# Protection Strategy for Edge-Weighted Graphs in Disease Spread
## 4 Simulation of Disease Spread with Protection and Analysis Results
### Scale-Free Network  
The scale-free networks are networks whose distribution degree follows a power law distribution with an exponent between 2 and 3. The study of epidemics and disease dynamics on scale-free networks is a relevant theoretical issue [47], because this networks are a model for the spread of sexually transmitted diseases (see for instance [48]) or a model to explain the early spread of COVID-19 in China (see [49]). Many works that address this type of network in spread of disease can be found in the literature. See for instance [50; 51; 52; 53].  
Following the previous structure, we analyze the protection on a Scale-free network using the model proposed by [PERSON] and [PERSON] in [54]. The variables size of the graph (\(N\)) and the average degree of the vertices (\(d\)) were studied with respect to the survival rate. The weights are uniformly distributed between 1 and 10. Finally, 5000 simulations on each network were done, considering \(\rho=0.011\), \(\delta=\dfrac{1}{14}\), and the initial population containing one infected vertex.  
Figure 3: \(\sigma\) as a function of \(k\) on each network.  
In the same direction as before, we analyze the variation of \(\sigma\) according to the change of \(k\). For that, we fix \(d=5\) and \(N=100\). The results obtained are the same as those obtained in the previous Section, this is to say, the ranking DIL-W\({}^{\alpha}\) performs the best. Indeed, with \(k=5\%\) the best is DIL-W\({}^{1}\). For the following protection values, DIL-W\({}^{0}\) performs the best very close to Strength ranking (see Figure 4). In Appendix A, Table A shows in detail \(\sigma\) according to \(k\).  
In the variation of the network size, we fix \(d=5\) and \(k=10\%\) of the network nodes. The DIL-W\({}^{\alpha}\) ranking performs the best as well as the Strength ranking when \(\alpha=1\) or \(\alpha=0\). It is clear that the survival rate is increasing along with the size of the network, when we consider DIL-W\({}^{\alpha}\). Figure 5 shows \(\sigma\) as a function of the size of the network.  
When \(\sigma\) is a function of the average contact, we fix \(N=100\) and \(k=10\%\) of the network nodes. The best performance is for the DIL-W\({}^{1}\) ranking up to the average of 30 contacts, that is, 30% of the total network nodes. The survival rate is decreasing because the higher the average number of contacts, the greater the probability of contagion. Figure 6 shows the results. (See Appendix A, Table A, to see in detail \(\sigma\) according to \(d\).)  
Figure 4: \(\sigma\) as a function of \(k\) on a Scale-free network with \(N=100\) and \(d=5\).  
Figure 5: \(\sigma\) as a function of \(N\) on a Scale-free network with \(d=5\).  
Figure 6: \(\sigma\) as a function of \(d\) on a Scale-free network.  
**Remark 1**.: _If the weights are uniformly distributed between \(\epsilon\) and \(1\), with \(\epsilon\) close to \(0\), then the results are equals to the above._  
Finally, in order to illustrate in a simple way why the DIL-W\({}^{\alpha}\) ranking has better performance in test networks and on free scale networks, let us consider a free scale network with \(N=15\), \(d=3\) and weights on the edges uniformly distributed between \(1\) and \(10\). When we apply the DIL-W\({}^{1}\) ranking, the first \(3\) places are occupied by nodes \(3\), \(5\) and \(1\) respectively. These nodes are precisely bridge nodes and when protecting them, according to Definition 10, the graph loses connectivity. If we apply the Strength ranking, the first \(3\) places are occupied by nodes \(3\), \(1\) and \(4\) respectively. Note that the order in which it positions the nodes and the importance it gives to node \(4\) makes the lost of network connectivity less than the lost when applying DIL-W\({}^{1}\). (See Figure 7).