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Stability and Bogdanov-Takens bifurcation of an SIS epidemic model with saturated treatment function. (English) Zbl 1394.92131

Summary: This paper introduces the global dynamics of an SIS model with bilinear incidence rate and saturated treatment function. The treatment function is a continuous and differential function which shows the effect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. Sufficient conditions for the existence and global asymptotic stability of the disease-free and endemic equilibria are given in this paper. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such as subcritical or supercritical. By some complex algebra, the Bogdanov-Takens normal form and the three types of bifurcation curves are derived. Finally, mathematical analysis and numerical simulations are given to support our theoretical results.

MSC:

92D30 Epidemiology
92C60 Medical epidemiology
34C23 Bifurcation theory for ordinary differential equations
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[1] Alexander, M. E.; Moghadas, S. M., Periodicity in an epidemic model with a generalized non-linear incidence, Mathematical Biosciences, 189, 1, 75-96, (2004) · Zbl 1073.92040 · doi:10.1016/j.mbs.2004.01.003
[2] Alexander, M. E.; Moghadas, S. M., Bifurcation analysis of SIRS epidemic model with generalized incidence, SIAM Journal on Applied Mathematics, 65, 5, 1794-1816, (2005) · Zbl 1088.34035 · doi:10.1137/040604947
[3] Arino, J.; McCluskey, C. C.; van den Driessche, P., Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM Journal on Applied Mathematics, 64, 1, 260-276, (2003) · Zbl 1034.92025 · doi:10.1137/S0036139902413829
[4] Derrick, W. R.; van den Driessche, P., Homoclinic orbits in a disease transmission model with nonlinear incidence and nonconstant population, Discrete and Continuous Dynamical Systems B, 3, 2, 299-309, (2003) · Zbl 1126.34337 · doi:10.3934/dcdsb.2003.3.299
[5] Brauer, F., Backward bifurcations in simple vaccination models, Journal of Mathematical Analysis and Applications, 298, 2, 418-431, (2004) · Zbl 1063.92037 · doi:10.1016/j.jmaa.2004.05.045
[6] Hadeler, K. P.; van den Driessche, P., Backward bifurcation in epidemic control, Mathematical Biosciences, 146, 1, 15-35, (1997) · Zbl 0904.92031 · doi:10.1016/S0025-5564(97)00027-8
[7] Hu, Z. X.; Liu, S.; Wang, H., Backward bifurcation of an epidemic model with standard incidence rate and treatment rate, Nonlinear Analysis: Real World Applications, 9, 5, 2302-2312, (2008) · Zbl 1156.34320 · doi:10.1016/j.nonrwa.2007.08.009
[8] Hui, J.; Zhu, D. M., Global stability and periodicity on SIS epidemic models with backward bifurcation, Computers & Mathematics with Applications, 50, 8-9, 1271-1290, (2005) · Zbl 1078.92058 · doi:10.1016/j.camwa.2005.06.003
[9] Jin, Y.; Wang, W. D.; Xiao, S. W., An SIRS model with a nonlinear incidence rate, Chaos, Solitons & Fractals, 34, 5, 1482-1497, (2007) · Zbl 1152.34339 · doi:10.1016/j.chaos.2006.04.022
[10] Li, G. H.; Wang, W. D., Bifurcation analysis of an epidemic model with nonlinear incidence, Applied Mathematics and Computation, 214, 2, 411-423, (2009) · Zbl 1168.92323 · doi:10.1016/j.amc.2009.04.012
[11] Li, X. Z.; Li, W. S.; Ghosh, M., Stability and bifurcation of an SIS epidemic model with treatment, Chaos, Solitons amp& Fractals, 42, 5, 2822-2832, (2009) · Zbl 1198.34060 · doi:10.1016/j.chaos.2009.04.024
[12] Liu, W. M.; Levin, S. A.; Iwasa, Y., Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, Journal of Mathematical Biology, 23, 2, 187-204, (1986) · Zbl 0582.92023 · doi:10.1007/BF00276956
[13] Liu, X. B.; Yang, L. J., Stability analysis of an SEIQV epidemic model with saturated incidence rate, Nonlinear Analysis: Real World Applications, 13, 6, 2671-2679, (2012) · Zbl 1254.92083 · doi:10.1016/j.nonrwa.2012.03.010
[14] Martcheva, M.; Thieme, H. R., Progression age enhanced backward bifurcation in an epidemic model with super-infection, Journal of Mathematical Biology, 46, 5, 385-424, (2003) · Zbl 1097.92046 · doi:10.1007/s00285-002-0181-7
[15] Ruan, S. G.; Wang, W. D., Dynamical behavior of an epidemic model with a nonlinear incidence rate, Journal of Differential Equations, 188, 1, 135-163, (2003) · Zbl 1028.34046 · doi:10.1016/S0022-0396(02)00089-X
