Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence

  • Received: 01 March 2010 Accepted: 29 June 2018 Published: 01 October 2010
  • MSC : Primary: 34K20, 92D30.

  • 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

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  • 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.


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