Finite time orbitally stabilizing synthesis of complex dynamic systems with bifurcations with application to biological systems

Lead Research Organisation: University of Exeter
Department Name: Engineering Computer Science and Maths


Stabilization, in finite time rather than asymptotically, of linear and non-linear dynamical systems is an active current area of research internationally. In much of the existing work finite time convergence of a Lyapunov function to the origin of the state space is achieved using an increasing condition on that Lyapunov function given by a differential inequality which is dependent upon the decay rate and both known and uncertain system parameters. The proof of finite time stability on the basis of such a strong Lyapunov function satisfying a differential inequality poses a challenge when compared to proofs of Lyapunov theorems relating to asymptotic stability considerations. The task can be further complicated when the paradigm requires not only a settling time estimate but also seeks to achieve parameter selections for a control strategy to ensure an apriori chosen settling time is achieved. Recent work by the investigators in the domain of mechanical systems has obtained corresponding results using a homogeneity approach where the methodology is founded on a quasihomogeneity principle of possibly discontinuous systems, and thus a broader range of uncertainty is permitted than in the existing literature. Finite time stability which is uniform in the initial data and in the uncertainty is possible, a feature that cannot be guaranteed using existing methods. A finite upper bound on the settling time is determined without the need to find a Lyapunov function satisfying a differential inequality. Work has developed a single Lyapunov function for uncertain, discontinuous mechanical systems to provide global finite time stability to the origin of the system in the presence of velocity jumps without having to analyze the Lyapunov function at the jump instants and has developed parameterisations of sliding mode controllers that ensure finite time stabilisation where the designer specifies a convergence time and controller parameters are explicitly computed as a function of the required convergence time. The current proof of concept demonstrates that finite time stability characteristics can be imposed in possibly discontinuous systems and provides an exciting platform to explore more complex practical scenarios of current interest. It is clear that current methods which analyse systems based upon an assumption of an infinite time horizon are frequently flawed. For example, individual clonal immune cell populations are required to expand and become activated for limited time. Further in the natural world, discontinuity is frequently found as a result of evolution. This project seeks to broaden the system class to which the developed theoretical framework can be applied to encompass such biological dynamics. One specific driver is to parameterise and assess the bifurcations present in the immune system, where a key paradigm is to investigate how a triggering event may move the immune system from the healthy to the autoimmune state and also how control paradigms can be used to postulate treatment to move the system back to the healthy state. Autoimmune disease affects 50 million people in the USA where it is one of the top ten causes of death in women under 65, is the second highest cause of chronic illness, and is the top cause of morbidity in women. The number of cases of autoimmune disease are rising across the world. This rise in the number of people affected and the absence of robust treatment regimes results in the incidence of autoimmune disease contributing significantly to the rise in healthcare spending as well as loss of productivity in the workforce and of course poor quality of life for those affected. There is currently no mechanism-based, conceptual understanding of autoimmune disease. This project seeks to develop and apply emerging methods from finite time stabilisation of uncertain possibly discontinuous dynamic systems to this problem.
Description During this grant we have developed a mathematical model to help us understand why plaques arise in the skin of people with Psoriasis. The model is novel in that it takes into account the ways in which different types of cells communicate with each other via signalling molecules called cytokines. Formulating the model in this way has allowed us to study the ways in which these communication pathways can cause imbalances in the number of cells that are present in the skin, which is a key feature of Psioratic plaques. The model further allows us to study how emerging treatment methods that target cytokines might alleviate symptoms of the disease.

