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Center for Nonlinear Studies |
Political destabilization and the emergence and re-emergence of deadly diseases such as HIV, SARS, Ebola and Zika highlight the danger posed by natural and designed biological threats to human welfare. The emergence of a new pandemic either in a natural way or through bio-terrorism represents one of the largest treats that will face our society within the next decades. It is therefore important to comprehend the rapid nonlinear dynamics of infectious diseases and understand how to respond to these threats at different levels. From the mathematical viewpoint, temporal dynamics of disease propagation within a population can be adequately described by population balance models (PBM). Such models have a rather distinctive non-linear structure which makes them an interesting object for research, but also adds to the complexity of the associated control problems. We will consider a particular example of population balance model and discuss a couple of basic concepts: the basic reproduction rate (R0), the notion of control for population balance models and others. Further, we mention several possible extensions of the described concepts, in particular bifurcation and sensitivity analysis of R0 . The second part of the talk will be devoted to the application of numerical optimal control for dynamical allocation of resources between different interventions. While there are many results on analytical or semi-analytical computation of optimal controls for special classes of PBM, less attention was paid to the numerical computation of optimal intervention profiles for realistic models. Recently, a variant of orthogonal collocations method was developed and successfully applied for solving the problem of resource allocation under different budgetary and programmatic restrictions. In conclusion, we consider some open problems that arise when introducing the developed method into the broader epidemiological practice and discuss possible applications of control theory within the epidemiological context.
Host: Ethan Obie Romero-Severson