Dangerous bacteria are an increasingly serious social problem. They occur most of all in hospitals and livestock farms, and in the latter among pigs and chickens in particular. These bacteria are so dangerous because antibiotics, our panaceas of many decades, are losing their edge as a weapon. Bacteria have grown resistant to these medicines. The term "multiresistant germs" has become a buzzword of our times. In 2013, nearly 31,000 people became infected with such germs in hospitals in the German federal state of North Rhine Westphalia alone. In terms of overall patient numbers, however, North Rhine Westphalia ranks "only" somewhere in the middle; in Thüringen, almost one in ten hospital patients fell ill with multiresistant germs, and in Saxony one in nine. Of the estimated total of 400,000 people annually infected, around 15,000 die from these germs. These are in fact conservative estimates; other estimates put the death rate much higher.

How is it that antibiotics, which have helped us for so long, are now losing their effectiveness increasingly often? Many experts ascribe it to excessive use of antibiotics in livestock because, by eating the animals treated with these medicines, people also regularly ingest the active substances without actually having an acute infection. In 2013, the Bundesinstitut für Risikobewertung, BfR (Federal Institute for Risk Assessment) ascertained that 90 percent of fattened poultry in Germany is contaminated with multiresistant germs.

The bacteria are happy! By now they have developed such efficient defence mechanisms against antibiotics that our medicines are simply ineffective. Apart from that, pressure to reduce antibiotic agents in fattening feed has met with hefty resistance amongst farmers since, without these agents, many animals would not survive on modern livestock farms, and so the margins of fattening farms are narrowing.

The extent of the threat becomes clear when one considers that, on the one hand, the number of multiresistant germs is rapidly rising and, on the other hand, the approval of new antibiotics has been steadily dropping in recent years. Is it a vicious circle? Is drug-based livestock farming really the way of the future, and do we have to simply accept there will be more people who will fall ill or even die? Carsten Hartmann and Max von Kleist see this as a real danger. The mathematicians at the Institute of Mathematics of the Freie Universität Berlin and of the Matheon research centre in Berlin have therefore come up with an approach that just might break this vicious circle. "Using mathematical approaches, we want to arrive at a more effective use of existing antibiotics because, aside from fattening feed, causes for the spread of multiresistant germs are the too-frequent and improper use of antibiotics," Carsten Hartmann says. Improper use means either the medicines are administered without there being a real need to do so, the wrong agents are used, or, as is frequently the case, the intake of antibiotics is stopped too early. If the medicine is not taken over the prescribed period, then some of the dangerous bacteria can survive and subsequently "remember" how to defend themselves against the medicine. In other words, the bacteria not only continue to multiply, they also mutate, after which the antibiotics can no longer be of any threat to them.

As it stands, there are now a great number of bacterial strains that respond only very slightly or, in some cases, not at all to a given antibiotic. It is being considered, therefore, whether a better approach is to administer a combination of different medicines: "In our research project, we are pondering how we can optimally apply and combine different agents so as to keep the number of bacteria as low as possible, and then kill off the targeted bacteria relatively quickly," the mathematicians explain. This requires a mathematical model and efficient algorithms by which to simulate on the computer how individual bacteria will respond to certain medicines. The mathematicians also want to understand how long the treatment must ideally be continued, and when the highest probability exists that all bacteria have actually been eliminated. "Normally, it should be obvious to take a medicine until the end of the course. It’s common sense, and you don't need mathematics to tell you. But, when the risk of resistance exists, then you should have scientifically substantiated methods that help you to minimise the corresponding risks," Prof. Hartmann advises. So the ultimate aim is mathematical optimisation.

The mathematicians accordingly start with a model that describes the number of bacteria present in a patient, which can constantly change during the course of a disease as well as due to random factors. "We are dealing here with a random variable, which normally varies from one patient to another," the researchers explain. Conventionally, the action of an antibiotic is researched by cultivating a bacterial strain in a petri dish and then seeing if the bacteria die off when the agent is added. If one has a relatively small number of bacteria, then one can still theoretically count them. That means one can actually see when a new bacterium has appeared or an older one has died off. If, however, one has a very large petri dish with very many bacteria, then it is no longer possible to count the pathogens. Yet, the process can still be reasonably approximated by averages and concentrations. One has then a stochastic model for the exact number of bacteria in the body on the one hand and a deterministic model for their averages over a population on the other.

Since the rough, deterministic model is easier to cope with mathematically, it would be nice if it could serve on its own to optimise the antibiotic treatment. It certainly works well when many bacteria are present in the body. Unfortunately, it does not work so well after a highly effective antibiotic has killed off most of the bacteria but left a few survivors. Then you have a case with a small number of bacteria. "With a small number of bacteria, it makes a big difference whether I use the stochastic or the deterministic model, since a single bacterium is either there or it is not, and a declaration of 'there is on average half a bacterium in the body' is not a useful description of the situation for an individuum," Hartmann says. The more precise, stochastic model is always needed when the question arises whether to continue administering the same medicine or instead change to a new one. "As scientists, we tend to prefer the higher precision model; but to calculate an optimum antibiotic therapy from it, this model simply has too many state variables. Therefore, in our project, we are studying approximations between the two models, and their effects on the optimal dose of medicine," von Kleist says.

Currently, Carsten Hartmann and Max von Kleist are investigating how close one can come to an optimum medicine dose using a deterministic model: "Unlike in the stochastic model where the bacteria can be killed off entirely, in the deterministic model, the concentrations of bacteria can ideally become arbitrarily small, and that has an impact on the optimal therapy." The calculations done so far, the two researchers report, show that a dose of medicine optimised on the basis of averages and concentrations achieves a greater probability of success compared to a standardised therapy, and that the optimised therapy can even cope with random fluctuations in the patients' state of health. The researchers call this property ‘robustness of the therapy towards random influences’. "It would naturally be excellent if we could measure the extent of the disease in individual patients and then have an individually adapted strategy at hand that is optimal for those patients. But that is very laborious and so also very expensive. So, we are following a path based on an approximated deterministic model and which leads to "rules of thumb", which give the treating physicians the greatest possible certainty of applying the correct treatment," the mathematicians say.

Their model ought to be useful both for treating with known medicines and for helping develop new antibiotics. For the latter to work, they require better data on the behaviour of certain molecules. The mathematicians are accordingly working with colleagues from the workgroup of Marcus Weber at the Konrad Zuse Centre for Information Technology Berlin. They also have cooperation partners from the medical profession, who provide the necessary patient data. "We also use insights gained from research on HIV patients. So there is justifiable hope that, building on many different parameters, we will come to an applicable model. But we still have a long way to go before we reach that goal," the mathematicians declare.

More information:

Prof. Dr. Carsten Hartmann,

Institute of Mathematics,

Free University of Berlin,

Email: chartman-aet-mi.fu-berlin-punkt-de

Dr. Max von Kleist,

Institute of Mathematics,

Free University of Berlin

Email: max.kleist-aet-fu-berlin-punkt-de

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