Page 86 - University of Pretoria RESEARCH REVIEW 2016

MATHEMATICAL MODELLING
FOR BIOLOGICAL SYSTEMS
Mathematical modelling is often understood
as statistical analyses of available data and inferences, based on the correlations found. However, correlations are not causations: they are based on assumptions that have to be tested.
Professor Jacek Banasiak, Chair of the DST-NRF South African Research Chair Initiative (SARChI) in Mathematical Models and Methods in Bioengineering and Biosciences (M3B2) at UP, writes
that there is a tendency to treat the results of modelling as universally valid whereas they are always conditional on the assumptions made while building the model. He cites a ‘spectacular example’ of a failure to understand the assumptions of a model, in the history of the Long Term Capital Management hedge fund, whose Board of Directors included Myron S Scholes and Robert C Merton, who shared the 1997 Nobel Memorial Prize in Economic Sciences. Their basis
of operation was the famous Black- Scholes equation for determining the
DISEASE PATHOGENICITY
The Allee effect is particularly important in epidemiological modelling as the introduction of a disease into a population increases the burden it faces and may push the survival threshold above the current size of the population, resulting in its extinction.
In this context, Dr Salisu Garba (left) has collaborated with Professor Alun Lloyd at the North Carolina State University (NCSU), United States. In work recently published, their focus is on the dynamical behaviour of an epidemiological model, with results underscoring earlier findings that the Allee effect and infectious disease are some of the extinction drivers of populations. The study
investigated the combined impact of the Allee effect and infectious disease at higher population levels, to determine which species are most vulnerable to extinction. Their research, using the susceptible-exposed-infectious (SEI) model, shows that the Allee effect, combined with disease pathogenicity, has a detrimental impact on the dynamics of a population at high population levels.
The value of epidemiological models in predicting the Allee effect was clearly demonstrated. For instance, it is well known that highly pathogenic species can cause their own extinction
but not that of their host. They found that there were two thresholds, one being the carrying capacity beyond which the population cannot grow, while the other is the survival threshold that determines the minimum size/density of the population that allows it to survive.
84 | UP Research Review 2016
value of derivatives. Initially successful, with annualised returns of over 40%, the hedge fund lost $4.6 billion in 1998 in
less than four months following the 1997 Asian financial crisis, and in 1998, the Russian financial crisis. The reason for the collapse was that the Black-Scholes model requires that the market fluctuations be small and this assumption failed during large crises.
Banasiak’s point is that while mathematical models are indispensable tools for cheaply testing various scenarios in engineering, or in the natural, medical, economic or social sciences, they only give simplified descriptions of real-life problems and should not be used without understanding their limitations.
A field where mathematical modelling has
been highly successful is in population
dynamics, where models can be used to
predict the size of populations in future.
The development of the population
models is a good example of the iterative
process of modelling. The first population
model, attributed to Thomas Malthus (An Essay on the Principle of Population, 1798) is the so-called exponential or geometric growth model, which correctly predicts development of many populations, as long as there is an abundance of resources to support their growth.
The model predicts fast growth of large populations, which contradicts the observation that the growth of such population slows down due to resource constraints and overcrowding, a shortcoming that was remedied by Pierre François Verhulst who in 1844–1845 introduced the so-called logistic model. As required by the principles of good modelling,
the model agrees with the Malthus model for small populations, predicting their exponential growth. Large populations, however, display much slower growth, finally reaching a stable population size that is referred to as the carrying capacity of the environment. The Verhulst model has been highly