Study of singularly perturbed models and its applications in ecology and epidemiology.

Doctoral degree. University of KwaZulu-Natal, Durban.In recent years the demand for a more accurate description of real life processes and advances in experimental techniques have resulted in construction of very complex mathematical models, consisting of tens, hundreds, if not thousands, of highly coupled di erential equations. The sheer size and complexity of such models often preclude any robust, theoretical or numerical, analysis of them. Fortunately, often such models describe phenomena occurring on vastly di erent time or size scales. We focused on complex processes with two time/size scales described by systems of ordinary di erential equations. In such a case, there is a small parameter that multiplies one or more derivatives. Using the Tikhonov Theorem, we have been able to understand the asymptotic behaviour of the solution to some complex epidemiological models. Furthermore, we present analysis based on the Butuzov theorem, which, for the purpose of the discussed models, was generalized to two dimensional non-autonomous problems. We applied the developed theory on an ecological model with interactions given by the mass action law

Orienting Future Trends in Local Ancestry Deconvolution Models to Optimally Decipher Admixed Individual Genome Variations

Rapid advances in sequencing and genotyping technologies have significantly contributed to shaping the area of medical and population genetics. Several thousand genomes are completed with millions of variants identified in the human deoxyribonucleic acid (DNA) sequences. These genomic variations highly influence changes in phenotypic manifestations and physiological functions of different individuals or population groups. Of particular importance are variations introduced by admixture event, contributing significantly to a remarkable phenotypic variability with medical and/or evolutionary implications. In this case, knowledge of local ancestry estimates and date of admixture is of utmost importance for a better understanding of genomic variation patterns throughout modern human evolution and adaptive processes. In this chapter, we survey existing local ancestry deconvolution and dating admixture event models to identify possible gaps that still need to be filled and orient future trends in designing more effective models, which account for current challenges and produce more accurate and biological relevant estimates

Designing Data-Driven Learning Algorithms: A Necessity to Ensure Effective Post-Genomic Medicine and Biomedical Research

Advances in sequencing technology have significantly contributed to shaping the area of genetics and enabled the identification of genetic variants associated with complex traits through genome-wide association studies. This has provided insights into genetic medicine, in which case, genetic factors influence variability in disease and treatment outcomes. On the other side, the missing or hidden heritability has suggested that the host quality of life and other environmental factors may also influence differences in disease risk and drug/treatment responses in genomic medicine, and orient biomedical research, even though this may be highly constrained by genetic capabilities. It is expected that combining these different factors can yield a paradigm-shift of personalized medicine and lead to a more effective medical treatment. With existing “big data” initiatives and high-performance computing infrastructures, there is a need for data-driven learning algorithms and models that enable the selection and prioritization of relevant genetic variants (post-genomic medicine) and trigger effective translation into clinical practice. In this chapter, we survey and discuss existing machine learning algorithms and post-genomic analysis models supporting the process of identifying valuable markers