A Statistical Model of Transformation David and Sarah Kerridge Transformation depends on changes in many things, each one interacting with the others. Similar situations often arise in biology and economics, and the general properties of such systems are well known. They were first studied in modelling infectious disease, but the conclusions are universal, and help us to understand the problem of transforming an organisation. "Infection" in general systems. Many things other than disease behave as if they are infectious. The more there is of something, the faster it grows, up to some natural limit. This gives a self-reinforcing, or "multiplicative" effect. Population is infectious in this sense. If people colonise a desert island, assuming that no-one can enter or leave it except by birth or death, the numbers will rise, until they reach a level determined by food supplies, disease, and cultural rules of behaviour. A change in any of these structural factors will change the equilibrium level, and population numbers will, given time, adjust to it. Equally, "special causes" such as war or natural disasters, may disturb the numbers temporarily, but they usually soon recover to the equilibrium level. Apparent exceptions merely confirm this. About a quarter of the inhabitants of Britain died in the Black Death of 1348-49[1]. In this case the population did not recover for a long time, because regular epidemics of plague had become part of a new system. Usually there is not one "infectious" variable, but many. Each one on its own may not be infectious, but provided that each affects one or more of the others, the overall effect is multiplicative. The first studies The first writer to discuss this multiplicative property was Malthus, in his "Essay on Population"[2]. Then Ross[3] studied the problem of malaria, and developed a mathematical theory. This disease is particularly interesting from our point of view. Malaria involves the interaction between three infectious compon- ents of the system, mosquitoes and parasites, and people. Ross showed that small changes in the system structure, either affecting the multiplication of mosquitoes, human population density, or the infectiousness of malaria, can produce a sudden change of equilibrium level. Two types of equilibrium There are, essentially, two possible ranges within which the equilibrium can exist. In one, malaria is everywhere, and most people are infected, or have already had it, and are less affected by infection, at least for some time. In the other, the equilibrium is at a very low level, which may be total extinction. In practice, a low level may be maintained by people entering the area, bringing infection with them, or by sub-populations in which the infectious rate is high. Moderate levels do not exist, except as a transient state, as the system changes between the extremes. He used his understanding, based on this model, to to eliminate malaria from an island population, without any of the modern aids such as DDT. The threshold theorem Later Kermack and McKendrick analysed in greater detail the simple case of a single infectious agent spreading in a popul- ation of fixed size, and derived the "threshold theorem". This is concerned with isolated epidemics of infection. This and other studies on infectious disease are conveniently summarised in [4]. Thinking about this simple problem shows why we have the two extremes in equilibrium levels. If conditions favour cross- infection, one case can lead to explosive spread throughout the population, until the majority are infected or immune. But if conditions do not favour spread, the epidemic caused by each introduced case soon dies out. There are many historical examples of a change of the equilibrium level for an infection. Tuberculosis was once the "Captain of all these men of death": everyone caught it, and thousands died, though many had mild infections which caused no symptoms and gave immunity. Now it survives at a low level in advanced countries, though it remains a major killer elsewhere. Recently there have been structural changes, such as new strains of tuberculosis resistant to existing treatments, and the decrease in resistance caused by HIV infection. These have raised the equilibrium level, but it still remains low in the West. The rise of HIV infection itself may not be due to the emergence of a new disease, but the result of a sharp upward change in its equilibrium level, due to new opportunities for spread. More complex models Of course, only the simplest mathematical models can be fully analysed. But so much data is available from records of infectious disease that we can see how well the model describes the behaviour of such systems. Some factors, such as geographical spread, are hard to analyse algebraically, but easy to simulate on a computer. Computer simulation is a very effective way to study waves of infection. Other things being equal, these arise where there is rapid spread, and a short infectious period, followed by a long immune period. So measles and influenza come in waves, but tuberculosis does not. The equilibrium we have been discussing refers to the average level over time, not the actual level, in the case of diseases which come in waves. The wide variety of infectious systems Many ecological systems illustrate the general properties. Small changes in human activity or pollution can produce catastrophic changes in animal or plant population numbers: perhaps driving a whole species to extinction. Waves are also common, as with locusts and lemmings. These biological systems are obviously "infectious", but physical systems may show infection too. For example, in an atomic pile, each neutron colliding