# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Anderson, R.M.; May, R.M., Population biology of infectious diseases: part I, Nature, 280, 361-367, (1979)  Beretta, E.; Capasso, V.; Rinaldi, F., Global stability results for a generalized lotka – volterra system with distributed delays: applications to predator – prey and epidemic systems, J. math. biol., 26, 661-668, (1988) · Zbl 0716.92020  Beretta, E.; Hara, T.; Ma, W.; Takeuchi, Y., Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear anal., 47, 4107-4115, (2001) · Zbl 1042.34585  Beretta, E.; Takeuchi, Y., Global stability of an SIR epidemic model with time delays, J. math. biol., 33, 250-260, (1995) · Zbl 0811.92019  Beretta, E.; Takeuchi, Y., Convergence results in SIR epidemic model with varying population sizes, Nonlinear anal., 28, 1909-1921, (1997) · Zbl 0879.34054  Capasso, V.; Serio, G., A generalization of the kermack – mckendrick deterministic epidemic model, Math. biosci., 42, 41-61, (1978) · Zbl 0398.92026  Cooke, K.L., Stability analysis for a vector disease model, Rocky mountain J. math., 9, 31-42, (1979) · Zbl 0423.92029  Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Dordrecht, Norwell, MA · Zbl 0752.34039  Hale, J., Theory of functional differential equations, (1977), Springer-Verlag Heidelberg  Hethcote, H.W., Qualitative analyses of communicable disease models, Math. biosci., 7, 335-356, (1976) · Zbl 0326.92017  Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press New York · Zbl 0777.34002  Ma, W.; Song, M.; Takeuchi, Y., Global stability of an SIR epidemic model with time delay, Appl. math. lett., 17, 1141-1145, (2004) · Zbl 1071.34082  Ma, W.; Takeuchi, Y.; Hara, T.; Beretta, E., Permanence of an SIR epidemic model with distributed time delays, Tohoku math. J., 54, 581-591, (2002) · Zbl 1014.92033  Takeuchi, Y.; Ma, W., Stability analysis on a delayed SIR epidemic model with density dependent birth process, Dynam. cont. discrete impul. syst., 5, 171-184, (1999) · Zbl 0937.92026  Takeuchi, Y.; Ma, W.; Beretta, E., Global asymptotic properties of a SIR epidemic model with night incubation time, Nonlinear anal., 42, 931-947, (2000) · Zbl 0967.34070
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.