# Exploring the multiscale character of mathematical models of infectious diseases.

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Dr S Tingas, Dr N Wheelhouse  No more applications being accepted  Self-Funded PhD Students Only

The origin of mathematical modeling in epidemiology dates back to the 18th century and since then it has undergone significant developments, thus, contributing to a better understanding of the transmission mechanisms of infectious diseases and to policy decisions. Epidemiological models are typically labelled as Compartmental Models (CM) and describe the interactions among population groups (i.e., compartments) such as Susceptible-Infected-Recovered (SIR), Susceptible-Infected- Susceptible (SIS) and numerous variations, where compartments (e.g., Diseased, Asymptomatic, Exposed, etc) are added, depending on the disease and the desired level of detail. In terms of the mathematical framework, the vast majority of such CMs are deterministic, thus, described by systems of differential equations (ordinary or partial, i.e., ODEs or PDEs, respectively), while stochastic CMs are more rarely used. In the community of deterministic CMs, point models (also called zero-dimensional, since the independent variable is only time) are described by ODEs and they are more largely used against those that carry spatial characteristics and described by PDEs.

Compartmental epidemiological models are multiscale systems, i.e., they consist of processes associated with different timescales. Such systems are typically encountered in many other scientific areas, such as celestial mechanics, fluid flows, electric circuits, biology etc, and their characteristics (from a mathematical perspective) have been investigated for more than a century. Asymptotic analysis has traditionally been used to tackle such systems, also called singularly perturbed. Such approach is significantly advantageous against other less rigorous methods, like sensitivity analysis, since it can lead to a deep understanding with respect to the underlying mechanisms that drive the system’s evolution.

In this research project, mathematical tools from asymptotic analysis will be used in order to explore the dynamics of virus spread of selected infectious diseases. The main objective of this project is to use timescale analysis and develop a rigorous mathematical toolset which will assist decision and policy makers in events of virus transmissions.

A first degree (at least a 2.1) ideally in Biological Sciences or Mechanical Engineering or Computer Science or closely related disciplines with a good fundamental knowledge of applied mathematics and programming.

English language requirement

IELTS score must be at least 6.5 (with not less than 6.0 in each of the four components). Other, equivalent qualifications will be accepted. Full details of the University’s policy are available online.

Essential attributes:

· Experience of fundamental research skills.

· Competent in programming (e.g., Fortran)

· Knowledge of deterministic mathematical models.

· Good written and oral communication skills

· Strong motivation, with evidence of independent research skills relevant to the project

· Good time management

Desirable attributes:

Fundamental knowledge of multiscale systems.

Fundamental knowledge of compartmental models of infectious diseases. Knowledge of machine learning algorithms. Experience of undertaking independent research. A completed (or nearing completion) MSc in a relevant subject area.

Self funded

## References

Keeling, M. J., & Rohani, P. (2011). Modeling infectious diseases in humans and animals. Princeton University Press. Strogatz, S. H. (1996). Nonlinear dynamics and chaos. Ginoux, J. M. (2009). Differential geometry applied to dynamical systems (Vol. 66). World Scientific.
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