The dynamics of epidemiological systems with nonautonomous and random coefficients

  • Peter E. Kloeden
  • Victor S. Kozyakin


Steady state solutions are important in characterizing the asymptotic behaviour of epidemiological systems such as the ubiquitous $\mathrm{SIR}$ system, but they need not exist when the coefficients vary with time. Recent developments in the theory of nonautonomous dynamical systems provide the appropriate counterpart, time varying nonautonomous equilibria, i.e., entire solutions determined by pullback attraction. These will be illustrated here through explicit solutions for the simpler $\mathrm{SI}$ system. For the more complicated $\mathrm{SIR}$ system it will be shown how pullback attractors provide additional insight into dynamics, especially the chaotic dynamics induced by periodical forcing.