We have shown that disease and healthy conditions exist in the model as "steady states". Typically, modelling approaches that aim to understand diseases in terms of steady states do not account for the possibility that such states may take a very long time to reach in the model. In our research we showed that the communication pathways in the model allow disease and healthy states to be reached in "finite time", thus increasing our confidence in the model as relevant for the study of Psoriasis. In studying the model we have demonstrated that certain communication pathways are particularly pertinent for Psoriasis. For example, we have shown in the model that varying levels of two cytokines, IL-23 and IL-17, can control transitions between a healthy and Psioratic state. This is in line with clinical use of drugs targeting these cytokines to help treat Psoriasis. We have further demonstrated that pathways involving other cytokines can cause alternative dynamic routes to emerge in the model, such as the coexistence of healthy and disease states (bistability). This paves the way for predictions regarding optimal, short-term treatments that may displace skin from its Plaque state into a healthy state.
Exploitation Route Our findings can be taken forwards in various ways, including theoretical and experimental research as well as translation into the clinical setting in the future. The model is complex and requires further analysis in order to fully explore the dynamic routes to the Psoriasis state and therefore uncover optimal interventions to switch back to the healthy state. The model presents a novel and succinct characterisation of cytokine signalling and autoimmune pathways involved in Psoriasis, and therefore could lead to further experimentation in order to quantify model parameters and test predictions. In particular it would be important in the future to test predictions regarding the ways in which changes in cytokine levels affect the size of populations of immune and intrinsic skin cells. Further into the future we envisage the use of our model to plan patient-specific interventions for Psoriasis. Parameterising the model using clinical data would allow to determine which Psoriasis pathway is pertinent for a given individual and further determine which of a suite of available treatment methods would be most appropriate to alleviate Psioratic plaques.
Sectors Healthcare,Pharmaceuticals and Medical Biotechnology

Description Work from this project was used in a science demonstration as part of the Sidmouth Festival of Science 2015. Dr Goodfellow alongside one of our clinical collaborators (Dr Al-Nuaimi) presented the concepts of how a mathematical model could be used to advance our understanding of psoriasis.
First Year Of Impact 2015
Sector Healthcare
Impact Types Societal

Description Impact Incubator
Amount £5,000 (GBP)
Organisation University of Exeter 
Sector Academic/University
Country United Kingdom
Start 06/2016 
End 07/2016
Title Psoriasis model 
Description We have developed a new mathematical model of psoriasis pathology that incorporates interactions between cells and cytokines. This model is being used to investigate mechanisms of disease and treatment efficacy. 
Type Of Material Computer model/algorithm 
Provided To Others? No  
Impact The model has allowed us to understand the role of cytokines as actuators in keratinocyte dynamics, that act in finite time, thus potentially leading to improved treatment of psoriasis 
Description Collaboration with Exeter Dermatology team 
Organisation Devon Partnership NHS Trust
Country United Kingdom 
Sector Public 
PI Contribution We are working with Dr. Y Al-Nuaimi to advance understanding of psoriasis pathogenesis and treatment efficacy
Collaborator Contribution Dr Al-Nuaimi has dedicated time to assist us in understanding aspects of psoriasis pathogenesis and to aid in model development. Further, Dr. Al-Nuaimi has arranged for R Pandey to sit in clinics to further our understanding of the disease.
Impact Collaboration is multi-disciplinary. Our recent outputs (Oza et al. article under review at International Journal of Control and article in preparation (R.Pandey et al.)) are derived from this collaboration. Further, R Pandey is planning to undertake sit-in sessions in the psoriasis clinic. Disciplines involved: medicine / healthcare, mathematics
Start Year 2014
Description Manchester Dermatology Department 
Organisation Salford Royal NHS Foundation Trust
Country United Kingdom 
Sector Public 
PI Contribution We developed a mathematical model of cellular cytokine mechanisms of psoriasis of potential clinical relevance.
Collaborator Contribution A member of the Manchester Dermatology Department dedicated 3 months of academic research time to provide assistance with model development and suggest avenues for future clinical relevance.
Impact Please see publications for conference paper relating to this work.
Start Year 2014
Description Sidmouth Science Festival 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Public/other audiences
Results and Impact We ran a stand at the Sidmouth Science Festival introducing the electrical rhythms of the brain through a wireless EEG set-up. During the day we had more than 30 volunteers and around 75 people visit the stand, to have the brain activity recorded or to discuss what the rhythm meant.
Year(s) Of Engagement Activity 2015,2016,2017