Abstracts Mathematics

Population Persistence and Disease Invasion in Heterogenous Networks

by Poroshat Yazdanbakhshghahyazi

The problem of understanding how biological species and infectious diseases can persist and spread in heterogeneous networks has brought a wide attention, recently highlighted due to the COVID-19 pandemic. This dissertation investigates the connection between the structures of heterogeneous networks and population persistence/disease invasion. To do so, we propose a new index for network heterogeneity by employing the Laplacian matrix of population dispersal and its corresponding group inverse. The network growth rate and reproduction number can be evaluated using the network average and the network heterogeneity index as the first and second order approximation, respectively. We also illustrate the impact of arrangement of ecological sources/sinks and disease hotspots/non-hotspots, which highlights the significance of the network structures on population persistence and disease invasion in heterogeneous environments. Mathematically, population and disease control strategies can be modeled via altering certain ecological and epidemiological parameters in the biological processes. To quantitatively measure the scale of the change in need, new indices and methods are introduced and developed to generalize the existing threshold parameters. Properties and implications of these are provided to demonstrate the applicability to infectious disease controls such as anthrax, cholera and Zika virus.