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Analysis of an SIR model with bilinear incidence rate. (English) Zbl 1203.34136
Under the assumption that the susceptible of host population satisfies the logistic equation and the incidence rate is the simple mass action incidence, the authors study an SIR vector disease model with incubation time. The threshold quantity $R_0$ is derived which determines whether the disease dies out or remains endemic. Using the time delay (i.e., incubation time) as a bifurcation parameter, the local stability of the endemic equilibrium is investigated, and the conditions for Hopf bifurcation to occur are derived. Numerical simulations are presented to illustrate the main results.

MSC:
34K60Qualitative investigation and simulation of models
92D30Epidemiology
34K18Bifurcation theory of functional differential equations
34K20Stability theory of functional-differential equations
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Full Text: DOI
References:
[1] Hethcote, H. W.: Qualitative analyses of communicable disease models. Mathematical biosciences 28, 335-356 (1976) · Zbl 0326.92017
[2] Cooke, K.: Stability analysis for a vector disease model. Rocky mountain journal of mathematics 9, 31-42 (1979) · Zbl 0423.92029
[3] Beretta, E.; Takeuchi, Y.: Global stability of an SIR epidemic model with time delays. Journal of mathematical biology 33, 250-260 (1995) · Zbl 0811.92019
[4] Beretta, E.; Takeuchi, Y.: Convergence results in SIR epidemic models with varying population sizes. Nonlinear analysis TMA 28, 1909-1921 (1997) · Zbl 0879.34054
[5] Takeuchi, Y.; Ma, W.; Beretta, E.: Global asymptotic properties of a delay SIR epidemic model with finite incubation times. Nonlinear analysis TMA 42, 931-947 (2000) · Zbl 0967.34070
[6] Ma, W.; Song, M.; Takeuchi, Y.: Global stability of an SIR epidemic model with time delay. Applied mathematics letters 17, 1141-1145 (2004) · Zbl 1071.34082
[7] Jin, Z.; Ma, Z.: The stability of an SIR epidemic model with time delays. Mathematical biosciences and engineering 3, 101-109 (2006) · Zbl 1089.92045
[8] Zhang, X.; Chen, L.: The periodic solution of a class of epidemic models. Computers and mathematics with applications 38, 61-71 (1999) · Zbl 0939.92031
[9] Yang, X.; Chen, L.: Permanence and positive periodic solution for the single-species nonautonomous delay diffusive model. Computers and mathematics with applications 32, 109-116 (1996) · Zbl 0873.34061
[10] Tchuenche, J. M.; Nwagwo, A.; Levins, R.: Global behaviour of an SIR epidemic model with time delay. Mathematical methods in the applied sciences 30, 733-749 (2007) · Zbl 1112.92055
[11] Kuang, Y.: Delay differential equations with applications in population dynamics. (1993) · Zbl 0777.34002
[12] Xiao, Y.; Chen, L.: Modeling and analysis of a predator--prey model with disease in the prey. Mathematical biosciences 171, 59-82 (2001) · Zbl 0978.92031
[13] Hale, J. K.; Waltman, P.: Persistence in infinite-dimensional systems. SIAM journal on mathematical analysis 20, 388-395 (1989) · Zbl 0692.34053
[14] Freedman, H. I.; Rao, V. S. H.: The trade-off between mutual interference and time lags in predator--prey systems. Bulletin of mathematical biology 45, 991-1004 (1983) · Zbl 0535.92024
[15] Hassard, B. D.; Kazarinoff, N.; Wan, Y. H.: Theory and applications of Hopf bifurcation. (1981) · Zbl 0474.34002