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Statistical and Dynamical Study of Disease Propagation in a Small World Network

We study numerically statistical properties and dynamical disease propagation using a percolation model on a one dimensional small world network. The parameters chosen correspond to a realistic network of school age children. We found that percolation threshold decreases as a power law as the short cut fluctuations increase. We found also the number of infected sites grows exponentially with time and its rate depends logarithmically on the density of susceptibles. This behavior provides an interesting way to estimate the serology for a given population from the measurement of the disease growing rate during an epidemic phase. We have also examined the case in which the infection probability of nearest neighbors is different from that of short cuts. We found a double diffusion behavior with a slower diffusion between the characteristic times.


Introduction
To model disease propagation, it is necessary to define the corresponding social network connecting any two individuals in the world. The expected properties of such a network should be both the clustering (which excludes models of disorder like the random graphs [1]), and to allow a connection between any two individuals within a finite number of steps (which excludes the regular networks with only nearest neighbor connections). Indeed, for the latter feature Milgram showed in 1967 that the average number of steps connecting any two individuals is six (called also six degrees of separation) [2]. This behavior led recently Watts and Strogatz to propose the model of Small World Network (SW N) [3,4]. They considered a low dimensional network with periodic boundary conditions for convenience (a ring for example) where they rewired some bonds with a probability φ to a new site randomly chosen from the network. For small values of φ this still corresponds to a regular network but with few long range connections called Short-Cuts (SC).
A more recent work on the SW N was proposed by Newman and Watts [5] where the number k of Nearest Neighbors (NN) is conserved but instead of rewiring, they added an average density φ of new bonds from each site i to other nodes randomly chosen (except its nearest neighbors).
A review of these models and their application to various fields and particularly epidemics can be found in the references [3,6]. In these networks the percolation threshold was extensively investigated and its dependence to the NN and SC was found to satisfy the following equation [5,7] This threshold corresponds in epidemics to the smallest concentration of susceptibles leading to the outbreak [5]. However, the statistical behavior of percolative SW N networks is different from that of regular systems [8]. In particular, at the percolation threshold, there is no diverging cluster for the SW N because a SC between the two ends of the system has a finite probability to occur for any non-vanishing number of these bonds. On the other hand, the characteristic length scale in such networks (which corresponds to the correlation length in regular lattices) behaves as φ −1/d , d being the euclidean dimension of the system [7]. It is then obvious that this characteristic length does not diverge at the percolation threshold for such networks. Therefore, a further investigation of the cluster statistics around this new percolation threshold for such networks and the phase transition seems to be necessary.
Let us now consider the application of this model to epidemics which seems to be one of its main aims. From the large amount of works using SW N, there is no direct comparison with the existing data may be due to the complexity of the diseases features (incubation and latent periods, birth and death rates etc.). Furthermore, the parameters used (mainly φ) are very small and do not simulate the real connections between individuals. They use also commonly average values of the NN and the SC while these quantities strongly fluctuate in the real live (the number of contacts, friends, family members etc. varies from 0 to few tens) which can influence sensitively the results on the density of susceptibles at the percolation threshold (epidemic outbreak). On the other hand, it is impossible in practice to measure the density of susceptibles systematically (it needs an extensive serological investigation in the epidemic phase). Generally, for large population samples epidemiologists measure the evolution with time of the number of cases for a given disease.
It is then necessary to study the dynamical behavior of the propagation of the disease and relate it to the density of susceptibles. There are only few works which examined (only qualitatively) the dynamical behavior of the disease on social networks [9,10,11]. The aim of these works was to show how the density of infected behaves in the endemic and epidemic phases.
In this article, we use a site percolation on a SW N with parameters (k and φ) representing a sample of school age children to study the effect of the fluctuations of NN and SC on the percolation threshold. Furthermore, in order to propose a formula for determining the serology of the sample from the rate of increase of the number of cases, we investigate also extensively the dynamical behavior of an infectious disease as well as its effect on the density of susceptibles below and above the percolation threshold. A new super-diffusion is found above the percolation threshold when the cluster is initially infected by one or a small number of infectious sites and its characteristic time dependence on the density of susceptibles is determined. We examined also the case where the infection probability of the NN is different from that of SC, showing a double diffusion with two characteristic times. In the next section we describe the model and then present the results on the statistics of the clusters and the percolation threshold. The results on the dynamical behavior of the disease are presented in section 4.

Model description
We consider the one dimensional SW N described by Newman and Watts [5] but φ represents the total number of SC generated for each site uniformly from all the other sites of the network.
In the case where k and φ are not fixed they are generated randomly within a normal distribution centered at their average values with fluctuations δk and δφ respectively. The coordination number is the total number of bonds to a given site (z = k +φ). We study in this network a site percolation problem [8] by assuming each susceptible site j (occupied) contracts the disease if it is connected with an ill site i (occupied also). The occupied sites (susceptibles) are randomly generated with a concentration p while the empty sites correspond to refracted individuals. For k and φ fixed, the percolation threshold p c is related to k and φ by Eq.(1) [7]. This threshold corresponds to a transition from the endemic phase below p c to the epidemic one above this point [10]. In SW N networks, p c is the minimum concentration of occupied sites above which the average largest cluster size ξ of the occupied sites becomes power-law increasing with the concentration (ξ = (p − p c ) x ), while it diverges in a regular network [8] (note here that the exponent x is positive). By analogy with the regular lattices [8], we will check the universality of the exponent x.
We are interested in the application of such a model to a childhood disease like measles.
In such diseases epidemiological investigations on school age children can be easily controlled and provide data with a minimum bias. We choose parameter values corresponding to such a disease by taking k = 2 to be the average number of brothers, sisters and neighbors, while φ = 30 represents the average number of children one can meet at the school. These parameters should correspond to a topology closer to that encountered in a real social network. Regarding the dynamical study, we assume the major contribution to the epidemics provided by the largest cluster. We restrict ourselves then to this cluster and start the infection with one or few infectious sites at time 0.
These sites will infect all the connected sites in the next step (after a time ∆t), which themselves infect their connected sites after 2∆t and so on. We assume the latent and incubation periods smaller than ∆t which is taken in the rest of this paper as a unit time. The number of infected sites in each step is averaged by varying the initial infectious site position through the whole cluster.

