關於多層次的傳染病模型

Abstract

這篇論文我們考慮在疾病傳染的實體世界中有susceptible-infected-susceptible (SIS) process 和在資訊傳播的虛擬世界中有unaware-aware-unaware (UAU) process 的多層 次傳染病模型，且其鄰接矩陣分別用B 和A 來表示。這樣的傳染病模型可能包含三 種平衡點: 無疾病和無資訊散佈平衡點(0; 0)、無疾病和資訊散佈平衡點(0; q) 和疾病 和資訊散佈平衡點(p; q)。我們得到的主要結果包含以下兩點: 第一，針對A 和B 都是不可約矩陣的情況下，如果在實體世界中疾病的基本再生數RP 0 小於等於1 和在 虛擬世界中資訊的基本再生數RV 0 小於等於1 ，則(0; 0) 是全域穩定。同樣地，若在 實體世界中疾病的基本再生數RP 0 小於等於1 和在虛擬世界中資訊的基本再生數RV 0 大於1 ，則(0; q) 是全域穩定除了(0; 0) 之外。第二，我們討論當A 和B 都以同 質網絡作為連結方式時的局部穩定性。特別地，當A 和B 都是all-to-all 的情況下， 如果1 < RP 0 RV 0 ，則(0; q) 是局部漸近穩定。也就是說，只要在虛擬世界中認知 層影響的效應比實體世界中感染層影響的效應還要來的大的話，則疾病的疫情幾乎是 可以被有效抑制住的。我們也發現到當RP 0 > 1 和RV 0 < RP 0 時，則(p; q) 存在。 此外，我們的結果也能刻劃出在參數空間中，疾病會不會爆發的臨界曲線，亦即決定 (p; q) 存在性或是無疾病平衡點((0; 0) 和(0; q))開始失去穩定性的分界。

In this thesis, an epidemic model in multiplex networks in which the dynamics of susceptible-infected-susceptible (SIS) process in the physical world coexists with that of a cyclic process of unaware-aware-unaware (UAU) in the virtual world is considered. The adjacency matrices that support the SIS and UAU processes are denoted by B and A, respectively. Such model may contain three possible equilibria: the disease and information free equilibrium (0; 0), the disease free and information saturated equilib- rium (0; q), and the endemic and information saturated equilibrium (p; q). Our main results contain the following. First, for any irreducible matrices A and B, if the basic reproduction number RP 0 in the physical world is less than or equal to 1, then (0; 0) (resp., (0; q)) is globally stable (resp., globally stable except at (0; 0)) provided that the basic reproduction number RV 0 in the virtual world is less than or equal to (resp., greater than) 1. Second, we investigate the local dynamics of the model and develop some strategies to study the case when the connecting networks are homoge- neous such as all-to-all. In particular, for A and B being all to all, we conclude that (0; q) is locally stable provided that 1 < RP 0 RV 0 . That is to say, the outbreak of the disease can always be prevented as long as the awareness level in the virtual world is better than the infectious level in the physical world. It is also showed that if RP 0 > 1 and RV 0 < RP 0 , then (p; q) exists. Our results also yield threshold curve of epidemic outbreak in the parameter space, which determines the existence of the endemic equi- librium or the onset of the loss of the stability of the disease free equilibria ((0; 0) and (0; q)).