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Rich dynamics of epidemic model with sub-optimal immunity and nonlinear recovery rate. (English) Zbl 1225.37108
Summary: We propose a simple epidemic model with sub-optimal immunity and an arbitrary recovery rate function to understand the impact of different forms of recovery rate. It shows that a nonlinear recovery rate leads to rich dynamic behaviors, such as multiple endemic equilibria, bistability, periodicity and bifurcations. Furthermore, we are pleasantly surprised to find that between the SIS and SIR models there are very similar dynamics with a special recovery rate function. Therefore, in order to simulate the disease trends more accurately, we need to construct a reasonable recovery rate, which is as important as the reasonable incidence rate.
MSC:
37N25Dynamical systems in biology
92D30Epidemiology
34C23Bifurcation (ODE)
34C25Periodic solutions of ODE
34D20Stability of ODE
References:
[1]Anderson, R. M.; May, R. M.: Infectious diseases of humans: dynamics and control, (1991)
[2]Hethcote, H. W.; Den Driessche, P. Van: Some epidemiological models with nonlinear incidence, J. math. Biol. 29, 271-287 (1991) · Zbl 0722.92015 · doi:10.1007/BF00160539
[3]Cui, J.; Xia, X.; Wan, H.: Saturation recovery leads to multiple endemic equilibria and backward bifurcation, J. theoret. Biol. 254, 275-283 (2008)
[4]Liu, Y.; Cui, J.: The impact of media coverage on the dynamics of infectious disease, Int. J. Biomath. 1, 65-74 (2008) · Zbl 1155.92343 · doi:10.1142/S1793524508000023
[5]Cui, J.; Tao, X.; Zhu, H.: An SIS epidemiological model incorporating media coverage, Rocky mountain J. Math. 28, 1323-1333 (2008) · Zbl 1170.92024 · doi:10.1216/RMJ-2008-38-5-1323
[6]Wang, W.: Backward bifurcation of an epidemic model with treatment, Math. biosci. 201, 58-71 (2006) · Zbl 1093.92054 · doi:10.1016/j.mbs.2005.12.022
[7]Liu, W.; Levin, S. A.; Iwasa, Y.: Influence of nonlinear incidence rates upon the behaviors of SIRS epidemiological models, J. math. Biol. 23, 187-204 (1986) · Zbl 0582.92023 · doi:10.1007/BF00276956
[8]Gomes, M. G. M.; Margheri, A.; Medley, G. F.; Rebelo, C.: Dynamical behaviour of epidemiological models with sub-optimal immunity and nonlinear incidence, J. math. Biol. 51, 414-430 (2005) · Zbl 1090.92043 · doi:10.1007/s00285-005-0331-9
[9]White, L. J.; Medley, G. F.: Microparasite population dynamics and continuous immunity, Proc. R. Soc. lond. Ser. B 265, 1977-1983 (1998)
[10]Gomes, M. G. M.; White, L. J.; Medley, G. F.: Infection, reinfection, and vaccination under suboptimal immune protection: epidemiological perspectives, J. theoret. Biol. 228, 539-549 (2004)
[11]Wang, W.; Ruan, S.: Bifurcations in an epidemic model with constant removal rate of the infectives, J. math. Anal. appl. 291, 775-793 (2004) · Zbl 1054.34071 · doi:10.1016/j.jmaa.2003.11.043
[12]Den Driessche, P. Van; Watmough, J.: Reproduction numbers and subthreshold endemic equilibrium for compartmental models of disease transmission, Math. biosci. 180, 29-48 (2002) · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6
[13]Murray, J. D.: Mathematical biology, (1998)
[14]Perko, L.: Differential equations and dynamic systems, (2000)
[15]Hethcote, H. W.; Lewis, M. A.; Den Driessche, P. Van: An epidemiological model with a delay and a nonlinear incidence rate, J. math. Biol. 27, 49-64 (1989) · Zbl 0714.92021 · doi:10.1007/BF00276080
[16]Liu, W.; Hethcote, H. W.; Levin, S. A.: Dynamical behavior of epidemiological models with nonlinear incidence rates, J. math. Biol. 25, 359-380 (1987) · Zbl 0621.92014 · doi:10.1007/BF00277162
[17]Ruan, S.; Wang, W.: Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. differential equations 188, 135-163 (2003) · Zbl 1028.34046 · doi:10.1016/S0022-0396(02)00089-X
[18]Meng, X.; Chen, L.; Cheng, H.: Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination, Appl. math. Comput. 186, 516-529 (2007) · Zbl 1111.92049 · doi:10.1016/j.amc.2006.07.124
[19]Kyrychkoa, Y. N.; Blyuss, K. B.: Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate, Nonlinear anal. RWA 6, 495-507 (2005) · Zbl 1144.34374 · doi:10.1016/j.nonrwa.2004.10.001
[20]Zou, W.; Xia, J.: An SI epidemic model with nonlinear infection rate and stage structure, Int. J. Biomath. 2, 19-27 (2009)