Bifurcation Analysis

We plot two bifurcation diagrams of the systems in Figure 3.10 and Figure 3.11. in order to explore the properties of the systems at equilibrium. As we mentioned before, it takes longer time for the cases ⌘= 1 to reach their stationary states so we plot these cases separately. In Figure 3.10,

the first row is the case⌘<1and the second row is the case⌘= 1. When⌘<1, in all three cases of

initial topologies, there are small bistable regions around = 0.16. The transition is discontinuous

when ⌘= 0as we show in the insets in the plots, and the gap between the transition starts shrinking

when we increase⌘. When⌘ = 0.8, the transition becomes continuous and this means ⌘ could make a qualitative change of the dynamical system. In the case ⌘ = 1, the systems do not reach their

stationary states in the given timet= 10,000we set. However they still show continuous transitions,

which are very different from the case⌘ = 0. Last but not least, the critical value of also decreases

0

0.5

1

0

0.25

0.5

0.75

1

η

I

∞

(a)

P at t = 0

TPL at t = 0

DR at t = 0

0

0.5

1

0

2

4

6

8

10

η

t

f

(1

0

3

)

(b)

DR at t = 0

Figure 3.9: (a) Disease prevalence I versus⌘ at time t= 10,000on networks with the same mean

degreehki= 2but different initial degree distributions (pP_k (Poisson): blue, pT P L_k (truncated power law): magenta, and pDR_k (degree regular): red). (b) tf versus ⌘ on networks with the same mean degree hk_i= 2 initial degree regular degree distribution. Dots correspond to the mean computed

0 0.5 1 (a) Poisson (η<1)

I

∞ 0 0.02 0.04 0 0.5 1 β I∞

(b) Truncated Power Law (η<1) 0 0.02 0.04 0 0.5 1 β I∞ (C) Degree Regular (η<1) 0 0.02 0.04 0 0.5 1 β I ∞ 0 0.01 0.02 0.03 0 0.5 1

β

(d) Poisson (η= 1)

I

0 0.01 0.02 0.03

(e) Truncated Power Law (η= 1)

β

0 0.01 0.02 0.03 (f) Degree Regular (η= 1)

β

η

0 0.2 0.4 0.6 0.8 1

Figure 3.10: Bifurcation diagrams of the disease prevalence I versus⌘ on networks with an initial (a) and (d): Poisson, (b) and (e): truncated power law, and (c) and (f): degree regular degree distribution. In (a), (b) and (c), we plot all the simulation results of 30 experiments. In (d), (e) and (f), dots and the error bar are the mean and standard deviation over 30 Monte Carlo simulations. The first row are the cases ⌘<1and the second row are the cases ⌘= 1. We vary the values of

and⌘. The other parameters of the system are = 0.02, and ↵ = 0.005. We run simulations for

each value✏= 0.001,0.01,0.05,0.99, and,0.999. We sett= 10,000, all networks with ⌘ not close to 1 reach their stationary states at this time.

Finally, using the same parameter set as before, we study the effect of⌘ on bifurcation diagrams on the three initially different networks. Instead of fixing the initial infected fraction ✏= 0.1, we

choose ✏to be0.001,0.01,0.2,0.4,0.6,0.8,0.99, and,0.999. We chooset= 10,000and at this time,

networks with⌘not close to1would reach their stationary states. In Figure 3.11, our semi-analytical

approximation gives us the same disease prevalence curves in the three cases. But in the degree regular case, there is also a horizontal line I = 0 as in Figure 3.9 (a). In this case, if the initial

infected fraction✏is not large enough, networks would stay disease free. And if✏is larger, networks would become endemic. On the other hand, in the truncated power law case, all networks with different✏would become endemic. The case in between is the Poisson case, when ✏= 0.001, about

5% of the networks would become disease free. The stochasticity of simulations plays a role here. In all networks with different initial topologies, when⌘ is closed to1, we could see the simulations

spread more and this shows these networks are still in their transient states. If the networks are endemic in their stationary states, our approximation gives us a good prediction of the disease prevalence levelI₁. Moreover, in all these three cases, they share the same disease prevalence curve even though they have different stationary degree distributions as we showed in Figure 3.5. Our conjecture here is given a set of parameters, when the networks reach their stationary states, the disease prevalence levelI₁ would always fall on the same universal curve independent of the network topologies, and that is the endemic state of the network. On the contrary, networks could go to disease free if the initial infected fraction✏is not large enough. Like we showed in Figure 3.10, these are the two stable regions or fixed points of the dynamics.

3.5 Conclusion

In this chapter we have introduced a model that includes reinforcement of clustering during the rewiring process, hence providing a unique opportunity to study the role of clustering in altering the dynamics of epidemic spread. In this new model clustering is controlled by combination of rewiring parameter and parameter⌘. We also saw that the initial topologies of networks play important roles on network dynamics.

We explored the parameter space of the rewiring parameters and ⌘, identified the regimes⌘ affects network dynamics the most. By extending the AME method to its second and third layers

0 0.4 0.8 0 0.25 0.5 0.75 1

η

I

(a) Poisson 0 0.4 0.8

η

(b) Truncated Power Law

0 0.4 0.8

η

(c) Degree Regular

ε = 0.999 ε = 0.99 ε = 0.8 ε = 0.6 ε = 0.4 ε = 0.2 ε = 0.01 ε = 0.001 approximation

Figure 3.11: Bifurcation diagrams of the disease prevalence I versus⌘ on networks with an initial (a) Poisson, (b) truncated power law, and (c) degree regular degree distribution. Lines are the predictions of our semi-analytical approach and dots are the outcome of 30 Monte Carlo simulations. We run simulations for each value ✏= 0.001,0.01,0.2,0.4,0.6,0.8,0.99, and, 0.999and plot them

with different colors. We set t= 10,000, all networks with⌘ not close to 1 reach their stationary states at this time.

we were able to capture the dynamics on the networks when⌘ is not very large. Our semi-analytical approximation gives a good prediction of the disease prevalence when networks reach their stationary states. In addition, this method also provides us a good estimation of network degree distribution as well.

We carried out a bifurcation analysis in order to understand the properties of the systems at equilibrium. We observed that introduction of the parameter⌘ brought several quantitative changes to the dynamics. We found there are bistable regions correspond to endemic and disease free states of the systems. Moreover, there is a universal disease prevalence curve independent of network topologies and our semi-analytical approximation could capture the disease prevalence accurately when ⌘ is not close to 1.

A key assumption of our model is that the networks we investigate are locally tree-like. When ⌘ is large, this assumption will no longer be valid and our approximate master equation method was not able to provide a good estimation. We showed that there exists an apparent phase transition at ⌘= 1. We also observed that for ⌘= 1, the time evolution of the system is slower than compared to

any ⌘<1 case.

We provided a simple scheme that reinforces the clustering effect on coevolving networks. Our semi-analytical method takes the information of the status and degree of nodes, their neighbor nodes, and their distance 2 or 3 neighbors. Using similar semi-analytical method, people could study more complicated rewiring methods based on the degree of a node or arbitrary distance from a node.

In document Lee_unc_0153D_16429.pdf (Page 59-65)