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.
\[ \dot S(t)= B-\mu_1 S(t)-{\beta S(t)I(t- \tau)\over 1+\alpha I(t-\tau)},\;\dot I(t)={\beta S(t)I(t-\tau)\over 1+\alpha I(t-\tau)}- (\mu_2+ v)I(t), \]
\[ \dot R(t)=\gamma I(t)- \mu_3R(t). \]
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.
Reviewer: Denis Khusainov (Kyïv)
MSC:
| 34K60 | Qualitative investigation and simulation of models involving functional-differential equations |
| 92D30 | Epidemiology |
| 34K20 | Stability theory of functional-differential equations |
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