An SIR model with distributed delay and a general incidence function is
studied. Conditions are given under which the system exhibits threshold
behaviour: the disease-free equilibrium is globally asymptotically stable
if R0<1 and globally attracting if R0=1; if R0>1,
then the unique endemic equilibrium is globally asymptotically stable.
The global stability proofs use a Lyapunov functional and do not require
uniform persistence to be shown a priori. It is shown that the
given conditions are satisfied by several common forms of the incidence
function.
Citation: C. Connell McCluskey. Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence[J]. Mathematical Biosciences and Engineering, 2010, 7(4): 837-850. doi: 10.3934/mbe.2010.7.837
Abstract
An SIR model with distributed delay and a general incidence function is
studied. Conditions are given under which the system exhibits threshold
behaviour: the disease-free equilibrium is globally asymptotically stable
if R0<1 and globally attracting if R0=1; if R0>1,
then the unique endemic equilibrium is globally asymptotically stable.
The global stability proofs use a Lyapunov functional and do not require
uniform persistence to be shown a priori. It is shown that the
given conditions are satisfied by several common forms of the incidence
function.