- UK/EU/International: Worldwide (International, UK and EU)
- Value: This project is open to self-funded students and is eligible for funding from the School of Mathematics Scholarships, EPSRC Doctoral Training Partnerships, and Leeds Doctoral Scholarships.
- Number of awards: 1
- Deadline: Applications accepted all year round
Contact Dr Martín López-García to discuss this project further informally.
During the last years, increasing efforts are being devoted to better understanding a wide range of processes arising in Health & Disease. In particular, mathematical modelling of infectious diseases is a research area which has been increasingly growing during the last years. When analysing a process in Health & Disease, both deterministic and stochastic approaches can be implemented. Deterministic approaches imply that, when analysing a, for example, Biological system, identical initial conditions would always lead to the same future dynamics of the corresponding process.
On the other hand, stochastic approaches are used to incorporate randomness when modelling these processes, so that two observations of the same process with identical initial conditions can still lead to different future dynamics or final outputs. Stochastic approaches are usually more appropriate when analysing the spread of a disease among individuals in a small or highly heterogeneous population, when considering Biological processes where extinction events play an important role, or for modelling processes which are intrinsically and naturally random.
The stochastic modelling of processes in Health & Disease usually leads to the analysis of continuous-time Markov chains, where one of the main assumptions is that events occur in these processes after Exponentially distributed random times. This hypothesis, which can be too strict when studying some particular systems, allows one to analyse these processes both from a mathematical and computational perspective.
The development and spread of infectious diseases is a combination of processes occurring at different scales: from the genetic and cellular, to the host and population levels; see, for example, the research paper , where our research group at Leeds focuses on analysing, at the cellular level, molecular processes involved in the immune response against infections, research paper  where we analyse processes at the within-host level related to the dynamics of cell populations which form part of our immune system during non-infection periods, or research paper  where we analyse the spread of an epidemic among a small group of highly heterogeneous individuals.
In recent years, several challenges have recently arisen in this area. A particular challenge is, when modelling the spread of an infectious disease among individuals in a population, to incorporate the heterogeneities at the individual level which usually exist in reality, and which can have a significant impact on the spread of an epidemic in this population. Incorporating heterogeneities at the individual level usually lead to analysing random processes (e.g., continuous-time Markov chains) defined on networks. The analysis of random processes on networks is not only relevant when analysing epidemic processes occurring among individuals in heterogeneous populations, but is also a powerful tool for analysing gene regulatory networks or signalling pathways involved in cellular processes.
A second challenge that has been recently identified in mathematical modelling in Health & Disease is to develop new tools in order to incorporate non-exponential events in Markov processes, since non-exponential events naturally arise in these processes in real life. In this project, we aim to analyse processes involved in the development and spread of infectious diseases at different scales, while addressing these and other challenges in this area. It will involve the analysis of continuous-time Markov chains [4, Chapter 6], birth-and-death processes [5, Chapter 6], and structured Markov processes [6, Chapter 3].
This project will involve the development of new and use of existing both analytical and computational approaches, and will thus include a significant amount of programming. The candidate should also be willing to participate in inter-disciplinary collaborations potentially arising during the project.
 de la Higuera L, López-García M, Lythe G, Molina-París C (2017) IL-2 stimulation of regulatory T cells: a stochastic and algorithmic approach. In: Pahle J, Matthäus F, Graw F (eds.) Modeling Cellular Systems, 81-105.
 Artalejo JR, Gómez-Corral A, López-García M, Molina-París C (2017) Stochastic descriptors to study the fate and potential of naive T cell clonotypes in the periphery. Journal of Mathematical Biology. 74: 673-708.
 López-García M (2016) Stochastic descriptors in an SIR epidemic model for heterogeneous individuals in small networks. Mathematical Biosciences 271: 42-61.
 Kulkarni VG (1995) Modeling and analysis of stochastic systems. Boca Raton, CRC Press.
 Allen LJS (2010) An introduction to stochastic processes with applications to biology. New Jersey, CRC Press.
 He QM (2014) Fundamentals of matrix-analytic methods. New York, Springer.
Applications are invited from candidates with or expecting a minimum of a UK upper second class honours degree (2:1), and/or a Master's degree in mathematics or relevant subject.
If English is not your first language, you must provide evidence that you meet the University's minimum English Language requirements.
How to apply
Formal applications for research degree study should be made online through the university's website. Please state clearly in the research information section that the PhD you wish to be considered for is 'Stochastic processes in Health & Disease' as well as Dr Martín López-García as your proposed supervisor.
We welcome scholarship applications from all suitably-qualified candidates, but UK black and minority ethnic (BME) researchers are currently under-represented in our Postgraduate Research community, and we would therefore particularly encourage applications from UK BME candidates. All scholarships will be awarded on the basis of merit.
If you require any further information please contact the Graduate School Office, e: firstname.lastname@example.org