Effects of Levy Flights Mobility Pattern on Epidemic Spreading

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:

Epidemic spreading processes always follow the mobility of human and animal. In this article, establish a network model with a Levy flight spatial structure to reflect the feature of human and animal mobility pattern and study the effects of Levy flights’ mobility pattern on epidemic spreading from a complex network perspective.

Spatial Network Model

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.

Network Model


Based on a uniform cycle, each node denotes a small group of people. Given a restriction on total energy, we get a one-dimensional weighted network with a Levy flight spatial structure. According to levi flight pattern (P(d) ~ d^alpha ), the weight w_ij on the link between node i and j should be proportional to d _ij^alpha, and for a given network size, the sum of all w_ijd_ij should be a constant which denotes the energy constraint. So, we can get an ensemble network model of these spatial weighted networks generated by many times of realization as:

 that solves into
    =>

 Diffusion


On this weighted network, the infected probability is related with the weight. 

A susceptive node i will be infected with a probability :

      where v is the spreading ratio, and I is the set of infected nodes.

Infected nodes become susceptible with rate delta.  The effective spreading rate is defined as :

SI Model

Randomly choose one node as an infected individual, others are susceptible.
Terminate until all the nodes are infected and register the steps have been taken as T .

Epidemic spreading speed with different exponents

The figure alongside is for n = 1000, V = 0.05.


The curve has a lowest point when alpha ~ 2.
This imples that the mobility pattern will drive the epidemic diffusion.

Spread ratio under different alpha and various network size n

The dependence of spread ratio on time under different alpha.
A. Spread ratio under alpha = 0 and alpha = −1, respectively.
B. Spread ration under = −2. We can see that when alpha = −2, the spread ratio grows much faster than other cases.