Data on healthcare is increasingly collected and processed electronically, creating both opportunities and challenges. Fortunately, recent years have seen major developments in the field of mathematical epidemiology, which systematically models the patterns of disease and health in the population to disentangle the complexities of the available data. This project seeks to drive some key insights from mathematical epidemiology into the healthcare system, particularly relating to: the analysis of infectious disease data collected at the household level; how different viruses linked to colds and 'flu hinder and help each others' spread through the population; understanding how people influence each others' health-related behaviours; and cancer radiotherapy. Other aspects of the project are technical, but seek to answer the questions: What should be done with data that are not experimental, but appear in an uncontrolled way such as during disease outbreaks? And what should we do with data that are constantly being collected through routine surveillance?

A few grams of any material contain a bewildering number of individual particles. Interactions between these particles give rise to a vast array of emergent phenomena which cannot be understood from looking at any of the particles in isolation. An important example of this is superconductivity, which enables materials to conduct electricity without resistance. Novel emergent states also occur out of equilibrium, due to the presence of large external forces or the occurrence of extreme events. Examples include turbulence in fluids and plasmas, the spreading of epidemics and diseases, and shocks in the stock market. The above examples illustrate the breadth of this nationally and internationally recognised "Grand Challenge" in Emergence and Physics Far From Equilibrium. Addressing this Grand Challenge requires a coordinated approach, spanning different areas of physics and related disciplines. The Network will facilitate cross-cutting workshops and advanced working groups to enable UK researchers to plan and carry out targeted research programmes. Pump-prime initiatives and interaction with industry will stimulate collaborative research, ensuring UK competitiveness in this far-reaching field.

We live in the "Era of Mathematics" (UKRI, 2018), in which mathematics research has deep and widespread impact. Medical imaging is enhanced using the theory of inverse problems. Predicting sewage contamination in waterways after storms requires solving complicated systems of hydrodynamic equations. Machine learning tools are revolutionising data-intensive computing and, handled with proper mathematical care, have vast potential benefits for science and society. These are examples of the ongoing explosion in mathematical innovation driving, and being driven by, the analysis and modelling of data running through every aspect of life. Cutting-edge research now sits at the interface of data science and mathematical modelling. Methods and fields such as compressed sensing, stochastic optimisation, neural networks, Bayesian hierarchical models - to name but a few - have become interwoven and contributed to the delivery of a new domain of research. We refer to this research interface as "statistical applied mathematics". Established in 2014, the Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa, samba.ac.uk) delivers leading research and training in this space. In the development of this bid, we have consulted widely with academic, industrial, and governmental partners, who consistently report a large and widening gap between demand and supply for highly skilled graduates. Our vision is to create a new generation of statistical applied mathematicians ready to lead high-impact, data-driven, mathematically-robust research in academia and industry. We will nurture a vibrant culture of cohort learning, enabling internationally-leading training in modern mathematical data science. A particularly important research focus will be the synthesis of data-driven methods with robust mathematical modelling frameworks. Tomorrow's industrial mathematicians and statisticians must understand when machine learning tools are (and are not) appropriate to use and be able to conduct the underpinning research to improve these tools by integrating scientific domain knowledge. This research challenge is informed by deep partnerships with a range of industry and government bodies. Our long-term partners such as BT, Syngenta, Novartis, the NHS, and the Environment Agency co-create our vision and our training. They are emphatic that we must address the urgent need for mathematical data science talent in this key strategic area for the UK economy. Many of our students will work directly on industry challenges during their PhD either in their core research or with internships. Our unique Integrative Think Tanks are the key mechanism for exploring new research ideas with industry. These are week-long events where SAMBa students, leading academics, and partners work together on industrial and societal problems. SAMBa graduates will be able to develop and apply new ideas and methods to harness the power of data to tackle challenges affecting society, the economy, and the environment. Our students will move into academia, providing sustainability to the UK's capacity in this field, as well as industry and government, providing impact through societal benefits and driving economic growth. Many alumni now hold permanent positions at leading UK universities and senior positions in a range of businesses. The CDT will be embedded within the University of Bath's Department of Mathematical Sciences, where 98% of the research is world leading or internationally excellent (REF2021). The CDT is supported by 58 academics in maths, with similar numbers of co-supervisors from industry and other departments. The centre will be co-delivered with 22 industry and government partners. A vital international perspective is provided by a worldwide network of 11 academic institutions sharing our scientific vision.

