School of Mathematical and Computer Sciences

In order to support the implementation of the project a Local Management Board (LMB) will be established at the University of Limpopo which will consist of the Deputy Vice Chancellor for Research and Innovation, the Director for Research, the Director for International Affairs, the University of Limpopo Project Coordinator, the Executive Dean of Faculty of Science and Agriculture and the Financial Administrator. The University of Sussex Projector Coordinator will also be part of the LMB representing the Consortium from an International perspective. The LMB will meet once a year and the board will be responsible for the financial management, administration of the grant and approval of funds for the various activities of the project.

The following activities will be funded:

Tuition fees, Intensive short courses and training, Mid-term workshop, International mini-symposia, UK visits, Local conferences, International conferences, Staff replacement costs, Administrative costs, Operational costs, Dissemination costs and External evaluation costs.

A rigorous doctoral candidate recruitment process will be undertaken with Holders of relevant Masters Degree. The doctoral candidate must be less than 45 years of age, black academic staff and should demonstrate ability to obtain the PhD degree within 5 years. Recruitment preferences will be given to previously disadvantaged black academic staff from the universities within the consortium, if the quota is not met, the opportunity will then be advertised to all 26 universities.

The doctoral candidate must meet all the admission requirements for doctoral studies for the institution he/she intends to enrol.

The doctoral candidate must meet all the admission requirements for doctoral studies for the institution he/she intends to enrol. The doctoral candidate must be less than 45 years of age, black academic staff and should demonstrate ability to obtain the PhD degree within 5 years. Recruitment preferences will be given to previously disadvantaged black academic staff from the universities within the consortium, if the quota is not met the opportunity will then be advertised to all 26 universities.

All applications for funding should be sent to Dr Farai Mhlanga (farai.mhlanga@ul.ac.za). Possible candidates must submit a brief Curriculum Vitae (CV), a certified copy of the Masters Degree Certificate and the transcripts. In the CV, please include 3 contactable referees.

The following benefits are envisaged:

• Enrolled academic staff members will acquire PhD Degree through international collaboration.
• Enrolled academic staff members will acquire expertise in Applied Mathematics from various institutions.
• Facilitation of research collaboration and mobility of both enrolled staff and their supervisors.
• Increase job opportunities for newly trained staff. The trained staff will recruit students for higher qualifications who will better serve as qualified personnel after the completion of their studies.
• Career enhancement such as promotions.
• International exposure through research visits, and summer schools.
• The South African institutions will benefit by having increased qualified permanent employees.
• Building supervision and mentoring capacity which will help increase research output through publications
• Quality of research is enhanced through mutual scrutiny from partner colleagues.
• Participation in collaborative peer-reviewed publication process across international institutions.
• The UK universities will benefit from staff and researcher exchange visits thereby fostering international relations with South African Universities.
• The South African researchers (not only within the network) but also those from other Universities will become aware of the International statue of the University and this will hopefully bring more postgraduate applications in Mathematical Sciences, thereby increasing student recruitment.
• South Africa has shortage of skilled work force in Epidemiology, Biophysics, Financial Mathematics and Ecology, both in academia as well as in the private sector, so this project will help alleviate the shortage by training staff at doctoral levels.
• The doctoral students will benefit from supervision by international experts in different fields of Applied Mathematics.
• The institutions will benefit as the graduates from the program will eventually solve the shortage in academia and contribute in the scientific growth in research and supervision capacity.

The project will consist of one cohort of 10 academic staff and will be implemented over a five-year period from 2020 until 2024. PhD co-supervision will be employed where a student is allocated a team of experts in the chosen area of research. Each supervision team will consist of at least one SA-based supervisor and one UK-based supervisor. The supervision will be carried out through various forms that include intensive short courses and training, proposal writing workshops, meetings and workshops, mid-term review workshop, writing retreats and international visits. Risk management and prevention strategies will be developed and agreed to ensure that most if not all the 10 PhD candidates successfully complete their PhDs without any withdrawals. The supervisors possess an excellent record on PhD supervision with almost 100% completion rates.

1. Project Title: Modelling biological invasions with the dynamic species distribution model

Project Description: Biological invasions are one of the main passengers and drivers of global change. The spreading dynamics of alien invasive species are often transient and at non-equilibrium in the invaded ecosystem, with the species continuously negotiating through environmental, dispersal and biotic barriers. Whilst there is a variety of mechanistic models to simulate the invasive species spread such as partial differential equation “reaction-diffusion” models and integral-difference models, a standard statistical tool to assess the potential distribution of alien species (as well as of native species in future climates) is the Species Distribution Model (SDM) that first matches the observed occurrences of focal species with realised environmental niches and then predicts potential distributions in novel environments. However, this obviously violates the key features of invasion dynamics. This project thus aims to design a dynamic Species Distribution Model (dSDM) that can capture the spatial dynamics of species invading heterogeneous landscapes to account for species dispersal by means of mechanistic models. This requires to formulate species recruitment and dispersal performance into the SDM framework, for instance using integral-difference or partial differential equations. The developed dSDM will be implemented and tested using existing data on the distribution of major invasive species in South Africa and Europe.

