@misc{Proell2013,
  type      = {Master Thesis},
  author    = {Pr{\"o}ll, Sebastian},
  title     = {Stability of Switched Epidemiological Models},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-108573},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2013},
  abstract  = {In this thesis it is shown how the spread of infectious diseases can be described via mathematical models that show the dynamic behavior of epidemics. Ordinary differential equations are used for the modeling process. SIR and SIRS models are distinguished, depending on whether a disease confers immunity to individuals after recovery or not. There are characteristic parameters for each disease like the infection rate or the recovery rate. These parameters indicate how aggressive a disease acts and how long it takes for an individual to recover, respectively. In general the parameters are time-varying and depend on population groups. For this reason, models with multiple subgroups are introduced, and switched systems are used to carry out time-variant parameters. When investigating such models, the so called disease-free equilibrium is of interest, where no infectives appear within the population. The question is whether there are conditions, under which this equilibrium is stable. Necessary mathematical tools for the stability analysis are presented. The theory of ordinary differential equations, including Lyapunov stability theory, is fundamental. Moreover, convex and nonsmooth analysis, positive systems and differential inclusions are introduced. With these tools, sufficient conditions are given for the disease-free equilibrium of SIS, SIR and SIRS systems to be asymptotically stable.},
  subject      = {Gew{\"o}hnliche Differentialgleichung},
  language  = {en}
}