The current AI paradigm at best reveals correlations between model input and output variables. This falls short of addressing health and healthcare challenges where knowing the causal relationship between interventions and outcomes is necessary and desirable. In addition, biases and vulnerability in AI systems arise, as models may pick up unwanted, spurious correlations from historic data, resulting in the widening of already existing health inequalities. Causal AI is the key to unlock robust, responsible and trustworthy AI and transform challenging tasks such as early prediction, diagnosis and prevention of disease. The Causality in Healthcare AI with Real Data (CHAI) Hub will bring together academia, industry, healthcare, and policy stakeholders to co-create the next-generation of world-leading artificial intelligence solutions that can predict outcomes of interventions and help choose personalised treatments, thus transforming health and healthcare. The CHAI Hub will develop novel methods to identify and account for causal relationships in complex data. The Hub will be built by the community for the community, amassing experts and stakeholders from across the UK to 1) push the boundaries of AI innovation; 2) develop cutting-edge solutions that drive desperately needed efficiency in resource-constrained healthcare systems; and 3) cement the UK's standing as a next-gen AI superpower. The data complexity in heterogeneous and distributed environments such as healthcare exacerbates the risks of bias and vulnerability and introduces additional challenges that must be addressed. Modern clinical investigations need to mix structured and unstructured data sources (e.g. patient health records, and medical imaging exams) which current AI cannot integrate effectively. These gaps in current AI technology must be addressed in order to develop algorithms that can help to better understand disease mechanisms, predict outcomes and estimate the effects of treatments. This is important if we want to ensure the safe and responsible use of AI in personalised decision making. Causal AI has the potential to unearth novel insights from observational data, formalise treatment effects, assess outcome likelihood, and estimate 'what-if' scenarios. Incorporating causal principles is critical for delivering on the National AI Strategy to ensure that AI is technically and clinically safe, transparent, fair and explainable. The CHAI Hub will be formed by a founding consortium of powerhouses in AI, healthcare, and data science throughout the UK in a hub-spoke model with geographic reach and diversity. The hub will be based in Edinburgh's Bayes Centre (leveraging world-class expertise in AI, data-driven innovation in health applications, a robust health data ecosystem, entrepreneurship, and translation). Regional spokes will be in Manchester (expertise in both methods and translation of AI through the Institute for Data Science and AI, and Pankhurst Institute), London (hosted at KCL, representing also UCL and Imperial, leveraging London's rapidly growing AI ecosystem) and Exeter (leveraging strengths in philosophy of causal inference and ethics of AI). The hub will develop a UK-wide multidisciplinary network for causal AI. Through extended collaborations with industry, policymakers and other stakeholders, we will expand the hub to deliver next-gen causal AI where it is needed most. We will work together to co-create, moving beyond co-ideation and co-design, to co-implementation, and co-evaluation where appropriate to ensure fit-for-purpose solutions Our programme will be flexible, will embed trusted, responsible innovation and environmental sustainability considerations, will ensure that equality diversity and inclusion principles are reflected through all activities, and will ensure that knowledge generated through CHAI will continue to have real-world impact beyond the initial 60 months.

We propose a new phase of the successful Mathematics for Real-World Systems (MathSys) Centre for Doctoral Training that will address the call priority area "Mathematical and Computational Modelling". Advanced quantitative skills and applied mathematical modelling are critical to address the contemporary challenges arising from biomedicine and health sectors, modern industry and the digital economy. The UK Commission for Employment and Skills as well as Tech City UK have identified that a skills shortage in this domain is one of the key challenges facing the UK technology sector: there is a severe lack of trained researchers with the technical skills and, importantly, the ability to translate these skills into effective solutions in collaboration with end-users. Our proposal addresses this need with a cross-disciplinary, cohort-based training programme that will equip the next generation of researchers with cutting-edge methodological toolkits and the experience of external end-user engagement to address a broad variety of real-world problems in fields ranging from mathematical biology to the high-tech sector. Our MSc training (and continued PhD development) will deliver a core of mathematical techniques relevant to all applied modelling, but will also focus on two cross-cutting methodological themes which we consider key to complex multi-scale systems prediction: modelling across spatial and temporal scales; and hybrid modelling integrating complex data and mechanistic models. These themes pervade many areas of active research and will shape mathematical and computational modelling for the coming decades. A core element of the CDT will be productive and impactful engagement with end-users throughout the teaching and research phases. This has been a distinguishing feature of the MathSys CDT and is further expanded in our new proposal. MSc Research Study Groups provide an ideal opportunity for MSc students to experience working in a collaborative environment and for our end-users to become actively involved. All PhD projects are expected to be co-supervised by an external partner, bringing knowledge, data and experience to the modelling of real-world problems; students will normally be expected to spend 2-4 weeks (or longer) with these end-users to better understand the case-specific challenges and motivate their research. The potential renewal of the MathSys CDT has provided us with the opportunity to expand our portfolio of external partners focusing on research challenges in four application areas: Quantitative biomedical research, (A2) Mathematical epidemiology, (A3) Socio-technical systems and (A4) Advanced modelling and optimization of industrial processes. We will retain the one-year MSc followed by three-year PhD format that has been successfully refined through staff experience and student feedback over more than a decade of previous Warwick doctoral training centres. However, both the training and research components of the programme will be thoroughly updated to reflect the evolving technical landscape of applied research and the changing priorities of end-users. At the same time, we have retained the flexibility that allows co-creation of activities with our end-users and allows us to respond to changes in the national and international research environments on an ongoing yearly basis. Students will share a dedicated space, with a lecture theatre and common area based in one of the UK's leading mathematical departments. The space is physically connected to the new Mathematical Sciences building, at the interface of Mathematics, Statistics and Computer Science, and provides a unique location for our interdisciplinary activities.