Supervision Team: Prof Cang Hui (Stellenbosch University, SA) and Prof Sergei Petrovskii (University of Leicester, UK)

2. Project Title: Continuous time principal agent models

Project Description: A principal agent model can be described informally as follows. The principal needs to implement a control strategy with a view to maximising their resulting payoff. The agent gets the task of implementing such a strategy on behalf of the principal for a given compensation payoff that reflects salary, effort, etc. In the context of a “first-best solution”, the principal has full information of the agent’s actions at all times. In this case, the principal’s problem amounts to determining a control strategy as well as a compensation strategy that maximise their own payoff subject to the constraint that they deliver an agreed compensation payoff to the agent. In the context of a “second-best solution”, the principal does not observe the agent’s actions. The resulting asymmetric information gives rise to the possibility of moral hazard: the agent might implement a control strategy that is different from the one agreed with the principal for extra personal benefit, e.g., by stealing or being lazy. In this case, the principal’s problem amounts to determining a control strategy as well as a compensation strategy that maximise their own payoff subject to the constraints that (a) they deliver an agreed compensation payoff to the agent, and (b) it is optimal for the agent to implement the control strategy agreed with the principal. There exists a most substantial literature on the development and the analysis of principal agent models in discrete time. The project will investigate continuous time versions of principal agent models that have been studied in discrete time as well as develop and study new continuous time models.

            Supervision Team: Prof Mihael Zervos (London School of Economics, UK) and Dr Farai J. Mhlanga (University of Limpopo, SA)

3. Project Title: Modeling, analysis and simulations of the spatiotemporal dynamics of RhoGTPases

Project Description: This project focuses on the derivation, from first principles, of various models describing excitable activator-inhibitor networks for RhoGTPases based on experimental observations. The project will be divided into two stages; the first stage will involve the formulation of temporal models (only ordinary differential equations) in the absence of spatial variations. For this part, rigorous mathematical analysis will be undertaken including bifurcation analysis, sensitivity analysis, and numerical bifurcation analysis. The aim would be to identify model parameter spaces as well as key bifurcation parameters. The second stage will include the introduction of spatial variations that will result partial differential equations. Here, again, rigorous analysis will be undertaken that includes linear and weakly nonlinear analysis. Similarly, parameter regimes will be derived, and bifurcation parameters identified. The project should be culminated with the development of robust, efficient and stable numerical solvers for the partial differential equations.

            Supervision Team: Prof Anotida Madzvamuse (University of Sussex, UK) and Prof Cang Hui (Stellenbosch University, SA) 

4. Project Title: Mathematical modelling for infectious disease control

Project Description: Using mathematics to understand infectious disease dynamics has transcended the disciplines to a point where the model parameter, which provides a measure of infectiousness is commonly used in the public health arena to understand the possible scale of an infection outbreak. By contrast, theoretical work on control of infections has yet to significantly impact decisions about how to contain infections. One reason for this, is that control is a highly complex problem. It combines intervention activities including vaccination and treatment strategies with public and individual perceptions and choice over whether to engage with the interventions. This PhD will combine elements of mathematical modelling with modern analytical techniques to provide opportunities for the student to gain deep insight into building relevant and useful mathematical models rather than simply using classic approaches that are of limited value in practice. Using techniques from dynamical systems and asymptotic analysis, the student will explore underlying model behaviours in the absence of any control and will appreciate the nuances of infectious disease dynamics in order to direct control efforts effectively. The control aspect will be studied from several angles, comparing and contrasting the potential of different approaches including optimal control, feedback and robust control. Depending on the availability of regional infectious disease data (HIV, Tuberculosis, Hepatitis A, for example), the student will learn simple techniques from inverse problems in order to estimate model parameters and will use this as a basis for their case study to propose effective control strategies.

