## Centre for Mathematical Biology

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 What is mathematical biology? Mathematical models have played an important role in our understanding of almost all areas of biology. A model is an abstraction of key components and processs of a system. We analyse these simplified versions of the world to understand the mechanisms that connect system components, examine the consequences of these connections, and predict how changes in one part of the system show up in other parts. A few examples of the many areas of biology in which mathematical models have been used are presented below.
 Controlling infectious disease The spread of diseases is governed by contact between people. The way in which epidemics unfold can be understood and, to some extent, predicted by modelling the processes of contact and transmission. The first systematic models were developed in the early twentieth century. These early models were highly stylised but provided invaluable insights. Notably, a quantity known as R0 was derived that depended on just a few  parameters such as population size, transmission probability, duration of infection and it was shown that a significant epidemic will only occur if R0 > 1. The modelling frameworks introduced in these pioneering studies underpin most modern models and the ultimate goal of most public health policy is still to reduce R0 below 1. Cell migration in embryo formation Neural crest cells in the early embryo form part of many diverse tissue types. Failure of neural crest cells to travel to their target locations can result in a wide range of human birth defects known as neurocristopathies. Mathematical models have been used to verify biological hypotheses about the cause of some neurocristopathies. These models often incorporate randomness at the cellular level as it can lead to interesting and counterintuitive phenomena. However, such models provide realism at the cost of significant computational resources. So mathematical techniques are used to make the models as efficient as possible. For example, hybrid (mixed) representations combine the best features of existing methodologies and algorithms to ensure feasible simulation times for complex biological models. Drug delivery and monitoring The success of many long term clinical treatments relies on the effective monitoring of drug levels in the patient to ensure they remain within a safe range. In conjunction with clinical trials, mathematical models are often used to predict these drug levels and inform dosage guidelines. Traditional methods of drug monitoring use blood samples from the patient. More recent developments use very small electric currents applied to the skin. With this type of monitoring machinery there is a time lag before a realistic estimate of drug levels in the body is obtained. Mathematical models can be used to predict this time lag and can also help us to understand how to use archive information in the skin barrier to determine whether patients have complied with their drug regime. Population cycles Many species show regular cycles in their population densities. Hare and lynx in North America are among the most prominent examples. Often these cycles cannot be explained by seasonality. But predator-prey interactions can intrinsically lead to population cycles: More prey (e.g. hares) allows the predator (e.g. lynx) population to grow. More predators diminish the prey population, upon which the predator population diminishes, the prey recovers, and the cycle starts anew. Nowadays this mechanism is well-known. But, in the 1920s, a set of equations developed independently by the American biophysicist Lotka and the Italian mathematician Volterra were crucial in elucidating the phenomenon of population cycles. These equations are now known as the Lotka-Volterra model. It has been tremendously extended by mathematical biologists to study many other driving forces of population cycles.

Centre for Mathematical Biology, Department of Mathematics, University of Bath, Bath BA2 7AY Tel: +44 (0) 1225 386320· E-mail: b.adams@bath.ac.uk, e.feil@bath.ac.uk

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