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.


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 predatorprey 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 wellknown. 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 LotkaVolterra model. It has been tremendously
extended by mathematical
biologists to study many other
driving forces of population cycles. 