            Supervision Team: Dr Jane White (University of Bath, UK), Prof Farai Nyabadza (University of Johannesburg, SA)

5. Project Title: Analysis and simulations of coupled bulk-surface reaction-diffusion systems for cell migration

Project Description: In this project, the aim is to derive from first principles coupled bulk-surface reaction-diffusion systems for cell migration where modelling is driven by             experimental observations which can be obtained from published data in the literature.We will focus on excitable network systems comprising Rho-Rac spatiotemporal dynamics. Once formulated, rigorous analysis of the models will be carried out focusing on existence and uniqueness of solutions, positivity and bounded of solutions, diffusion-driven instability analysis, parameter identification, and numerical methods for solving such systems in 2- and 3-dimensions using finite elements.  This topic is timely since there is a surge in models that couple bulk (interior) and surface (membrane) cell dynamics giving rise to cell migration.

            Supervision Team: Prof Anotida Madzvamuse (University of Sussex, UK) Prof Farai Nyabadza (University of Johannesburg, SA), Prof. Sergei Petrovskii (University of Leicester, UK) and Dr Farai J. Mhlanga (University of Limpopo, SA)

6. Project Title: Properties of model-free price paths in Mathematical Finance -instant enforcement, volatility and local times

Project Description: In this project, the aim is to investigate properties of model-free price paths in mathematical finance. In particular, we consider the concepts of instant enforcement, volatility and local times. We investigate how a trader can instantly enforce some of the best-known properties of Brownian motion such as the existence of quadratic variation. The quadratic variation which do not depend on any partition will be constructed for both cadlag typical price paths and cadlag semimartingales following Follmer’s and the truncated variation approaches. Typical price paths are those trajectories representing possible evolution of prices of some asset which do not allow to obtain infinite wealth by risking small amount and trading this asset. Several consequences of these results will be derived and comparison will be made. Further, a new approach to define Follmer’s pathwise integral will be proposed. The local times for cadlag typical price paths will be constructed and various pathwise change of variables which generalizes Follmer’s pathwise Ito formula in the same way that the classical Tanaka-Meyer formula generalizes the classical Ito formula will be derived.

Supervision Team: Dr Farai J. Mhlanga (University of Limpopo, SA) and Professor Rafal M Lochowski (Warsaw School of economics, Poland)

7. Project Title:  Lumping of epidemic models on networks

Project Description: The exact representation of a stochastic epidemic on a network leads to a continuous time Markov Chain with a state space whose size scales as m^N, where m is the number of states a node can be in (e.g. m=2 for the susceptible-infected-susceptible (SIS) dynamics) and N is the number of nodes in the network [1]. Working with such a system is not feasible and techniques to reduce the system size will be studied. Some of these are based on some form of lumping/grouping where similar states, in some sense, are grouped together to form new variables and thus reduce the size of the system size [2]. Such techniques explore symmetries in the network or alternative ways of keeping track of the epidemic. A particularly promising direction is to restrict the Markov Chain to a state space with two entries (I,SI), that is the number of infected nodes and the number of SI link when I infected node are present in the network. Knowing these it is sufficient to work out the time to next event ans the type of transition. The potential state for (I,SI) is of O(N^2) but in practice is much smaller. The project will initially carry out a systematic feasibility study of this approach by starting from the simplest networks, such two nodes connected by a link, three node connected by two links (a line network with 3 nodes), a fully connected network with three nodes and so on. For all such simple networks, the full state space will be mapped out as well as all possible transitions. We will aim to understand the gain compared to the explicit exact model over the whole network and the complexity of and the potential to automatise the process of reducing the model to the (I,SI) state space for some more general networks. Elements from the study of imprecise lumping of Continuous-time Markov-Chains will also be considered [3].

 [1] Kiss, I.Z., Miller, J.C. and Simon, P.L., 2017. Mathematics of epidemics on networks. Cham: Springer.

 [2] Simon, P.L., Taylor, M. and Kiss, I.Z., 2011. Exact epidemic models on graphs using graph-automorphism driven lumping. Journal of mathematical biology, 62(4), pp.479-508.

 [3] Erreygers, A. and De Bock, J., 2017. Imprecise continuous-time Markov chains: Efficient computational methods with guaranteed error bounds. arXiv preprint arXiv:1702.07150.

