What is one of the most efficient ways that developing countries can improve health? You may think of health education, hygiene, mass vaccinations, health screening, prevention of diseases by fighting risk factors in our environment or food, and many more. All of these belong to what is commonly being called public health. But then, what is the main tool of these various components of public health? The answer may surprise even health policy experts:

That tool is mathematics.

Mathematicians like to work with clear definitions. Public health is, like medicine, thousands of years old but its practitioners have not yet agreed on its definition. I use the following:

*Public health is the entirety of theoretical and practical activities that are related to health and deal with populations as a whole but not specifically with their individual members.*

Do policymakers have a clear idea about what mathematics is? My own idea looks like this:

Mathematics is the study of formal, abstract, structures, which can be applied in many different concrete settings.

For example, every starter in elementary school learns that 1+1 = 2. This abstract result can be applied to find concretely that 1egg + 1egg = 2eggs but just as well that 1patient + 1 patient = 2 patients. It is this flexibility, and the possibility of discarding irrelevant elements in a study, which makes the strength of mathematics.

The literature on the early history of public health in the South Asian, East Asian, Near-Eastern and European regions describes the planning and building of public health care institutions and of hygienic measures such as public baths, clean water supply and sewers. The mathematical methods were elementary predecessors of those used in the present-day public health fields of health planning and health economy. They are not much different from those employed in other planning or economic activities.

**Maths, health and old graves**

There has been a challenging mathematical problem regarding a particular aspect of public health in former times, namely, to estimate life expectancy from the information on old graves, in particular from the indication of the age at death. It belongs to the subject area demography, which still plays an important role in public health. Demography is a largely mathematical subject, its main tools stemming from probability theory and mathematical statistics. Demographers use these tools for *adjusting *for instance incomes of people to their age but exactly the same procedure is employed in public health by epidemiologists to *standardize *for example prevalence of tuberculosis to housing conditions. This illustrates again how flexible and universal mathematics is.

This has brought us to contemporary public health. The "four arithmetic" are a simple mathematical tool but they usually suffice for running health registries and health statistics, which are classical and indispensable parts of public health, required in health planning and economy, for example.

Much information on numerical indicators like disease frequencies is today obtained by sample surveys. But many organizations do expensive sample surveys on fairly irrelevant subjects. Moreover, many sample surveys are superfluous because their results can be derived more easily and less expensively from existing registries.

Let me sketch the two main types of deep mathematics in modern public health, namely mathematical modelling of the evolution of epidemics and statistical models to investigate the influence of certain factors on a disease.

**Modelling an epidemic's evolution**

Mathematical modelling started in the year 1916 with a study of the evolution of malaria. This was particularly complicated because of the malaria cycle. The pathogen, which belongs to the genus *Plasmodium*, passes from a human host to a mosquito host of the genus *Anopheles* and back.

It is easier to model the evolution of a measles epidemic, for example, because there is only the human host. Suppose we know the average number B of persons infected during a given period by a given already infected person. Mathematical modelling then answers questions like the following ones: What will be the likely evolution of the epidemic? What happens if B is replaced by a smaller value as the result of a vaccination campaign? How much smaller must this new value be in order to lead to the extinction of the epidemic? Thus, planning a vaccination campaign is today mainly a mathematical affair, based on more advanced mathematical models. The mathematical tools are usually advanced and diverse, for example systems of partial differential equations.

The rigorous investigation of the influence of factors on diseases by using a statistical model comprises a very large variety of problems, depending on the type of factors, on the diseases involved and on the kind of influence to be studied. However, the intuitive idea is always the same: we compare the influence on the disease in different populations where the factor has different levels.

In clinical trials, the factors are medical treatments of a disease. Their influence on the disease may have results ranging from complete recovery to improvement to dangerous side effects – or no effect at all. The method of study is to compare the result of the treatment with the result when there is no treatment, and the mathematical tool is a statistical analysis via a suitable model.

