Phase Transition of Epidemic Dynamics on Adaptive Networks
Condensed Matter Theory I

02110902

Yuta MURAKAMI

1 Introduction

3 Simulations and Results

Mathematical models of disease propagation have been studied for a long time, and various models have been proposed. In
the last decade, much attention has been paid to the epidemic
dynamics on a network, and those researches have revealed
that the network structure aﬀects the epidemic dynamics on
the network. Similarly, the spreading of an infectious diseaseon the network can have a strong eﬀect on the dynamics ofthe network structure. For example, people tend to avoid contact with infectious individuals in real networks. Therefore, anepidemic model on an adaptive network has been introduced
as a model that takes into account the interaction between thedynamics of networks and the dynamics on networks [1, 2].
The study of these models develops our understanding of theepidemic spreading on networks.
In the previous study [2], the moment-closure approximation (MCA) and the degree-based mean-field (DBMF) theory
have been used to analyze the epidemic dynamics and the network dynamics. These are common analytical approaches for
dynamics on networks, however, these are often inaccurate
due to the following assumptions. The DBMF theory is basedon the assumption that there is no dynamical correlation. TheMCA includes dynamical correlations at a pairwise level only.
In order to improve accuracy, the approximate master equation (AME) approach, which is the extension of the DBMF
theory, has been proposed [3, 4]. The variables of the AME
have full information of those neighbors, namely, the dynamical correlation is taken into account. The AME approach provides a good prediction of the time evolution and good estimates of critical values for binary-state dynamics on a network.
In this study, we extend the AME approach to threestate dynamics and analyze the adaptive SIRS (susceptibleinfected-recovered-susceptible) model of Shaw et al. [2].

Let Sklm .t/ .Iklm .t/; Rklm .t// be the fractions of susceptible(infected, recovered) nodes that have degree k, l infectious
neighbors and m recovered neighbors at time t. We derive
the AMEs for these variables, and obtain time evolution ofthe system by numerical integration. Similarly, we obtain solutions of the MCA and the DBMF theory. In addition, We
performed Monte Carlo simulations.
First, we examined the time evolution of infection, definedas the fraction of infected individuals I.t /. The AME solutionagrees with the result of the simulation (Fig. 1).
Next, we studied the epidemic threshold c . The SIRS
model exhibits a phase transition between the healthy state andthe endemic state. These states are characterized by a parameter  and an order parameter I1 that expresses the fractionof infected nodes at steady state. If  < c , the system stays
healthy and I1 is zero. In contrast, the endemic phase appears
and I1 > 0 at  > c . The threshold obtained from AME
approach is diﬀerent from that obtained from MCA or DBMF
theory, and match the result of simulations.

We consider a network with N nodes and K undirected links.
The nodes correspond to individuals, and the links represent relationships between them. We assume that the networks have no degree correlation. The degree distributionof the network
is pk , and the mean degree of the network isP
hki D k kpk D 2K=N . Self-loops and multiple links are
not allowed.
At any time, each node is in a specific state, either susceptible (S), infected (I) or recovered (R). A susceptible node
which has l infectious neighbors becomes infected with rate
(probability per unit time) l. An infected node becomes recovered with rate . A recovered node becomes susceptible
again with rate . In addition, for every link between a noninfected node (S or R) and an infected node (I), the noninfected
node cuts the link with rate ! and forms a new link with another randomly selected noninfected node. These rules guarantee that N and K remain constant over time.

Summary

We have studied the adaptive SIRS model using the AMEapproach. By integrating AMEs numerically, the AMEs reproduce the time evolution of the model with good accuracy.
Moreover, the epidemic threshold obtained from the AME approach is in good agreement with that found by the simulations.

References[1] T. Gloss, C. J. D. D’Lima, and B. Blasius, Phys. Rev. Lett.
96, 208701 (2006).
[2] L. B. Shaw, and I. B. Schwartz, Phys. Rev. E 77, 066101(2008).
[3] V. Marceau et al., Phys. Rev. E 82, 036116 (2010).
[4] J. P. Gleeson, Phys. Rev. X 3, 021004 (2013).

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2 SIRS Model on an Adaptive Network

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MF
MCA
AME
Simulation

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Fig. 1 Time evolution of the fraction of infected nodes
I.t/. The initial degree distribution is regular-random network with hki D 5, and parameters are N D 104 ,  D 2:0,
 D 1:0,  D 1:0, ! D 5:0.