Geometric Singular Perturbation Analysis of a Multiple Time-scale Model for Diabetes and COVID-19 Comorbidity

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More and more information on mortality and morbidity indicates that in order to fight the COVID-19 pandemic, it is important to focus our attention on comorbidities. Several reports evidence of how many elderly patients who become severely ill exhibit underlying illness such as cardiovascular disease, kidney disease, tumor, and more to our special attention here, type 2 diabetes. Better understanding of the mechanism underlying the comorbidity between different diseases requires merging models of systems across different time-scales. The model homogenization across multiple spatial and time scales poses an important challenge to researchers in the field of medical science. An approach that has been found relatively efficient in the analysis of such models is the use of singular perturbation technique. Here, we study a differential equation model system with multiple time scales which describes the diabetes and COVID-19 comorbidity. It tracks the changes in levels of plasma glucose, insulin, and functional-cells, incorporating insulin resistance and inflammation responses. The model is analyzed with the geometric singular perturbation technique, by which conditions on the system parameters may be derived to identify regions in which the system exhibits different dynamic behavior, whether the system would be stable, or eventually oscillate in a sustained fashion. Discussion of these conditions allows us to better understand how comorbidity mediates the development of life-threatening symptoms in a diabetic patient in order that proper care and treatment may be prescribed.