Preserving Bifurcations through Moment Closures

Abstract

Moment systems arise in a wide range of contexts and applications, e.g. in network modeling of complex systems. Since moment systems consist of a high or even infinite number of coupled equations, an indispensable step in obtaining a low-dimensional representation that is amenable to further analysis is, in many cases, to select a moment closure. A moment closure consists of a set of approximations that express certain higher-order moments in terms of lower-order ones, so that applying those leads to a closed system of equations for only the lower-order moments. Closures are frequently found drawing on intuition and heuristics to come up with quantitatively good approximations. In contrast to that, we propose an alternative approach where we instead focus on closures giving rise to certain qualitative features, such as bifurcations. Importantly, this fundamental change of perspective provides one with the possibility of classifying moment closures rigorously in regard to these features. This makes the design and selection of closures more algorithmic, precise, and reliable. In this work, we carefully study the moment systems that arise in the mean-field descriptions of two widely known network dynamical systems, the SIS epidemic and the adaptive voter model. We derive conditions that any moment closure has to satisfy so that the corresponding closed systems exhibit the transcritical bifurcation that one expects in these systems coming from the stochastic particle model.

Figures (3)

• Figure 1: Indications of a bifurcation in the SIS epidemic and the adaptive voter model. Both the SIS epidemic (a) and the adaptive voter model (b) were simulated on a network with regular topology for different values of the infection rate and the rewiring probability, respectively. Depending on those, the prevalence of infected and active edges, respectively, either vanishes or remains positive after some finite time (left column). In fact, when considering this prevalence after some finite time in expectation, in both cases the system seems to exhibit a (supercritical) transcritical bifurcation (right column).
• Figure 2: Bifurcation diagrams for the SIS epidemic model under different closures. Depending on the closure, the SIS epidemic model can exhibit very different bifurcation behaviours. Under the frequently used closure $[ \mathrm{X}\mathrm{S}\mathrm{I} ] = \zeta^{( 2 )} \frac{[ \mathrm{S}\mathrm{X} ] [ \mathrm{S}\mathrm{I} ]}{[ \mathrm{S} ]}$ ($\zeta^{( 2 )} > 0$) (a), and depending on whether $\zeta^{( 2 )} > \zeta^{( 2 )}_{*}$ or $\zeta^{( 2 )} < \zeta^{( 2 )}_{*}$ for $\zeta^{( 2 )}_{*} = \frac{1}{2} ( 1 - \frac{N}{2 M} )$, we can observe a super- or subcritical transcritical bifurcation, respectively. Alternative closures (b) can give rise to the same, a different, or no bifurcation. For instance, the closure $[ \mathrm{X}\mathrm{S}\mathrm{I} ] = \xi [ \mathrm{X} ] [ \mathrm{S}\mathrm{I} ]$ (top left) gives rise to a supercritical transcritical bifurcation, the closure $[ \mathrm{X}\mathrm{S}\mathrm{I} ] = \xi ( [ \mathrm{X} ] + [ \mathrm{I}\mathrm{I} ] )$ (bottom right) to a subcritical pitchfork bifurcation, and the closure $[ \mathrm{X}\mathrm{S}\mathrm{I} ] = 0$ (top right) to no bifurcation.
• Figure 3: Schematic of a degenerate transcritical bifurcation. In an open neighbourhood $U$, there exists a manifold of codimension $1$ of equilibria, $\Omega_{0}$, as well as an equilibrium point. Upon variation of the bifurcation parameter $\lambda$, the equilibrium point eventually collides with and crosses the manifold of equilibria. At the bifurcation, when $\lambda = \lambda_{*}$ and the bifurcation lies in the manifold, there occurs an exchange of stability.

Theorems & Definitions (19)

• Theorem I.1
• Theorem I.2
• Remark
• Theorem II
• Remark
• Lemma 1
• proof
• Theorem : Crandall--Rabinowitz kielhoefer2012bifurcation
• proof : Proof of Theorem \ref{['thm:sis-epidemic-model-order-2-closure']}
• Lemma 2