A diffusive epidemic model with two delays subjecting to Neumann boundary conditions is considered. First we obtain the existence and the stability of the positive constant steady state. Then we investigate the existence of Hopf bifurcations by analyzing the distribution of the eigenvalues. Furthermore, we derive the normal form on the center manifold near the Hopf bifurcation singularity. Finally, some numerical simulations are carried out to illustrate the theoretical results.
Citation: Huan Dai, Yuying Liu, Junjie Wei. Stability analysis and Hopf bifurcation in a diffusive epidemic model with two delays[J]. Mathematical Biosciences and Engineering, 2020, 17(4): 4127-4146. doi: 10.3934/mbe.2020229
A diffusive epidemic model with two delays subjecting to Neumann boundary conditions is considered. First we obtain the existence and the stability of the positive constant steady state. Then we investigate the existence of Hopf bifurcations by analyzing the distribution of the eigenvalues. Furthermore, we derive the normal form on the center manifold near the Hopf bifurcation singularity. Finally, some numerical simulations are carried out to illustrate the theoretical results.
| [1] | W. Kermack, A. Mckendrick, Contributions to the mathematical theory of epidemics. Ⅱ. The problem of endemicity, Bull. Math. Biol., 53 (1991), 57-87. |
| [2] |
H. Smith, Subharmonic bifurcation in an S-I-R epidemic model, J. Math. Biol., 17 (1983), 163-177. doi: 10.1007/BF00305757
|
| [3] | E. Beretta, Y. Takeuchi, Global stability of an SIR epidemic model with time delays, J. Math. Biol., 33 (1995), 250-260. |
| [4] |
H. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907
|
| [5] | M. Delasen, P. Agarwal, A. Ibeas, S. Alonso-Quesada, On a generalized time-varying SEIR epidemic model with mixed point and distributed time-varying delays and combined regular and impulsive vaccination controls, Adv. Differ. Equations, 2010 (2010), 1-42. |
| [6] | V. Capasso, S. Paverifontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, SIAM J. Appl. Math., 27 (1979), 121-132. |
| [7] |
M. Fan, M. Li, K. Wang, Global stability of an SEIS epidemic model with recruitment and a varying total population size, Math. Biosci., 170 (2001), 199-208. doi: 10.1016/S0025-5564(00)00067-5
|
| [8] | X. Mi, Global dynamics of an SEIR epidemic model with vertical transmission, J. Shanxi Normal Univ., 62 (2013), 58-69. |
| [9] | W. Wang, S. Ruan, Bifurcations in an epidemic model with constant removal rate of the infectives, J. Math. Anal. Appl., 291 (2015), 775-793. |
| [10] |
Y. Zhou, Z. Ma, F. Brauer, A discrete epidemic model for SARS transmission and control in China, Math. Comput. Modell., 40 (2004), 1491-1506. doi: 10.1016/j.mcm.2005.01.007
|
| [11] |
V. Capasso, R. Wilson, Analysis of a reaction-diffusion system modeling man-environment-man epidemics, SIAM J. Appl. Math., 57 (1997), 327-346. doi: 10.1137/S0036139995284681
|
| [12] |
V. Capasso, K. Kunisch, A reaction-diffusion system arising in modelling man-environment diseases, Q. Appl. Math., 46 (1988), 431-450. doi: 10.1090/qam/963580
|
| [13] | T. Zhang, J. Liu, Z. Teng, Dynamic behavior for a nonautonomous SIRS epidemic model with distributed delays, Appl. Math. Comput., 214 (2009), 624-631. |
| [14] |
K. Wang, Z. Teng, X. Zhang, Dynamical behaviors of an Echinococcosis epidemic model with distributed delays, Math. Biosci. Eng., 14 (2017), 1425-1445. doi: 10.3934/mbe.2017074
|
| [15] |
T. Zhao, Z. Zhang, R. Upadhyay, Delay-induced Hopf bifurcation of an SVEIR computer virus model with nonlinear incidence rate, Adv. Differ. Equations, 2018 (2018), 256. doi: 10.1186/s13662-018-1698-4
|
| [16] | J. Zhang, Z. Jin, The analysis of epidemic network model with infectious force in latent and infected period, Discrete Dyn. Nat. Soc., 2010 (2010), 1-12. |
