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Stability analysis in a class of discrete SIRS epidemic models. (English) Zbl 1254.92082
Summary: The dynamical behavior of a class of discrete-time SIRS epidemic models is discussed. Conditions for the existence and local stability of the disease-free equilibrium and endemic equilibrium are obtained. The numerical simulations not only illustrate the validity of our results, but also exhibit more complex dynamical behavior, such as flip bifurcation, Hopf bifurcation and chaos phenomena. These results reveal far richer dynamical behaviors of the discrete epidemic model compared with continuous epidemic models.
MSC:
92D30Epidemiology
39A30Stability theory (difference equations)
65C60Computational problems in statistics
39A60Applications of difference equations
References:
[1]Zhou, L.; Fan, M.: Dynamics of an SIR epidemic model with limited medical resources revisited, Nonlinear anal. RWA 13, 312-324 (2012)
[2]Alexanderian, A.; Gobbert, M. K.; Fister, K. R.; Gaff, H.; Lenhart, S.; Schaefe, E.: An age-structured model for the spread of epidemic cholera: analysis and simulation, Nonlinear anal. RWA 12, 3483-3498 (2011) · Zbl 1231.35268 · doi:10.1016/j.nonrwa.2011.06.009
[3]Zhang, H.; Chen, L.; Nieto, J. J.: A delayed epidemic model with stage-structure and pulses for pest management strategy, Nonlinear anal. RWA 9, 1714-1726 (2008) · Zbl 1154.34394 · doi:10.1016/j.nonrwa.2007.05.004
[4]Robledoa, G.; Grognardb, F.; Gouz, J. L.: Global stability for a model of competition in the chemostat with microbial inputs, Nonlinear anal. RWA 13, 582-598 (2012)
[5]Mena-Lorca, J.; Hethcote, H. W.: Dynamica models of infectious disease as regulations of population sizes, J. math. Biol. 30, 693-716 (1992) · Zbl 0748.92012
[6]Zhang, T.; Teng, Z.: Global behavior and permanence of SIRS epidemic model with time delay, Nonlinear anal. RWA 9, 1409-1424 (2008) · Zbl 1154.34390 · doi:10.1016/j.nonrwa.2007.03.010
[7]Wang, L.; Chen, L.; Nieto, J.: The dynamics of an epidemic model for pest control with impulsive effect, Nonlinear anal. RWA 11, 1374-1386 (2010) · Zbl 1188.93038 · doi:10.1016/j.nonrwa.2009.02.027
[8]Gao, S.; Liu, Y.; Nieto, J.; Andrade, H.: Seasonality and mixed vaccination stategy in an epidemic model with vertical transmission, Math. comput. Simulation 81, 1855-1868 (2011) · Zbl 1217.92066 · doi:10.1016/j.matcom.2010.10.032
[9]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 · doi:10.1016/j.nonrwa.2008.10.014
[10]Muroya, Y.; Enatsu, Y.; Nakata, Y.: Montone iterative technique to SIRS epidemic models with nonlinear incidence rates and distributed delays, Nonlinear anal. RWA 12, 1897-1910 (2011) · Zbl 1225.93091 · doi:10.1016/j.nonrwa.2010.12.002
[11]Franke, J. E.; Yakubu, A. -A.: Disease-induced mortality in density-dependent discrete-time S-I-S epidemic models, J. math. Biol. 57, 755-790 (2008) · Zbl 1161.92046 · doi:10.1007/s00285-008-0188-9
[12]Castillo-Chavez, C.; Yakubu, A. -A.: Discrete-time SIS models with complex dynamics, Nonlinear anal. 47, 4753-4762 (2001) · Zbl 1042.37544 · doi:10.1016/S0362-546X(01)00587-9
[13]Sekiguchi, M.; Ishiwata, E.: Global dynamics of a discretized SIRS epidemic model with time delay, J. math. Anal. appl. 371, 195-202 (2010) · Zbl 1193.92081 · doi:10.1016/j.jmaa.2010.05.007
[14]Allen, L. J. S.; Burgin, A. M.: Comparison of deterministic and stochastic SIS and SIR models in discrete time, Math. biosci. 163, 1-33 (2000) · Zbl 0978.92024 · doi:10.1016/S0025-5564(99)00047-4