with a uranium nucleus produces more neutrons. Normal working produces an equilibrium which is safe and controllable. But increase the rate at which neutrons multiply beyond a certain point and there is an explosion. Another example is a forest fire, which tends to blaze uncontrollably or smoulder. These varied systems are linked solely by the multiplicative form of the model. The properties which the widely different systems share, chiefly the abrupt change in equilibrium level, and the possibility of waves, have been amply confirmed by observation. Both properties come from the tendency of "infectious" systems to go to extremes. We may not always see waves, but but there will always be a narrow range over which a small change in structure produces a sharp change in equilibrium level. Economic development Another example of a system of this type is provided by the economic state of a whole country or region. Poverty, unstable government, ill-health, and ignorance reinforce each other to hold a country back: while investment, education, and confidence interact to promote development. Each culture has its own equil- ibrium level, which is usually high or low. Galbraith[5] in a comparison of various countries, remarks that the relationship between the economic levels of different countries is remarkably stable. Outside aid appears to do very little, unless the system is changed, This is what we would expect if the equilibrium level depends strongly on the national structure, determined by politics, culture and geography. Countries like Japan, which break out of this trap, are exceptional. Yet the same people, who can apparently achieve so little in their own country, are often outstandingly successful when they emigrate. The Chinese of Singapore and Hong Kong are obvious examples. The new "Free Economic Zones" in China itself may do equally well. Application to business "Nothing succeeds like success". In other words, success is infectious. This is why we should not copy examples of success. A business may be successful just because it is successful, or get worse because it is unsuccessful. Inevitable chaotic cause variation is exaggerated by the multiplicative property of the infectious process, to give impressive but deceptive peaks of success. In a business, many and varied infectious factors underly success. Profits, confidence, and growth all promote each other, in a complex system. Applying the principles we have already seen, we expect a high failure rate among small businesses, but most that survive will settle at an equilibrium level in each factor, with chance variation about it. This suggests that attempts to force any factor artificially away from its natural level, without changing the system structure which fixes the level, will be self-defeating in the long run, and will in fact increase variability. To understand transformation, we must study both positive and negative factors. Positive factors include cooperation, quality, productivity, and understanding of the system, and of scientific method. These may not all be individually infectious, but each promotes the others: a subtle form of multiplicative model. The negative factors, such as fear, conflict, stress, waste, frustration, chaotic variation, and fire-fighting, are easier to understand. A manager who has no time, cannot learn how to save time. Each of these is separately, infectious, and each reinforces the others. So these powerful but intangible forces form a complex infectious system, and the basic principles apply. Long term transformation If this model is correct, it explains why so many "Quality Initiatives" fail. In our culture it is natural to want to take dramatic action, and see dramatic effects. So most quality initiatives try to push the system away from its equilibrium level, without making any structural change. This search for "instant pudding" may bring short term gains, but equilibrium soon re-asserts itself. The waves that result from these false starts simply increase chaotic variation. Instead, we should work on the system. Effective action includes changing reward systems to remove fear and frustration, reducing variation, simplifying systems, and above all increasing education. These undramatic actions have dramatic long-term effects. At first, the equilibrium level changes slowly, but when a certain point is reached, the equilibrium level rises sharply. Of course, outcomes do not respond instantly to each change of equilibrium level. So the manager, eager for change, may see little at first, and may give up before the critical point is reached. Conclusion Ross's discovery that mosquitoes spread malaria was accepted at once. Unfortunately his statistical modelling, which in some ways was an even greater achievement, was almost completely ignored. We may wonder why this was, when his model made accurate predictions. Yet system thinking, and statistical modelling, were both entirely new. If they are not widely understood now, resistance then was inevitable. We must make sure that the theory of transformation does not meet the same fate. At present, the world economy is subject to epidemic waves of growth and recession. We must hope that wider understanding of the New Economics[6] will lead at last to a more stable, and more efficient, system. References [1] William H Mc Neill Plagues and People Blackwell, Oxford, 1977 [2] Thomas Malthus Essay on Population London, 1798 [3] R Ross The Prevention of Malaria 2nd edition London 1911 [4] Norman Bailey The Mathematical Theory of Infectious Diseases 2nd edition Griffin, London 1975 [5] John Kenneth Galbraith The Nature of Mass Poverty Harvard, 1979 [6] W Edwards Deming The New Economics MIT 1993