            Supervision Team: Prof Istvan Zoltan Kiss (University of Sussex, UK) and Prof Cang Hui (Stellenbosch University, SA)

8. Project Title: Variation of the global oxygen production by marine phytoplankton in response to global warming

Project Description: Ocean dynamics is known to have a strong effect on the global climate change and on the composition of the atmosphere. It is estimated that about 70% of the atmospheric oxygen is produced in the oceans due to the photosynthetic activity of phytoplankton. The project will build on the recent discovery of a new type of a global ecological catastrophe resulting from the global warming [1,2]. A sustainable oxygen production is only possible in an intermediate range of the production rates, which in their turn depend on the oxygen concentration in water. If, in the course of time, the oxygen production rate becomes too low or too high, the system’s dynamics bifurcates and may evolve to the regime with the oxygen depletion and plankton extinction. Obviously, this catastrophic scenario implies a drastic decrease of oxygen concentration in the Earth atmosphere and a vital hazard for most of the life on the Earth. Hence, the depletion of atmospheric is a possible ecological disaster, accompanying the global warming, that has been overlooked. The project will extend the above results onto more realistic models of natural system and also link them to some available data. In the real ocean, the rate of oxygen production depends on many factors, including water temperature and its acidity, the latter being directly related to the concentration of carbon dioxide in the atmosphere. The increasing temperature leads to the decreasing oxygen production rate and solubility of oxygen in water. In contrast, the increasing concentration of carbon dioxide in the atmosphere enhances the oxygen production rate, but is accompanied with the increasing acidity of water; the later may be harmful for the growth activity of the phytoplankton’s cells. The global warming is thought to be provoked by the drastic increase of the carbon dioxide concentration in the atmosphere, therefore the both processes — of the ocean warming and the increase of the CO2 concentration are to be addressed together.  

[1] Sekerci, Y., Petrovskii, S.V. Mathematical modelling of plankton-oxygen dynamics under the climate change. Bull. Math. Biol. 77, 2325 (2015).

[2] Petrovskii, S.V., Sekerci, Y., and Venturino, E. (2017) Regime shifts and ecological catastrophes in a model of plankton-oxygen dynamics under the climate change. J. Theor. Biol. 424, 91-109.

            Supervision Team: Prof Sergei Petrovskii (University of Leicester, UK), Professor  Anotida Madzvamuse (University of Sussex, UK) and Prof Cang Hui (Stellenbosch    University, SA)

9. Project Title: Complexities of multi-infections model and mental health roles in diseases control

Project Description: Multi-infections are now very common in Sub-Sahara Africa, imposing tremendous diseases burdens. While mental health involves psychological, emotional and social well-being which affects how we think, feel and act. Also, showing how stress is handled, relate to other people and the choices we make. Mental health is very important at every stage of human development that is, from childhood and adolescence to adulthood. There is urgent need to understand mental health roles in the prevention, control of multi-infections and their complexities.

            Supervision Team: Dr Kazeem Okosun (Vaal University of Technology, SA) and Dr Konstantin Blyuss (University of Sussex, UK)

10. Project Title: Complex dynamics of epidemics with awareness

Project Description: Awareness about ongoing epidemics has a profound effect on their dynamics, and it is essential to correctly account for it when developing the corresponding mathematical models. Earlier work has shown how explicit inclusion of awareness in the model can result in the onset of sustained oscillations in the population level of infection. This project will investigate the role of the distribution of response time (to be modelled with discrete or distributed time delays), multiple levels of awareness, as well as more realistic functional forms of the disease transmission term.

            Supervision Team: Dr Konstantin Blyuss (University of Sussex, UK) and Dr Kazeem Okosun (Vaal University of Technology, SA)

11. Project Title: Using mechanistic models to understand and ultimately control criminal behaviour and activity

Project Description: Over the past decade, there has been growing interest in using mechanistic and stochastic modelling to understand the highly complex social dynamics which underlie criminal behaviour and activity. Preliminary work undertaken in a collaboration involving Dr White & Professor Nyabadza has resulted in a novel mathematical formulation which uses criminal activity data to parameterise a dynamical system describing criminal behaviour in a spatial context. This PhD project will use that modelling paradigm to test hypotheses about criminal behaviour, particularly as it might be described in terms of epidemics associated with infectious diseases. To do this, the student will develop and analyse a deterministic spatio-temporal model in the context of a particular criminal behaviour in South Africa. Building from the dynamical system, the research will focus on methods of control and will provide opportunities for the student to explore a range of mathematical control techniques, including optimal control, robust control and feedback control. At every stage, the theoretical results will be interpreted in the context of the original problem so that the student develops a clear understanding of the mathematical modelling paradigm and how to make effective use of the approach.