Percolation threshold and cluster distribution
In this section, we realize 100 configurations of the network described in the previous section with a size fixed at 100000 sites. We examine the effects of φ and its fluctuations on the average cluster sizes, p c and x. Finally, we investigate the cluster size distribution around p c in order to determine the main contribution to the propagation of the disease.
In figure 1a we From this behavior, we can estimate the percolation threshold in a real sample of school age children to be in the range 2.3% to 2.8%. Now let us restrict ourselves to the case of fixed k = 2 and φ = 6 in order to examine the statistical behavior of the clusters around p c (without loss of generality, these values are chosen only because p c is large enough to enable sufficient cluster statistics for such a sample size) . We found that the cluster size fluctuations are maximum at this threshold (see Fig.2a) implying a divergence of this quantity at p c which seems to be a signature of a phase transition. The cluster size distribution (see figure 2b) confirms this divergence since it decreases exponentially below p c while it is power-law decreasing at this threshold (this power law behavior is in agreement with the results of Castellano et al. [12] on other systems). Indeed, at p c this corresponds to a Lévy distribution [13] with an exponent of 2.13 indicating the divergence of all its moments. We notice here that only the higher sizes (rare events) contribute to the outbreak at p c (as expected in such distributions). Above p c the small size clusters are absorbed by the largest one and we have again an exponentially decreasing distribution for small clusters while there is only one very large cluster (not shown in Fig.2b).
Since the cluster size does not diverge at p c , it is obvious that x is not universal (because it is not a critical exponent), but it is interesting to know how it depends on φ in such lattices. In figure 2c, the exponent x seems to vary only linearly for larger values of φ but with a very small slope (about 5.6 10 −3 ). It is difficult to predict its behavior for very small values of φ because in this case the network tends to a regular one and the cluster size becomes very large so that the sample sizes taken here do not allow us to measure this exponent accurately.
However, even if the parameters chosen in this model are close to those of a real social network, it seems impossible for epidemiologists to check these results. Indeed, as explained below, they cannot measure the density of susceptibles, except if they investigate systematically the serology of a sufficiently large sample of school age children (e.,e.g. for a city sample). Therefore, the behavior of p c should be checked for measurable quantities. In the case of disease propagation, the time dependence of the number of cases can be directly measured by epidemiological techniques. We will investigate this dynamical behavior in the following section.  figure 3b). This exponential growth is also observed for SIR models [15] where the rate is proportional to the basic reproduction rate R 0 which correponds in our case to the average number of connections in the cluster. We have also performed a Monte-Carlo simulation to the measles propagation in a more realistic sample (births, deaths, incubation and latent periods etc.) where the average infections is 2 for each infectious individual and found also an exponential growth of the infected cases [16]. Therefore, this exponential growth does not seem to depend on the topology of the sample but the rate is sensitive to the geometry of the network. Note in the present work that in the case of more than one initial infectious site (see figure 3c) the exponential growth behavior does not change but the growing rate fluctuates due to the fluctuating number of connections. The average rate of the exponential growth γ (corresponding to the characteristic time of the epidemics) is shown in figure 4 to increases as Ln(p) above p c while the period of this epidemic behavior decreases. From this figure we can conclude that when the characteristic time decreases below 5 (or γ increases above 0.2), the epidemic behavior takes place. This behavior seems to have a direct application in epidemiology since it provides a method for the estimation of the serological situation (density of susceptibles) from the characteristic time which is easily measurable. Therefore, this result stimulates a proposal for a serological examination for a given childhood disease in a sample of age school children, but during an epidemic period to compare a realistic behavior with that obtained in this paper. with the same rate. The slow diffusion is due to the small contact probability for the neighbors (p n = 0.1) and has been observed in other fields [17]. This slow diffusion appears very short because the number of NN is very small (k = 2). It should be interesting to investigate this double diffusion for larger k (which is the case of animal diseases).

Conclusion
We have investigated in this article the statistics of the cluster sizes in a one dimensional SW N by taking into account the NN and SC fluctuations. We found that these fluctuations decrease p c as a power law with a small exponent leading to a new expression for the percolation threshold. We found also that cluster size fluctuations is the quantity governing the phase transition in such a network. On the other hand, in order to apply our results to the measured quantities in epidemiology, we have studied the dynamics of the disease propagation in such clusters. We     c) The rate of the exponential growth (in arbitrary units) versus number of initial infectious sites. The horizontal line is the average rate.

Figure 4
The rate of the exponential growth versus p. The solid line is a fit of the curve linearly with Ln(p). Fitting to N α t 2

Number of cases
Time