Global properties of epidemiological models and virus dynamics models with nonlinear incidence rates by Lyapunov direct method
 
 
Description:  Classical models of infectious diseases postulate that the spread of an infection occurs according to the principle of mass action and associated with that an incidence rate which is bilinear with respect to the interacting populations (the numbers of susceptible and infective individuals in this case). However, there is a number of reasons why this standard bilinear incidence rate may require modification. For example, the underlying assumption of homogeneous mixing and homogeneous environment may be invalid. In this case the necessary population structure and heterogeneous mixing may be incorporated into a model with a specific form of non-linear interaction. A nonlinear incidence rate also arises from saturation effects: if the proportion of the infective hosts in a population, or the concentration of the pathogen is very high, so that exposure to the disease agent is virtually certain, then the transmission rate may respond more slowly than linear to the increase in the number of infectives. This effect was encountered in clinical observations as well as in laboratory experiments. We consider the impact of the nonlinearity of the incidence rate on the dynamics of a variety of the models in epidemiology and virus dynamics. We consider global properties for the classical compartmental models of infectious diseases with a very general form of the nonlinear incidence rate; in fact, we assumed that the incidence rate is given by an unspecified function \( f(S,I) \) constrained by a few biologically feasible conditions. For this rather general case, the direct Lyapunov method enables us to find biologically realistic conditions that are sufficient to ensure existence and uniqueness of a globally asymptotically stable equilibrium state.
Area(s):
Speaker:  Andrei Korobeinikov (University of Limerick)
Place:  5.5
Research Groups: -Numerical Analysis and Optimization
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