Application and evaluation of local and global analysis for dynamic models of infectious disease spread

Abstract

In this thesis, we applied local analysis tools (eigenvalue and eigenvalue elasticity analysis, global function elasticity/sensitivity analysis), and global analysis tools (deriving the location and stability of fixed points) to both aggregate and individual-level dynamic models of infectious diseases. We sought to use these methods to gain insight into the models and to evaluate the use of these methods to study their short-term and long-term dynamics and the influences of arameters on the models.

We found that eigenvalues are effective for understanding short-term behaviours of a nonlinear system, but less effective in providing insights of the long-term impacts of a parameter change on its behaviours. In term of disease control, local changes of behaviours, yielded from the changes of parameters based on eigenvalue elasticity, are able to alter behaviours in a short-term, especially in the period of a disease outbreak. While eigenvalue elasticity analysis can be helpful for understanding the impact of parameter changes for simple aggregate models, such analyses prove unwieldy and complicated, particularly for models with large number of state variables; and easily fall prey to eigenvalue multiplicity problems for large individual-based models, and istracting artifacts associated with small denominators. In response to these concerns, we introduced other local methods (global function elasticity/sensitivity analyses) that capture many of the advantages of eigenvalue elasticity methods with much greater simplicity. Unfortunately, parameter changes guided by these local analysis techniques are often insufficient to alter behaviours in the longer-term, such as when system behaviours approach stable endemic equilibria. By contrast, the global analytic tools, such as fixed point location and stability analysis, are effective for providing insights into the global behaviours of disease spread in the long-term, as well as their dependence on parameters. Using all of the above analysis as a toolset, we gained some possible insights into combination of local and global approaches. Choice of applying local or global analysis tools to infectious disease models is dependent on the specific target of policy makers as well as model type.