The Symbiotic Contact Process on Non-Lattice Structures

The symbiotic contact process (SCP) was introduced by de Oliviera, Dos Santos, and Dickman (2012) and is an interacting particle system for modelling two different species. The study of the SCP on the lattice was extended by Durrett and Yao (2020) who gave results for the critical value for the infection parameter between extinction and survival. Our aim will be to further this research by studying the process on non-lattice structures. Structures such as random graphs with power law behavior are frequently being used to model human interactions and relationships such as modelling connections people have on social media sites. Surprising results have recently been found for the contact process on these random structures (Chatterjee and Durrett (2009), Huang and Durrett (2020)) and we wish to extend this area of research.

The process can be seen as an adaptation of the contact process with two particle types 'A' and 'B'. Both particle types infect their neighbours at a rate lambda. The particles of both types die at rate one unless both types are present at the same site and then they die at rate mu, strictly less than one, giving the symbiotic nature of the model. This model is motivated by symbiotic relationships that naturally occur with the two most useful motivational examples being the symbiotic survival of two species, and the worse recovery rate of a patient with two diseases. Our short term aim is to study the critical values for infection parameter, as a function of mu, for this process on a Galton-Watson tree, the starting point for the research on the contact process on random structures. More specifically, we will be aiming to compare these values with the corresponding critical values for the standard contact process; we hope that the symbiotic nature of our model will lead to critical values that are smaller than those for the corresponding contact process. We will aim to extend our study to other random structures including Erdos-Renyi random graphs and dynamic random graphs with power law behavior. The overall goal is to fully characterise phase transitions of the process on these non-lattice structures. The mathematical theory that we will use to help analyse the process on these structures frequently hail from percolation theory and martingale theory.

Combining specialised modelling techniques with complex data analysis in order to deliver prediction with quantified uncertainties lies at the heart of many of the major challenges facing UK industry and society over the next decades. Indeed, the recent Government Office for Science report "Computational Modelling, Technological Futures, 2018" specifies putting the UK at the forefront of the data revolution as one of their Grand Challenges.

The beneficiaries of our research portfolio will include a wide range of UK industrial sectors such as the pharmaceutical industry, risk consultancy, telecommunications and advanced materials, as well as government bodies, including the NHS, the Met Office and the Environment Agency.

Examples of current impactful projects pursued by students and in collaboration with stake-holders include:

- Using machine learning techniques to develop automated assessment of psoriatic arthritis from hand X-Rays, freeing up consultants' time (with the NHS).

- Uncertainty quantification for the Neutron Transport Equation improving nuclear reactor safety (co-funded by Wood).

- Optimising the resilience and self-configuration of communication networks with the help of random graph colouring problems (co-funded by BT).

- Risk quantification of failure cascades on oil platforms by using Bayesian networks to improve safety assessment for certification (co-funded by DNV-GL).

- Krylov regularisation in a Bayesian framework for low-resolution Nuclear Magnetic Resonance to assess properties of porous media for real-time exploration (co-funded by Schlumberger).

- Machine learning methods to untangle oceanographic sound data for a variety of goals in including the protection of wildlife in shipping lanes (with the Department of Physics).

Future committed partners for SAMBa 2.0 are: BT, Syngenta, Schlumberger, DNV GL, Wood, ONS, AstraZeneca, Roche, Diamond Light Source, GKN, NHS, NPL, Environment Agency, Novartis, Cytel, Mango, Moogsoft, Willis Towers Watson.

SAMBa's core mission is to train the next generation of academic and industrial researchers with the breadth and depth of skills necessary to address these challenges. SAMBa's most sustained impact will be through the contributions these researchers make over the longer term of their careers. To set the students up with the skills needed to maximise this impact, SAMBa has developed a bespoke training experience in collaboration with industry, at the heart of its activities. Integrative Think Tanks (ITTs) are week-long workshops in which industrial partners present high-level research challenges to students and academics. All participants work collaboratively to formulate mathematical
models and questions that address the challenges. These outputs are meaningful both to the non-academic partner, and as a mechanism for identifying mathematical topics which are suitable for PhD research. Through the co-ownership of collaboratively developed projects, SAMBa has the capacity to lead industry in capitalising on recent advances in mathematics. ITTs occur twice a year and excel in the process of problem distillation and formulation, resulting in an exemplary environment for developing impactful projects.

SAMBa's impact on the student experience will be profound, with training in a broad range of mathematical areas, in team working, in academic-industrial collaborations, and in developing skills in communicating with specialist and generalist audiences about their research. Experience with current SAMBa students has proven that these skills are highly prized: "The SAMBa approach was a great template for setting up a productive, creative and collaborative atmosphere. The commitment of the students in getting involved with unfamiliar areas of research and applying their experience towards producing solutions was very impressive." - Dr Mike Marsh, Space weather researcher, Met Office.