[16] Wang, W. D.; Ruan, S. G., Bifurcation in an epidemic model with constant removal rate of the infectives, Journal of Mathematical Analysis and Applications, 291, 2, 775-793, (2004) · Zbl 1054.34071 · doi:10.1016/j.jmaa.2003.11.043
[17] Tang, Y. L.; Huang, D. Q.; Ruan, S. G.; Zhang, W. N., Coexistence of limit cycles and homoclinic loops in a SIRS model with a nonlinear incidence rate, SIAM Journal on Applied Mathematics, 69, 2, 621-639, (2008) · Zbl 1171.34033 · doi:10.1137/070700966
[18] van den Driessche, P.; Watmough, J., A simple SIS epidemic model with a backward bifurcation, Journal of Mathematical Biology, 40, 6, 525-540, (2000) · Zbl 0961.92029 · doi:10.1007/s002850000032
[19] Wei, J. J.; Cui, J. A., Dynamics of SIS epidemic model with the standard incidence rate and saturated treatment function, International Journal of Biomathematics, 5, 3, (2012) · Zbl 1280.92065 · doi:10.1142/S1793524512600030
[20] Wang, W. D., Backward bifurcation of an epidemic model with treatment, Mathematical Biosciences, 201, 1-2, 58-71, (2006) · Zbl 1093.92054 · doi:10.1016/j.mbs.2005.12.022
[21] Wang, Z. W., Backward bifurcation in simple SIS model, Acta Mathematicae Applicatae Sinica, 25, 1, 127-136, (2009) · Zbl 1189.34086 · doi:10.1007/s10255-006-6160-9
[22] Xue, L.; Scoglio, C., The network level reproduction number for infectious diseases with both vertical and horizontal transmission, Mathematical Biosciences, 243, 1, 67-80, (2013) · Zbl 1279.92057 · doi:10.1016/j.mbs.2013.02.004
[23] Zhang, X.; Liu, X. N., Backward bifurcation of an epidemic model with saturated treatment function, Journal of Mathematical Analysis and Applications, 348, 1, 433-443, (2008) · Zbl 1144.92038 · doi:10.1016/j.jmaa.2008.07.042
[24] Guckenheimer, J.; Holmes, P. J., Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, (1996), New York, NY, USA: Springer, New York, NY, USA
[25] Zhang, X.; Liu, X. N., Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment, Nonlinear Analysis: Real World Applications, 10, 2, 565-575, (2009) · Zbl 1167.34338 · doi:10.1016/j.nonrwa.2007.10.011
[26] Safi, M. A.; Gumel, A. B.; Elbasha, E. H., Qualitative analysis of an age-structured SEIR epidemic model with treatment, Applied Mathematics and Computation, 219, 22, 10627-10642, (2013) · Zbl 1298.92105 · doi:10.1016/j.amc.2013.03.126
[27] Zhou, L. H.; Fan, M., Dynamics of an SIR epidemic model with limited medical resources revisited, Nonlinear Analysis: Real World Applications, 13, 1, 312-324, (2012) · Zbl 1238.37041 · doi:10.1016/j.nonrwa.2011.07.036
[28] Zhou, X. Y.; Cui, J. A., Analysis of stability and bifurcation for an SEIR epidemic model with saturated recovery rate, Communications in Nonlinear Science and Numerical Simulation, 16, 11, 4438-4450, (2011) · Zbl 1219.92060 · doi:10.1016/j.cnsns.2011.03.026
[29] Lacitignola, D., Saturated treatments and measles resurgence episodes in South Africa: a possible linkage, Mathematical Biosciences and Engineering: MBE, 10, 4, 1135-1157, (2013) · Zbl 1273.92054 · doi:10.3934/mbe.2013.10.1135
[30] Eckalbar, J. C.; Eckalbar, W. L., Dynamics of an epidemic model with quadratic treatment, Nonlinear Analysis: Real World Applications, 12, 1, 320-332, (2011) · Zbl 1204.92056 · doi:10.1016/j.nonrwa.2010.06.018
[31] Brauer, F., Backward bifurcations in simple vaccination/treatment models, Journal of Biological Dynamics, 5, 5, 410-418, (2011) · Zbl 1225.92029 · doi:10.1080/17513758.2010.510584
[32] Hirsch, M. W.; Smale, S.; Devaney, R. L., Differential Equations, Dynamical Systems, and an Introduction to Chaos, (2003), New York, NY, USA: Academic Press, New York, NY, USA
[33] Bogdanov, R., Bifurcations of a limit cycle for a family of vector fields on the plan, Selecta Mathematica, 1, 373-388, (1981)
[34] Bogdanov, R., Versal deformations of a singular point on the plan in the case of zero eigen-values, Selecta Mathematica: Sovietica, 1, 389-421, (1981)
[35] Takens, F., Forced oscillations and bifurcation, Applications of Global Analysis I, 3, 1-59, (1974), Communications of the Mathematical Institute, Rijksuniversiteit Utrecht
[36] Kuznetsov, Y. A., Elements of Applied Bifurcation Theory, (1998), New York, NY, USA: Springer, New York, NY, USA · Zbl 0914.58025
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