| [17] |
Q. Khan, E. Krishnan, An epidemic model with a time delay in transmission, Appl. Math., 48 (2003), 193-203. doi: 10.1023/A:1026002429257
|
| [18] |
J. Wei, J. Zhou, W. Chen, Z. Zhen, L. Tian, Traveling waves in a nonlocal dispersal epidemic model with spatio-temporal delay, Commun. Pure Appl. Anal., 19 (2020), 2853-2886. doi: 10.3934/cpaa.2020125
|
| [19] |
Z. Guo, Y. Li, Z. Feng, Exponential stability of traveling waves in a nonlocal dispersal epidemic model with delay, J. Comput. Appl. Math., 344 (2018), 47-72. doi: 10.1016/j.cam.2018.05.018
|
| [20] |
M. Delasen, On some structures of stabilizing control laws for linear and time-invariant systems with bounded point delays and unmeasurable states, Int. J. Control, 59 (1994), 529-541. doi: 10.1080/00207179408923091
|
| [21] |
R. M. Nguimdo, Constructing Hopf bifurcation lines for the stability of nonlinear systems with two time delays, Phys. Rev. E, 97 (2018), 032211. doi: 10.1103/PhysRevE.97.032211
|
| [22] |
C. Shen, Y. Li, X. Zhu, W. Duan, Improved stability criteria for linear systems with two additive time-varying delays via a novel Lyapunov functional, J. Comput. Appl. Math., 363 (2020), 312- 324. doi: 10.1016/j.cam.2019.06.010
|
| [23] |
Y. Du, B. Niu, J. Wei, Two delays induce Hopf bifurcation and double Hopf bifurcation in a diffusive Leslie-Gower predator-prey system, Chaos: Interdiscip. J. Nonlinear Sci., 29 (2019), 013101. doi: 10.1063/1.5078814
|
| [24] | S. Wu, C. Hsu, Existence of entire solutions for delayed monostable epidemic models, Trans.Am. Math. Soc., 368 (2016), 6033-6062. |
| [25] | X. Zhao, W. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 1117-1128. |
| [26] |
D. Xu, X. Zhao, Bistable waves in an epidemic model, J. Dyn. Differ. Equations, 16 (2004), 679-707. doi: 10.1007/s10884-004-6113-z
|
| [27] |
C. Hsu, T. Yang, Z. Yu, Existence and exponential stability of traveling waves for delayed reactiondiffusion systems, Nonlinearity, 31 (2018), 838-863. doi: 10.1088/1361-6544/aa99a1
|
| [28] |
S. Marialisa, A. Di Stefano, L. Pietro, A. La Corte, The impact of heterogeneity and awareness in modeling epidemic spreading on multiplex networks, Sci. Rep., 6 (2016), 37105. doi: 10.1038/srep37105
|
| [29] | L. Zuo, M. Liu, J. Wang, The impact of awareness programs with recruitment and delay on the spread of an epidemic, Math. Probl. Eng., 2015 (2015), 1-10. |
| [30] | Y. Qin, X. Zhong, H. Jiang, Y. Ye, An environment aware epidemic spreading model and immune strategy in complex networks, Appl. Math. Comput., 261 (2015), 206-215. |
| [31] | S. Ruan, J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin., Discrete Impulsive Syst. Ser. A: Math. Anal., 10 (2003), 863-874. |
| [32] |
X. Wei, J. Wei, The effect of delayed feedback on the dynamics of an autocatalysis reactiondiffusion system, Nonlinear Anal.: Modell. Control, 23 (2018), 749-770. doi: 10.15388/NA.2018.5.7
|
| [33] |
Y. Song, J. Wei, Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system, J. Math. Anal. Appl., 301 (2005), 1-21. doi: 10.1016/j.jmaa.2004.06.056
|
| [34] | J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. |
| [35] |
J. Zhao, J. Wei, Dynamics in a diffusive plankton system with delay and toxic substances effect, Nonlinear Anal.: Real World Appl., 22 (2015), 66-83. doi: 10.1016/j.nonrwa.2014.07.010
|
| [36] | B. Hassard, N. Kazarinoff, Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University-Verlag, Cambridge, 1981. |
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Huan Dai, Yuying Liu, Junjie Wei. Stability analysis and Hopf bifurcation in a diffusive epidemic model with two delays[J]. Mathematical Biosciences and Engineering, 2020, 17(4): 4127-4146. doi: 10.3934/mbe.2020229
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