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It is well known that cycles can also appear in models that involve an R-stage of
the epidemic, such as SIRS models. It appears likely that in an adaptive SIRS model
the interaction of rewiring-induced oscillations and recovery-induced oscillations can
give rise to more complex, i.e., chaotic or quasiperiodic, dynamics.
18.6. Summary and Conclusions
In this chapter we have investigated a simple conceptual model of the topological
response of contact networks to the emergence of an infectious disease and the
feedback of topological change on the epidemic. The analysis of this model has
revealed that state-dependent rewiring is likely to increase the invasion threshold for
diseases but at the same time introduces a persistence threshold below the invasion
threshold. The consequence are discontinuous transitions, bistability, hysteresis and
the existence of an oscillatory phase.
From an abstract point of view rewiring is certainly benecial as it generally
reduces the number of infected and can potentially drive the disease to extinction.
However, from another perspective a word of caution is in order. If we observe small
sub-threshold outbreaks of an epidemic in the real world we should ask whether
the low prevalence is maybe a result of rewiring and not of low infectiousness of
the disease as such. If rewiring plays a signicant role in keeping the epidemic
below threshold then a discontinuous transition can be expected if the threshold is
eventually crossed. This may result in a large outbreak which can be dicult to
combat because of hysteresis.
Also, if network properties are taken into account in the planning of vaccination
campaigns, one should consider that these properties can change in response to
emergence of a major epidemic. In particular the widening of the degree distribu-
tion and the appearance of strong degree correlations are detrimental for targeted
vaccination.
Let me emphasize that the concerns stated above are based on the analysis of
very simple models and may therefore prove to be unfounded. To assess the role
of state-topology interplay in epidemic dynamics in the real world, more research
is certainly necessary. The structure of real contact networks is shaped by many
social processes and constraints not related to epidemics. For minor epidemics,
disease-induced changes in the contact network are therefore probably negligible.
However, if a disease is perceived as a major threat, it may induce risk-avoidance
behavior that has a strong impact on the contact network.
The moment closure approximation that has been discussed here in detail, ap-
pears to be a powerful tool for future investigations. It is interesting to note that
this tool is also applicable to certain simple models of social processes which could
therefore be included in models of epidemics. A discussion of the dierences and
similarities between epidemic and social contact processes on adaptive networks is
presented in Do and Gross (2009). Constraints that cannot be captured easily by
the moment closure approach arise from the physical space in which personal con-
