2, pp. 67–106, 1992.
[4] H. W. Hethcote, J. A. Yorke, H. W. Hethcote, and J. A. Yorke, “A simple model for
gonorrhea dynamics,” Gonorrhea Transmission Dynamics and Control, pp. 18–
24, 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,
no. 11-12, pp. 1235–1243, 2002.
[6] L. Jianquan and M. Zhien, “Global analysis of sis epidemic models with variable
total population size,” Mathematical and computer modelling, vol. 39, no. 11-12,
pp. 1231–1242, 2004.
[7] Y. Zhou, S. Yuan, and D. Zhao, “Threshold behavior of a stochastic sis model
with le ́ vy jumps,” Applied Mathematics and Computation, vol. 275, pp. 255–
267, 2016.
[8] Y. Zhao, D. Jiang, and D. O’Regan, “The extinction and persistence of the stochas-
tic sis epidemic model with vaccination,” Physica A: Statistical Mechanics and Its
Applications, vol. 392, no. 20, pp. 4916–4927, 2013.
[9] X. Mao, “Stochastic differential equations and applications,” 2008.
[10] X. Mao, Stochastic differential equations and applications. Elsevier, 2007.
[11] X. Mao and C. Yuan, Stochastic differential equations with Markovian switching.
Imperial college press, 2006.
[12] J. O. Rawlings, S. G. Pantula, and D. A. Dickey, Applied regression analysis: a
research tool. Springer, 1998.
[13] N. R. Kristensen, H. Madsen, and S. B. Jørgensen, “Parameter estimation in
stochastic grey-box models,” Automatica, vol. 40, no. 2, pp. 225–237, 2004.
[14] J. K. Lindsey, Parametric statistical inference. Oxford University Press, 1996.