Analytic Study of Bifurcations of the Pairwise Model for SIS Epidemic Propagation on an Adaptive Network
- PDF / 987,228 Bytes
- 20 Pages / 439.37 x 666.142 pts Page_size
- 87 Downloads / 209 Views
Analytic Study of Bifurcations of the Pairwise Model for SIS Epidemic Propagation on an Adaptive Network Ágnes Bodó1,2 · Péter L. Simon1,2
© Foundation for Scientific Research and Technological Innovation 2017
Abstract The pairwise ODE model for SIS epidemic propagation on an adaptive network with link number preserving rewiring is studied. The model, introduced by Gross et al. (Phys Rev Lett 96:208701, 2006), consists of four ODEs and contains three parameters, the infection rate τ , the recovery rate γ and the rewiring rate w. It is proved that transcritical, saddlenode and Andronov–Hopf bifurcations may occur. These bifurcation curves are determined analytically in the (τ, w) parameter plane by using the parametric representation method, together with the two co-dimensional Takens–Bogdanov bifurcation point. It is shown that this parameter plane is divided into four regions by the above bifurcation curves. The possible behaviours are as follows: (a) globally stable disease-free steady state, (b) stable disease-free steady state with two unstable endemic equilibria and a stable periodic orbit, (c) stable diseasefree steady state with a stable and an unstable endemic equilibrium and (d) a globally stable endemic equilibrium. Numerical evidence is shown that homoclinic bifurcation, giving rise to an unstable periodic orbit, and cycle-fold bifurcation also occur. Keywords SIS epidemic · Dynamic graph · Oscillation · Bifurcation Mathematics Subject Classification 34C23 · 34D23 · 37G10 · 90B10 · 92D30
Introduction Spreading processes on adaptive networks have been intensively studied recently, motivated by the observation that the network itself is changing during the process, leading to the concurrent investigation of the dynamic on and of the network [5,11]. Among many other
B
Péter L. Simon [email protected]
1
Institute of Mathematics, Eötvös Loránd University, Budapest, Hungary
2
Numerical Analysis and Large Networks Research Group, Hungarian Academy of Sciences, Budapest, Hungary
123
Differ Equ Dyn Syst
processes [2,3,11], susceptible-infected-susceptible (SIS) epidemic propagation on adaptive networks has been widely studied [4,6,7,12], because of the obvious reason that susceptible nodes try to cut their links to infected ones and at the same time they create new links to susceptible nodes, in order not to be separated from the network. While the process on the network is well-defined, there are several ways for modeling the creation and deletion of links in the network, called rewiring mechanisms. The most frequently used rewiring models are: random link activation and deletion [7,14], link number preserving rewiring [4], linktype dependent activation and deletion [7,14,15] and link de-activation and activation on a fixed network [12]. These scenarios have been modeled by individual-based stochastic simulation and mean-field ordinary differential equation (ODE) models, such as pairwise [4,7,14] and effective degree models [9,17]. We note that edge-based compartmental models are developed for SIR e
Data Loading...
