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Global stability of a SIR epidemic model with nonlinear incidence rate and time delay. (English) Zbl 1183.34131
Consider the following delay differential equations
$\dot S(t)= B-\mu_1 S(t)-{\beta S(t)I(t- \tau)\over 1+\alpha I(t-\tau)},\;\dot I(t)={\beta S(t)I(t-\tau)\over 1+\alpha I(t-\tau)}- (\mu_2+ v)I(t),$
$\dot R(t)=\gamma I(t)- \mu_3R(t).$ Let $$R_0= {B\beta\over \mu_1(\mu_2+v)}$$. Consider the situations $$R_0> 1$$, $$R_0< 1$$. It is proved that for $$R_0> 1$$ the equilibrium $$E_1(B/\mu_1,0,0)$$ is locally asymptotically stable. In case $$R_0> 1$$, the equilibrium point $$E^*(S^*, I^*,R^*)$$, where
$S^*= {B\alpha+ \mu_2+\gamma\over\beta+ \alpha\mu_1},\;I^*= {B\beta- \mu_1(\mu_2+ \gamma)\over (\mu_2+ \gamma)(\beta+ \alpha\mu_1)},\;R^*= {\gamma[B\beta- \mu_1(\mu_2+ \gamma)]\over \mu_3(\mu_2+ \gamma)(\beta+ \alpha\mu_1)},$ exists and is locally asymptotically stable while $$E_1$$ is unstable.

##### MSC:
 34K60 Qualitative investigation and simulation of models involving functional-differential equations 92D30 Epidemiology 34K20 Stability theory of functional-differential equations
##### Keywords:
SIR epidemic model; nonlinear incidence; time delay; stability
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##### References:
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