Complexity, self-organised criticality and fractal analysis

Echinococcus spp transmission (as with many systems in epidemiology and ecology (Anderson, 1994; Horwitz and Wilcox, 2005)) has a number of characteristics of a ‘complex system’, with numerous interconnected objects and processes operating at different scales in a nonlinear manner (for example, through nesting and feedback loops) (Anderson, 1994; Goldenfeld and Kadanoff, 1999; Horwitz and Wilcox, 2005; Pearce and Merletti, 2006). Parasites exist within their hosts, which themselves exist within a local ecosystem, which exists within the regional ecosystem, and so on. Each of these levels do not exist in isolation, and changes at one level can have repercussions in the others. As such, study of the constituent components in isolation is unlikely to be able to fully capture the full dynamics of transmission, even if relatively simple rules and patterns are apparent in the system when viewed at an appropriate scale. This “nested” relationship has been demonstrated through spatial analysis of risk factors for E. multilocularis infection in people: at the continental scale, climatic conditions and availability of grassland is of importance; at the local scale (kilometres), proximity of human populations to suitable landscapes is important; at the patch scale (villages/households), human behaviour is important; and at the

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individual person level, genetic and immunological factors are important (Danson et al., 2003). Considering this, attributing any form of ‘causality’ to infection with E. multilocularis is challenging, as it would be expected to differ at different spatial levels. Indeed, the concept of ‘causality’ in epidemiological studies is currently undergoing a transition, with a move away from the identification of individual-level exposure-outcome relationships towards a more ‘ecological’ interpretation, set in a wider scale (such as at the societal level) (Susser and Susser, 1996a; b; Rothman and Greenland, 2005). This paradigm shift is of particular relevance in the presence of complexity, where traditional techniques are likely to give variable and potentially misleading results (Glattre and Nygård, 2004; Glattre et al., 2012).

The concept of ‘criticality’ is increasingly being applied to epidemiological and ecological scenarios, since it provides a possible explantion for two commonly observed characteristics: threshold behaviour and spatial ‘patchiness’ in distribution (Pascual and Guichard, 2005). Criticality describes the situation in which large systemic changes can occur in a system in response to small changes in the underlying system conditions, and commonly results in a scale-invariant distribution of outcomes (that is, a power law relationship between the size of event and the frequency of event). Different forms of criticality have been identified, and the form may have considerable repercussions for potential control measures (Zinck et al., 2011). For example, ‘self-organising criticality’ was introduced by Per Bak and others (Bak et al., 1987; Bak and Chen, 1991) as a possible method whereby complexity arises in nature. Under this theory, dynamic systems naturally evolve into a ‘critical’ state, which is only barely stable, and can therefore destabilise given particular conditions. The example given in the original paper was that of a pile of sand, with grains being continually added to it. The pile will tend to exist at a particular height and slope (any less than which, and more sand can be added; any greater than which, and ‘avalanches’ of sand of varying magnitudes will occur). It can be shown that a power law relationship exists between the size of ‘avalanches’ and the frequency of these events (Bak et al., 1987). However, other forms of criticality have been proposed which are not self-organising, and which may develop in response to changes or variation in

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underlying parameters and conditions (Pascual and Guichard, 2005; Zinck et al., 2011). This difference is of great importance to the predicted efficacy of control measures, since a system governed by self-organising criticality would tend towards the critical state regardless of intervention, whereas other systems may be more conducive to particular interventions, depending upon their position in relation to the criticality ‘threshold’. This issue is discussed largely in relation to wildfire dynamics in Zinck et al. (2011), but equally could apply to epidemiological issues. There are possible parallels between the concept of criticality and that of the endemic persistence of a pathogen in a population (including ‘endemic stability’, whereby high levels of infection conversely can result in lower incidence of clinical disease (Gemmell, 1978; Coleman et al., 2001)). In the endemic/hyperendemic situation, there is little change in levels of infection over time, yet a disturbance to the system (for example, with praziquantel dosing, in the case of hyperendemic cysticercosis in New Zealand described above (Gemmell, 1978)) can lead to large changes in infection levels.

One method of investigating possible complexity is through fractal analysis, which is based upon investigation of the ‘scale invariance’ often seen in complex systems. This entails identifying a pattern within the ‘fractal dimension’ which may be a more appropriate method of description of the situation than those available using more traditional Euclidean or Gaussian techniques. The fractal dimension can be viewed as a form of scaling parameter which measures the ‘complexity’ in a system – whether the scaling is on a spatial, temporal or some other level. The first example of this concept was described by Benoit Mandelbrot, in an investigation of the ‘self-similar’ nature of the British coastline. The central concept identified here was that the measured length of the British coastline will vary according to the length of the item used to measure it, due to differences in fine ‘resolution’ at different spatial scales. The fractal dimension describes the relationship between the measured length of the coastline and the length of the measuring tool used, with higher values (closer to 2) indicating a greater effect of the length of the tool on the measured length (and therefore greater ‘complexity’), and lower values (closer to 1) suggesting lower complexity (Mandelbrot, 1967). This same concept can also be applied in a non-geographical context, for

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example to epidemiology (Skjerve and Glattre, 2006; Glattre et al., 2008, 2012) and physiology and medicine (Goldberger, 1996).

One characteristic of fractal processes is self-similarity, which may present as a power law relationship: a scale-invariant relationship between two quantities which presents as a linear function on the logarithmic scale. The possible power law relationship governing parasite distributions in hosts has been known for some time, with Anderson and May reporting that ‘it is not uncommon to find 80 per cent or more of the macroparasites contained within 20 per cent or fewer of their human hosts’ (Anderson and May, 1991c), followed by the assertion being substantiated through empirical analysis (Woolhouse et al., 1997; Perkins et al., 2003). This ‘80:20’ pattern is characteristic of a Pareto distribution: a type of power law relationship. Identification of the form of these relationships may be of great relevance to statistical testing and modelling of Echinococcus spp, and may also assist in establishing whether there is evidence of self-organised criticality in Echinococcus transmission. Attempts were made during the current thesis to investigate some of these concepts further, especially in the context of coproELISA OD distributions. However, time constraints prevented comprehensive investigation, and therefore this area of exploration is briefly mentioned as an area worthy of potential further investigation in chapter 8 only.