**Leeches: an early clinical trial**

In the year 1828 the French physician Pierre Charles Alexandre Louis published the method and the result of one of the first clinical trials in the modern sense. He showed that bloodletting by leeches as a remedy of many kinds of inflammatory diseases, which had been done since antiquity, was completely inefficient. His idea of looking at the disease in a population instead of studying single cases and that of using statistical methods was much attacked. Even today physicians all over the world apply many treatments that have not been submitted to a correct clinical trial; they are not "evidence-based". Many of them are inefficient or even harmful. Diagnostic rules, too, can be studied from the same angle.

In a clinical trial, the factors are created by a physician or by other components of the health system, for example by nurses. Clinical trials are needed for efficient case-management, that is anamnesis, diagnosis and treatment. To investigate the action of factors that exist independently of this system is just as important. These factors fall into categories:

- Lifestyle, for example physical work, smoking, personal hygiene and stress;
- Environment, including the workplace but also biologic factors such as infective microorganisms;
- Nutrition;
- Social and economic factors; and
- Genetic factors.

Again, public health professionals study the action of such factors in populations, not in single individuals; they use statistical models. The results form the basis of prevention, which consists of reducing or eliminating harmful factors. Health education means above all making these results known and applied.

The antibiotics scandal is a more involved issue of public health. In 1928, Alexander Fleming made his famous observation that led to the discovery of penicillin, but by 1945 he was warning that its uncontrolled overuse might render the targeted microbes resistant against it by a process of mutations and selections. This ought to have been investigated by methods of mathematical statistics, too. His warnings were ignored during almost a half-century in which physicians felt a false sense of security and the pharmaceutical industry scored vast profits. Now multidrug resistance is widespread. Some new antibiotics developed to overcome it have serious side effects.

**Re-imagining the health care system**

The mathematical way of looking at a public health issue may offer a guide for reorganizing part of the health system. The confrontation of infectious with non-infectious diseases furnishes an illuminating example. As noted above, we deal with infectious diseases in populations by mathematical modelling, whereas we discover risk factors for non-infectious diseases with the help of models from mathematical statistics.

Therefore an institution to control non-infectious diseases should not be a simple appendix to hygiene institutes, which still form the bulk of public health in most developing countries; there ought to be an independent institution of equal weight to control diseases like cardio-vascular ailments, obesity or diabetes. Cancer is an interesting case because some cancers are infectious, that is, caused by a microorganism, whereas others are not. Dealing with these two forms of cancer in two separate institutions would appear strange to a physician but very natural to public health professionals, especially when they are mathematicians.

What are the general implications for public health policy? In several so-called developed countries mathematics and mathematicians have "invaded" public health long ago. In France the scientific director of the INSERM, which is the research institution for all of health, medicine and public health, was for many years a mathematician. The foremost German research institute BIPS, focused on the central parts of public health disease prevention and epidemiology, has been built up and directed since 2004 by a mathematician. In the Americas, many mathematicians have written excellent books on many aspects of public health. Among them the 2010 book "Design and analysis of vaccine studies", has been very influential; it is "dedicated to those lives saved by vaccination".

In the public health policy of developing countries, mathematics generally still plays a very minor role, to the detriment of the health of the population. We may advance some obvious recommendations. Every practitioner of public health on any level ought to know suitable basic mathematics. High-level practitioners and policymakers ought to see the value of cooperating with mathematicians, unless they are mathematicians themselves.

*Klaus Krickeberg*

*Klaus Krickeberg was born 1 March 1929 in Ludwigslust, Germany, and is now both a German and French citizen. After studying mathematics at the Humboldt University of Berlin, he was a professor in various universities. He has been working extensively on public health in addition to mathematics. He was elected a TWAS Fellow in 1994. In 2018, Vietnamese President Trần Đại Quang awarded him the Friend of Vietnam medal for "positive essential contributions to the development of the Vietnamese health sector". *