從隨機性到確定性的傳染病模型

From stochastic to deterministic epidemic models

鄧德明(De-Ming Deng)

指導教授 ： 莊重

國立交通大學/理學院/應用數學系暨研究所/碩士(2019年)

摘要

根據不同的研究目的，疾病傳染模型主要可分為隨機性模型和確定性模型兩大類。隨機性模型能夠反映真實世界中疾病在人群間傳染的隨機性；而確定性模型的主要好處在於它有很多有用的常微分方程工具來分析疾病傳播的動態行為。
　　在本論文中，我們著眼於如何連結描述疾病傳染過程的隨機性和確定性模型。對於充分混合模型，我們介紹了兩種方法―直接計算法和Kurtz法來連結隨機性和確定性模型。直接計算法的核心概念在於先對隨機性模型的主方程取平均後，再收集一階矩的項來得出確定性方程。Kurtz法則利用由躍遷強度組成的漂移函數來當作隨機性和確定性模型之間的橋樑。對於網路為基模型，我們考慮各節點可交互作用的確定性多群模型和隨機性SIS模型兩種模型。在強大數法則下，我們證明了前者可條件收斂至確定性充分混合模型，並且找出其收斂的充分條件；而後者總是可以收斂到充分混合模型而無須任何條件。我們所有的理論結果都能用蒙地卡羅模擬來加以驗證。此外，用來產生充分混合模型和網路為基模型的樣本路徑的演算法也會在本文中提及。

關鍵字

強大數法則 ； SIS模型 ； 漂移函數 ； 蒙地卡羅模擬

並列摘要

Depending on the purpose of theoretical study, disease spreading model consists of two main categories, stochastic model and deterministic model. Stochastic model can reflect stochasticity of disease spreading among individuals in real world and deterministic model has benefit of possessing many useful ODE tools for analyzing the dynamics of disease spreading.
In this thesis, we focus on how to link stochastic and deterministic models for disease spreading process. For well-mixed models, we introduce two methods, direct method and Kurtz’s method, to link stochastic and deterministic SIS models. The core of direct method is computing the expectations of master equations of stochastic model and collecting the first moment terms as deterministic equations. Kurtz’s method introduce the drift function, formed by the jump intensities of transition probabilities, acting as a bridge between stochastic and deterministic models. For network-based models, we consider two basic models, deterministic multi-group model and stochastic SIS model, with all-to-all interaction for disease spreading. Under the strong law of large numbers, we prove that the former model conditionally converges to a deterministic well-mixed model and identify sufficient conditions for its convergence. On the other hand, the latter model always converges to a well-mixed SIS model without imposing any conditions. All the theoretical results are confirmed by Monte Carlo simulations. Moreover, the algorithms for generating sample paths of well-mixed model and network-based model are mentioned.

並列關鍵字

Strong law of large numbers ； SIS model ； Drift function ； Monte Carlo simulations

參考文獻

[1] Pastor-Satorras, R., Castellano, C., Van Mieghem, P., 　Vespignani, A., “Epidemic processes in complex networks”, Reviews of modern physics, vol 87, pp.925-979, 2015.

[2] Kuniya, T., and Muroya, Y., “Global stability of a multi-group SIS epidemic model with varying total population size”, Applied Mathematics and Computation, vol. 265, pp.785–798, 2015.

[3] Kiss, I. Z., Miller, J. C., Simon, P. L., Mathematics of epidemics on networks: from exact to approximate models, vol. 46. IAM Series. Berlin, Germany: Springer, 2017.

[4] Van Kampen, N. G., Stochastic process in physics and chemistry, Amsterdam: Elsevier, 2006.

[5] Kurtz, T. G., “Solutions of ordinary differential equations as limits of pure jump Markov processes”, Journal of Applied Probability, vol 7, pp.49-58, 1970.