@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}
}
@phdthesis{Dirr2001,
  author    = {Dirr, Gunther},
  title     = {Differentialgleichungen in Fr{\´e}chetr{\"a}umen},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-1180417},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2001},
  abstract  = {Teil 1 der Arbeit beinhaltet eine Zusammenfassung grundlegender funktionalanalytischer Ergebnisse sowie eine Einf{\"u}hrung in die Integral- und Differentialrechnung in Fr{\´e}chetr{\"a}umen. Insbesondere wird in Kapitel 2 eine ausf{\"u}hrliche Darstellung des Lebesgue-Bochner-Integrals auf Fr{\´e}chetr{\"a}umen geliefert. Teil 2 behandelt die Theorie der linearen Differentialgleichungen auf Fr{\´e}chetr{\"a}umen. Dazu werden in Kapitel 3 stark differenzierbare Halbgruppen und deren infinitesimale Generatoren charakterisiert. In Kapitel 4 werden diese Ergebnisse benutzt, um lineare Evolutionsgleichungen (von hyperbolischem oder parabolischem Typ) zu untersuchen. Teil 3 enth{\"a}lt die zentralen Resultate der Arbeit. In Kapitel 5 werden zwei Existenz- und Eindeutigkeitss{\"a}tze f{\"u}r nichtlineare gew{\"o}hnliche Differentialgleichungen in zahmen Fr{\´e}chetr{\"a}umen bewiesen. Kapitel 6 liefert eine Anwendung der Ergebnisse aus Kapitel 5 auf nichtlineare partielle Differentialgleichungen erster Ordnung.},
  subject      = {Differentialgleichung},
  language  = {de}
}