Spreading of Epidemic Based on Human and Animal Mobility Pattern Yanqing Hu, Dan Luo, Xiaoke Xu, Zhangang Han, Zengru Di Department of Systems Science, Beijing Normal University 2009-12-22
Background & Motivation In the recent 10 years, the worldwide deluges of epidemics happened in human societies have been more frequent, e. g., the SARS in 2003, the H5N1 in 2006 and H1N1 in 2009.
Pattern Levy flight Most of the studies on human and animal mobility pattern including experimental data and theoretic analysis found that their mobility pattern follows the Levy flight: 2 Pr( d ) d Scaling laws of marine predator search behaviour, Nature (2008 ) D. Brockmann, L. Hufnagel and T. Geisel, The scaling laws of human travel, Nature, 439, 462-465, (2006).
1. Epidemic spreading processes always follow the mobility of human and animal. 2. Network based epidemic model should reflect the feature of human and animal mobility. Previous works focus on the topology of network based epidemic models. 3. We want to establish a network model with a Levy flight spatial structure to reflect the feature of human and animal mobility pattern. 1. How to describe the Levy flight spatial structure of epidemic spreading? 2. Whether the exponential of the Levy flight on epidemic spreading process will show the same character as the mobility patterns?
How does the mobility pattern impact on epidemical diffusion? 1. SI model for extremely rapid spreading epidemic. 2. SIS model for investigating the diffusion ability.
Spatial Network Model and Property Energy To establish a Levy flight spatial network, as all the individuals only have limited energy, there must be a cost constraint on the mobility. Hence, we give a restriction on total energy. The frequency distribution of sum of walk distances in a day for deer and sheep. Both distribution are vary narrow, which represents the energy distribution is homogenous. Source: These data were obtained at the Macaulay Institute s Glensaugh Research Station, Aberdeenshire, in north-east Scotland.
Network model Based on a uniform cycle, each node denotes a small group of people. Given a restriction on Ω = total k nenergy, we get a onedimensional weighted network with a Levy flight spatial structure. α wij dij n wd ij i= 1, j= 1 ij = W w = ij Wd 2 ne( d) α ij n/2 d = 1 d α 1. Full connected weighted network and each node is same W w with the k = degree =. E(d) denotes the expectation of ne( d) E( d) one Levy flight distance. 2. Each node has limited energy. 3. The weight of edge means the expectation of communication times.
Diffusion On this weighted network, the infected probability is related with the weight. j I 1 (1 ) w ij A susceptive node i will be infected with a v probability, where v is the spreading ratio, and I is the set of infected nodes. Infected nodes become susceptive with rate. The effective spreading rate is defined as. 1. SI model Randomly choose one node as an infected individual, others are susceptible. δ v λ = δ Terminate until all the nodes are infected and register the steps have been taken as T.
Results of SI Model 1. Epidemic spreading speed with different exponent Epidemic spreading speed with different exponent on SI model. The curve has a lowest point when a 2 Which implies that the mobility pattern will driven the epidemic diffusion. n=1000 V=0.05
2. Spread ratio under different a and various network size n
3.Terminate time with population size under different α and various size n There is a log-log linear dependence between terminate time and the population size. When a=-2, the slope is 0.5
2. SIS model α 2 On SIS model, there is a phase transition when. Using the mean field theory, we analyzed the critical threshold of SIS model on this network. Reaction diffusion equation : ρ() t = ρ() t + λ< k> ρ()[1 t ρ()] t t Our Model: The reaction equation of SIS model w ρ E( d) [1 (1 ) ](1 ) dρ = ρ + λ ρ dt 1 α 2 λc = 1 Ed ( ) Z( α) = α > 2 k c cz( α 1) 1 λ c = < k >
There is a phase transition α 2when, this conclusion can be extended to the cases of high-dimensional. The simulated and analytical critical threshold with different exponent on SIS model. w=10
The End, Thanks!