Mathematical modelling

Mathematical models require the combination of the highly sought after skills of mathematicians, computer technologists and research scientists. Models provide a platform to study the mechanisms that enable diseases to spread, and for predicting outcomes and patterns that are highly complex.

“Mathematical modelling enables us to make predictions under various regimes. In particular, we can estimate the impact of global warming on spread of endemic zones of disease [...] We are resuming this research. This autumn, we will run a series of experiments to study in more detail how agricultural chemicals affect the level of phages in the soil, how phages and bacteria interact in it, and which limiting factors are at play there" Dr A. Morozov

Mathematical modelling in infectious disease can identify:

  • Disease progression
  • Outcome of epidemics
  • Inform public health interventions

Parameters are determined mathematically, and these are manipulated to calculate effects of interventions such as vaccinations in large.

Phase variation in Campylobacter jejuni and Neisseria meningitidis 

Phase variation is a method by which bacterial populations rapidly generate genotypic and phenotypic heterogeneity. At Leicester we study two major mechanisms of phase variation - (a) the switching of gene expression from an ON to an OFF phase by reversible insertion or deletion mutations in hyper-mutatable simple sequence repeats, and (b) epigenetic phase variation driven by recombination events in methyl transferase genes, leading to altered site specificity of methylation, and hence gene expression.

Dr Chris Bayliss and colleagues have assigned a mathematical model to bacteria that show phase variation, such as Campylobacter jejuni and Neisseria meningitidis.

Dr Bayliss’s group work on a mechanism of phase variation in Gram-negative bacteria afforded by insertions or deletions (InDels) of ‘repeat units’ in simple sequence repeats (SSRs). SSRs are prone to hypermutation, and if they are present inside reading frames, InDels can rapidly lead to the ON/OFF switching of gene expression (B).

The stochastic mathematical model can be ‘used to examine experimentally observed populations and determine whether mutation rate alone or mutation rate and selection for changes in expression of one or more loci were driving changes in bacterial population structure.’

The randomness of mutation, the changes in rate of mutation and the number of genes that are phase variable mean that the predictions are much more complex than seen in the simple schematic above, and this is where Dr Bayliss’ and team apply their mathematical models.

As survival of phase variable bacteria is not in response to environmental change but due to the hyper-mutation that involves phase variation, increasing the chances of survival in the environment.

A ‘mutation-selection model’ was proposed, where some phasotypes are favoured over others in the promotion of bacterial growth. Using adequate assumptions the team were able to apply algorithms

This is particularly useful for Neisseria meningitidis as phase variation of its surface antigens can help the bacteria to evade the immune response.

The seasonal and daily dynamics of Burkholderia pseudomallei

Dr Ed Galyov and Dr Andrew Morozov collaborated to model the outcomes of bacteriophage interactions to asses the potential impact of phages on seasonality of Melioidosis (caused by the bacteria Burkholderia pseudomallei), which is a serious tropical disease.

Melioidosis is a serious infectious disease manifesting itself as multiple abscesses of various organs due to severe sepsis. In 40% of the cases, the disease leads to death. Patients with diabetes are at a higher risk of being infected. Melioidosis, which is also known as Whitmore’s disease, is caused by the bacterium Burkholderia pseudomallei, which is particularly active in soil and water. This pathogen is widespread in Southeast Asia, Australia, West and East Africa.

Study co-author Dr Andrew Morozov, from the University of Leicester explains:

“Melioidosis is a severe and dangerous disease, but much of the research into it focuses on the bacterium that causes. Meanwhile, the phages abundantly found in the habitat of the pathogen are not getting enough attention. We wondered whether we could predict the variation in the number of pathogenic bacteria and the impact of phages on it depending on season and environmental conditions.”

To achieve this goal, the researchers developed several mathematical models predicting the seasonal and daily dynamics of the size of B. pseudomallei populations. They focused on the bacterial populations on rice fields in two Thailand provinces — Nakhon Phanom and Sa Kaeo — and how the pathogen is affected by temperature-dependent phages, viruses selectively killing bacteria. The variation was explored in terms of the change in temperatures and ultraviolet radiation levels.

Simulation results show that the period from March to September poses the greatest threat, because that is when phages killing B. pseudomallei are at their lowest due to high levels of UV radiation. The drop in the number of phages in spring and summer means there are more bacteria unaffected by the virus. At constant UV exposure, temperature takes over as the decisive factor. Specifically, at more than 35 degrees Celsius, the phage enters the so-called lytic cycle when it destroys bacterial cells, reducing the pathogen population. The model indicates phage-free bacteria numbers are at their highest around 9 a.m. and 8 p.m.

Besides UV rays, the number of phage-free bacteria is affected by the use of fertilizers. Recent studies indicate that iron-based chemicals kill off phages, but the effects of other fertilizers are unknown. Since they result in unpredictable dynamics of B. pseudomallei populations, they might pose a risk of more frequent infections in humans.