Analytic Study of Bifurcations of the Pairwise Model for SIS Epidemic Propagation on an Adaptive Network

• Published in: Differential equations and dynamical systems Vol. 28; no. 4; pp. 807 - 826
• Main Authors: Bodó, Ágnes, Simon, Péter L.
• Format: Journal Article
• Language: English
• Published: New Delhi Springer India 01.10.2020; Springer Nature B.V
• Subjects: Bifurcations; Computer Science; Engineering; Epidemics; Hopf bifurcation; Mathematical models; Mathematics; Mathematics and Statistics; Orbital stability; Orbits; Original Research; Parameters; Propagation; Rewiring; Steady state; SIS epidemic; 90B10; 92D30; 34D23; 34C23; Oscillation; Bifurcation; 37G10; Dynamic graph
• ISSN: 0971-3514; 0974-6870
• DOI: 10.1007/s12591-017-0348-8

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, saddle-node 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 disease-free 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.