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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.
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