36

4

Evolution

New genomic epidemiological modelling tools have been developed for infectious

disease research. ²²

4.2

Evolutionary Systems

Equilibrium models, which are traditionally often used to model systems, are char-

acterized by the following assumptions (Allen 2007):

1. Entities of a given type are identical, or their characteristics are normally dis-

tributed around a well-defined mean

2. Microscopic events occur at their average rate

3. The system will move rapidly to a stationary (equilibrium) state (this movement

is enhanced if all agents are assumed to perfectly anticipate what the others will

do).

Hence, only simultaneous, not dynamical, equations need be considered, and the

effect of any change can be evaluated by comparing the stationary states before and

after the change.

The next level in sophistication is reached by abandoning Assumption 3. Now,

several stationary states may be possible, including cyclical and chaotic ones (strange

attractors).

If Assumption 2 is abandoned, nonaverage fluctuations are permitted, and

behaviour becomes much richer. In particular, external noise may allow the system

to cross separatrices. The system is then enabled to adopt new régimes of behaviour,

exploring regions of phase space inaccessible to the lower-level systems, ²³ which

can be seen as a kind of collective adaptive response, requiring noise, to changing

external conditions.

The fourth and most sophisticated level is achieved by abandoning the remaining

Assumption 1. Local dynamics cause the microdiversity of the entities themselves to

change. Certain attributes may be selected by the system and others may disappear.

These systems are called evolutionary. Their structures reorganize, and the equa-

tions themselves may change. Most natural systems seem to belong to this category.

Rational prediction of their future is extremely difficult.

The evolutionary process is often analysed as a game, in which alternative strate-

gies invade an extant one. Ferrière and Gatto (1995) have shown how the Lyapunov

exponent (Sect. 12.3) can be useful for tracking invasion. To properly understand

invasion, however, spatial organization must also be taken into account, and this

requires modelling; cellular automata (Sect. 12.1.2) are useful. ²⁴

22 Cárdenas et al. (2022).

23 This type of behaviour is sometimes called “self-organization”; cf. Érdi and Barna (1984).

24 Galam et al. (1998).