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Stability analysis of delayed SIR epidemic models with a class of nonlinear incidence rates. (English) Zbl 1244.92048

Summary: We analyze the stability of equilibria for a delayed SIR epidemic model, in which population growth is subject to logistic growth in absence of disease, with a nonlinear incidence rate satisfying suitable monotonicity conditions. The model admits a unique endemic equilibrium if and only if the basic reproduction number \(R_{0}\) exceeds one, while the trivial equilibrium and the disease-free equilibrium always exist.
First we show that the disease-free equilibrium is globally asymptotically stable if and only if \(R_{0} \leqslant 1\). Second we show that the model is permanent if and only if \(R_{0} > 1\). Moreover, using a threshold parameter \(\bar R_0\) characterized by the nonlinear incidence function, we establish that the endemic equilibrium is locally asymptotically stable for \(1 < R_0 \leqslant \bar R_0\) and it loses stability as the length of the delay increases past a critical value for \(1< \bar R_0 < R_0\). Our result is an extension of the stability results of J.-J. Wang, J.-Z. Zhang and Z. Jin [Analysis of an SIR model with bilinear incidence rate. Nonlinear Anal., Real World Appl. 11, No. 4, 2390–2402 (2010; Zbl 1203.34136)].

MSC:

92D30 Epidemiology
34K20 Stability theory of functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations

Citations:

Zbl 1203.34136
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References:

[1] Anderson, R.M.; May, R.M., Population biology of infectious diseases: part I, Nature, 280, 361-367, (1979)
[2] Capasso, V.; Serio, G., A generalization of the kermack-mckendrick deterministic epidemic model, Math. biosci., 42, 43-61, (1978) · Zbl 0398.92026
[3] Cooke, K.L., Stability analysis for a vector disease model, Rocky mountain J. math., 9, 31-42, (1979) · Zbl 0423.92029
[4] Enatsu, Y.; Nakata, Y.; Muroya, Y., Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays, Dis. contin. dyn. syst. ser. B, 15, 61-74, (2011) · Zbl 1217.34127
[5] Hethcote, H.W., The mathematics of infectious diseases, SIAM rev., 42, 599-653, (2000) · Zbl 0993.92033
[6] Huang, G.; Takeuchi, Y., Global analysis on delay epidemiological dynamics models with nonlinear incidence, J. math. biol., 63, 125-139, (2011) · Zbl 1230.92048
[7] Korobeinikov, A., Global properties of infectious disease models with nonlinear incidence, Bull. math. biol., 69, 1871-1886, (2007) · Zbl 1298.92101
[8] McCluskey, C.C., Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear anal. RWA, 11, 55-59, (2010) · Zbl 1185.37209
[9] McCluskey, C.C., Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear anal. RWA, 11, 3106-3109, (2010) · Zbl 1197.34166
[10] McCluskey, C.C., Global stability of an SIR epidemic model with delay and general nonlinear incidence, Math. biosci. eng., 7, 837-850, (2010) · Zbl 1259.34067
[11] Takeuchi, Y.; Ma, W.; Beretta, E., Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear anal., 42, 931-947, (2000) · Zbl 0967.34070
[12] Wang, J-J.; Zhang, J-Z.; Jin, Z., Analysis of an SIR model with bilinear incidence rate, Nonlinear anal. RWA, 11, 2390-2402, (2009) · Zbl 1203.34136
[13] Xu, R.; Ma, Z., Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear anal. RWA, 10, 3175-3189, (2009) · Zbl 1183.34131
[14] Zhang, X.; Chen, L., The periodic solution of a class of epidemic models, Comput. math. appl., 38, 61-71, (1999) · Zbl 0939.92031
[15] Zhou, X.; Cui, J., Stability and Hopf bifurcation of a delay eco-epidemiological model with nonlinear incidence rate, Math. model. anal., 15, 547-569, (2010) · Zbl 1234.34054
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