Influence of infection rate and migration on extinction of disease in spatial epidemics

孙桂全中北大学理学院

“ 第六届全国复杂网络会议”最佳学生论文答辩

合作者：靳祯 ( 中北大学 ) 刘权兴 ( 荷兰生态院 ) Amit Chakraborty ( 加州大学河滨分校 ) Bai-Lian Li ( 加州大学河滨分校 )

背景

主要工作

结论

背 景 There are newly emerging and high fatal infectious diseases,

such as SARS (Guan et al., 2003; Marra et al., 2003; Riley et al., 2003), the spread of H1N1 (Garten et al., 2009; Liu et al., 2009a; Smith et al., 2009), the H5N1 strain of avian influenza (Li et al., 2004; Shortridge et al., 1998; Ungchusak et al., 2005), and RABIES (Smith et al., 2002; Fabera et al., 2009).

Epidemic spreading have been modeled by reaction–diffusion models (Li et al., 2008; Sun et al., 2007, 2008), and patch-structured models, which include meta-populations (Rossi et al., 2007), coupled-map lattices (Dorogovtsev et al., 2008; Shirley and Rushton, 2005) and demestructured populations (Newman, 2002).

All these models are amenable to mathematical analysis, but these limit their applicability, in particular in evolutionary contexts.

Extinction may occur where the infection rate is so small that there is insufficient transmission to keep the disease in its endemic state (Dykman et al., 2008; Keeling, 2004).

It is natural to ask, whether the disease will persist if the infection rate is large enough.

Several theoretical models have shown that migration of population is more stable (Rohani and Miramontes, 1995), which means that migration can promote the high density of the disease, and thus lead to the persistence of the disease. We will address the effect of the migration between two patches, especially to check that if migration may result in the extinction of the disease.

主要工作

Model

S: susceptible; I: infected; E: empty. \beta: infection rate; d: death rate due to diease; b: birth rate.

Rules (i) Initially, the populations are randomly located in the spatial

domain (note that the spatial structure at the equilibrium of the system is insensitive to the initial condition) (Hiebeler and Morin, 2007).

(ii) The susceptible can be infected at a probability of through the contact between the susceptible and infected individuals. As a result, a susceptible can become infected at a probability (Rhodes and Anderson, 1997; Tilman, 1994), where is the sum of infected in the neighborhood.

(iii) There is only disease-related death, no natural death. When death events occur, the infected state becomes the empty one (E).

(iv) Only the susceptible (S) can colonize the empty suitable patches at a probability of b in each time step. Specifically, if the chosen patch is empty (E), it becomes occupied by the offspring reproduced from the neighboring (Bairagi et al., 2007; Deredec and Courchamp, 2006; Packer et al., 2003), where is the sum of susceptible in the neighborhood.

(I) 发生率对疾病的影响

Case a: infection rate is small

病人的数目在减少，不足以维持疾病的持续。

Case b: infection rate is moderate

振荡且疾病持续。

Case c: infection rate is large

疾病会形成簇类 (clustered group) 分布，而由于因病死亡的存在，疾病也会灭绝

该结果很好为 1918 年世界性大流行的“西班牙流感”灭绝，提供了一个可能的理论解释。

In 1918, influenza out-broke in Spain firstly, called as ‘‘Spanish Flu’’ and spread all over the world (Corral-Corral and RodriguezNavarro, 2007; Llorca et al., 1998; Trilla et al., 2008). Until 1919, there were 50 million people died due to the disease. However, the obtained spatio-temporal data revealed that when the infected rate reached (or more than) 0.5, the immune barrier would emerge, and thus led to disease to be disappeared (Dory,1977; Gregorio et al., 1993; Oxford et al., 2001, 2002; Schoenbaum,2001).

(II) 迁移对疾病的影响

Case a: infection rate is small

a: 迁移率为 0.2; b: 迁移率为 0.8.

(A) 和 (B) 是不同斑块

当发生率很小时，由于斑块的耦合作用，若迁移率大于一个值，则疾病不会灭绝。

Case b: infection rate is moderate

迁移率低于某个值，疾病总是持续。

迁移率高于某个值，第一个斑块的疾病会灭绝。原因： emergence of anti-phase synchronization

Case c: infection rate is large

迁移率大于某个非常小的值，疾病就会持续。

发生率和迁移率的参数空间

I ： 灭绝区域II ：持续区域

结 论 1) The results showed that if the infection rate is large

enough, the disease will disappear, which can well explain the extinction of ‘‘Spanish Flu’’;

2) The mechanisms of the disease extinction for small and lager infection rate are different. If we want to control the disease, we should pay attention to estimating its infection rate of the disease and use different methods to control disease.

3) Migration has dual effects on the epidemic spreading. On one hand, in the extinction region of single patch, if the migration rate is large enough, there is a phase transition from the disease free to endemic state. On the other hand, migration will induce extinction in the regime, which can ensure the persistence of the disease in single patch, due to emergence of anti-phase synchrony.

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