[15]Li, J.; Ma, Z.; Brauer, F.: Global analysis of discrete-time SI and SIS epidemic models, Math. biosci. Eng. 4, 699-710 (2007) · Zbl 1142.92038 · doi:10.3934/mbe.2007.4.699
[16]Emmert, K. E.; Allen, L. J. S.: Population extinction in deterministic and stochastic discrete-time epidemic models with periodic coefficients with applications to amphibian populations, Nat. resour. Model. 19, 117-164 (2006) · Zbl 1157.92323 · doi:10.1111/j.1939-7445.2006.tb00178.x
[17]Li, J.; Lou, J.; Lou, M.: Some discrete SI and SIS epidemic models, Appl. math. Mech. (English ed.) 29, 113-119 (2008)
[18]Ramani, A.; Carstea, A. S.; Willox, R.; Grammaticos, B.: Oscillating epidemics: a discrete-time model, Physica A 333, 278-292 (2004)
[19]Satsuma, J.; Willox, R.; Ramani, A.; Grammaticos, B.; Carstea, A. S.: Extending the SIR epidemic model, Physica A 336, 369-375 (2004)
[20]Zhang, D.; Shi, B.: Oscillation and global asymptotic stability in a discrete epidemic model, J. math. Anal. appl. 278, 194-202 (2003) · Zbl 1025.39013 · doi:10.1016/S0022-247X(02)00717-5
[21]D’innocenzo, A.; Paladini, F.; Renna, L.: A numerical investigation of discrete oscillating epidemic models, Physica A 364, 497-512 (2006)
[22]Willoxa, R.; Grammaticosa, B.; Carsteab, A. S.; Ramani, A.: Epidemic dynamics: discrete-time and cellular automaton models, Physica A 328, 13-22 (2003) · Zbl 1026.92042 · doi:10.1016/S0378-4371(03)00552-1
[23]Allen, L. J. S.; Driessche, P.: The basic reproduction number in some discrete-time epidemic models, J. difference equ. Appl. 14, 1127-1147 (2008) · Zbl 1147.92032 · doi:10.1080/10236190802332308
[24]Li, X.; Wang, W.: A discrete epidemic model with stage structure, Chaos solitons fractals 26, 947-958 (2005) · Zbl 1066.92045 · doi:10.1016/j.chaos.2005.01.063
[25]Li, L.; Sun, G.; Jin, Z.: Bifurcation and chaos in an epidemic model with nonlinear incidence rates, Appl. math. Comput. 216, 1226-1234 (2010) · Zbl 1187.92073 · doi:10.1016/j.amc.2010.02.014
[26]Allen, L. J. S.: Some discrete-time SI, SIR, and SIS epidemic models, Math. biosci. 124, 83-105 (1994) · Zbl 0807.92022 · doi:10.1016/0025-5564(94)90025-6
[27]Allen, L. J. S.; Lou, Y.; Nevai, A. L.: Spatial patterns in a discrete-time SIS patch model, J. math. Biol. 58, 339-375 (2009) · Zbl 1162.92033 · doi:10.1007/s00285-008-0194-y
[28]Franke, J. E.; Yakubu, A. -A.: Discrete-time SIS epidemic model in a seasonal environment, SIAM J. Appl. math. 66, 1563-1587 (2006) · Zbl 1108.37303 · doi:10.1137/050638345
[29]Mendez, V.; Fort, J.: Dynamical evolution of discrete epidemic models, Physica A 284, 309-317 (2000)
[30]Sekiguchi, M.: Permanence of a discrete SIRS epidemic model with time delays, Appl. math. Lett. 23, 1280-1285 (2010) · Zbl 1194.92065 · doi:10.1016/j.aml.2010.06.013
[31]Muroya, Y.; Bellen, A.; Enatsu, Y.; Nakata, Y.: Global stability for a discrete epidemic model for disease with immunity and latency spreading in a heterogeneous host population, Nonlinear anal. RWA 13, 258-274 (2012)
[32]Muroya, Y.; Nakata, Y.; Izzo, G.; Vecchio, A.: Permanence and global stability of a class of discrete epidemic models, Nonlinear anal. RWA 12, 2105-2117 (2011) · Zbl 1216.92055 · doi:10.1155/2009/143019
[33]Fan, S.: A new extracting formula and a new distinguishing means on the one variable cubic equation, J. Hainan teach. College 2, No. 2, 91-98 (1989)
[34]Li, C.; Zhou, H.: Studies of elementary mathematics, (2007)
[35]Zhang, H.; Du, X.; Xia, M.; Zheng, C.: Mathematical encyclopaedia, Mathematical encyclopaedia 1 (1994)