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Zbl 1183.34131
Xu, Rui; Ma, Zhien
Global stability of a SIR epidemic model with nonlinear incidence rate and time delay. (English)
[J] Nonlinear Anal., Real World Appl. 10, No. 5, 3175-3189 (2009). ISSN 1468-1218

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.
[Denis Khusainov (Ky{\"\i}v)]

MSC 2000:: *34K60 Applications of 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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