            Supervision Team: Dr Jane White (University of Bath, UK), Prof Farai Nyabadza (University of Johannesburg, SA) and Dr Konstantin Blyuss (University of Sussex, UK)

12. Project Title: Modeling of sexually transmitted diseases by incorporating assessment of knowledge, attitude, and preventive practices

Project Description: Sexually Transmitted diseases (STDs) (including chlamydia, gonorrhoea, syphilis, trichomoniasis, human immunodeficiency virus (HIV), etc.) have long been an underestimated opponent in the public health protection policies. It is a challenge for both developed and developing countries. Sexually transmitted diseases constitute a huge health and economic burden for developing countries: 75–85% of the estimated 340 million annual new cases of curable STDs occur in these countries, and STDs account for 17% economic losses because of ill health (Mayaud and Mabey, (2004)). In South Africa, STDs contribute largely to the burden of health and are recognized as major contributors to the HIV epidemic (Naidoo et al. 2014). According to figures released by the Department of Health (South Africa), more than one million people have been treated for sexually transmitted diseases in the year 2015/2016 where in the same year, HIV prevalence in Limpopo province was estimated 8.3 % among adults between the ages 15-49. Department of Health (South Africa) sets the National Development Plan (NDP) 2030 to achieve “a long and healthy life for all South Africans”. This includes a life expectancy rate of at least 70 years for men and women and a generation of under-20s largely free from HIV. Mathematical models with their analysis usually assist these policies to simulate the possible outcomes of the control measures (Bacaer N., 2011). Models designed in this manner will help to understand the disease transmission dynamics and to project the likely impact of proposed interventions before they are implemented. To add input in this regard, we will design and analysis mathematical models, which take into account the complex properties of some STDs as well as their co-infection with other sexually transmitting diseases such as HIV.

            Supervision Team: Dr Y. Terefe (University of Limpopo, SA) and

            Dr Konstantin Blyuss (University of Sussex, UK)

13. Project Title: Mathematical modelling of tumour growth using poro-morpho-elastic models

Project Description: Cancer development involves initiation and growth of tumours, as well as metastasis of cancer to other parts of the (human) body. Cancerous tissue consists of (cancer) cells and extracellular material. This extracellular material consists of collagen fibers, but also of liquidised materials, which is often described as a porous medium. Tumours often experience and exert mechanical interactions with its immediate environment, and hence mathematical models need to incorporate these mechanical interactions and the porous nature of the medium. Since tumours may grow under the influence of oxygen and glucosis, we model the expansion of the tumour by the use of a morfo-elastic formulation. In this project, a morfo-poro-elastic formulation that is able to incorporate large deformations will be set up. Further, numerical strategies for the approximation of the solution from the resulting set of partial differential equations will be developed. The current framework will also be applicable in a wider range of applications from mathematical biology, such as wound closure and scar formation.

            Supervision Team: Prof Fred Vermolen (Delft University of Technology, Netherlands) and Prof Farai Nyabadza (University of Johannesburg, SA)

14. Project Title: Modelling transmission infection dynamics: From models to public health policy

Project Description: Significant research efforts in sub-Saharan Africa has been done in the area of modelling but, application of mathematical modelling in public health decision making and policy designs is still to reach its full potential. The objective of this project is to develop mathematical models that strengthen the connections between mathematical modelling and public health policy, with linkages to data on infectious diseases that are a challenge to the continent. The methods include the use of differential equations (both stochastic and ordinary differential equations). Functions that model different transmission dynamics will be formulated while aligning to various scenarios that will be modelled.  Numerical simulations and fitting of the models to data will be carried out.

            Supervision Team: Prof Farai Nyabadza (University of Johannesburg, SA) and Dr Konstantin Blyuss (University of Sussex, UK)

Students will be encouraged to publish their work in both local and international journals. This will help to disseminate research outcomes. Published work will also be uploaded on this website.

The postgraduate intensive research workshop is geared at in-depth introduction of the research students to material in Applied Mathematics. These are introductory courses to bring up to speed students into the subject areas they will be studying.

There will be a mid-term review workshop on which all students will present their work to the supervisors. The program will be reviewed by both students and the supervisory board during this workshop. In addition, heads of departments will also be invited to get an update on this programme.

The aim of these international mini symposia is to bring together researchers and students from this consortium to organise an international meeting involving two other researchers whereby research outcomes from this programme will be disseminated. The dissemination will be mainly carried through presentations by research students from this programme.

The supervisors from South Africa and the students will be mandated to undertake an international exchange research visit to the UK that could include summer schools where appropriate. The visit is expected to last up to a maximum of one week

All the Doctoral Researchers will be required to attend one local and one international conference in an appropriate area of their research with the aim of giving students exposure to recent developments and advances in Applied Mathematics. Student will get constructive comments on their research and will establish collaborative network with those working in similar type of research.

Lead University Information

Dr F J Mhlanga
Department of Mathematics and Applied Mathematics
University of Limpopo
Private bag X1106
Sovenga
0727
South Africa

Email: farai.mhlanga@ul.ac.za
Tel: +27 15 268 3815
Cell: +27 82 351 9914