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doi: 10.3934/dcdsb.2021016

Complex dynamics of a SIRS epidemic model with the influence of hospital bed number

Yancong Xu 1,, , Lijun Wei 1, , Xiaoyu Jiang 1, and  Zirui Zhu 2,

1. 

Department of Mathematics, Hangzhou Normal University, Hangzhou, 310036, China
2. 

School of Mathematical Sciences, East China Normal University, Shanghai 200241, China

* Corresponding author: Yancong Xu, Email: Yancongx@hznu.edu.cn

Received  May 2020 Revised  September 2020 Published  January 2021

Fund Project: The first author was supported by the National NSF of China (No. 11671114, 11871022) and NSF of Zhejiang (LY20A010002); the second author was supported by the National NSF of China (No. 11901144) and NSF of Zhejiang (Y201840020)

In this paper, the nonlinear dynamics of a SIRS epidemic model with vertical transmission rate of neonates, nonlinear incidence rate and nonlinear recovery rate are investigated. We focus on the influence of public available resources (especially the number of hospital beds) on disease control and transmission. The existence and stability of equilibria are analyzed with the basic reproduction number as the threshold value. The conditions for the existence of transcritical bifurcation, Hopf bifurcation, saddle-node bifurcation, backward bifurcation and the normal form of Bogdanov-Takens bifurcation are obtained. In particular, the coexistence of limit cycle and homoclinic cycle, and the coexistence of stable limit cycle and unstable limit cycle are also obtained. This study indicates that maintaining enough number of hospital beds is very crucial to the control of the infectious diseases no matter whether the immunity loss population are involved or not. Finally, numerical simulations are also given to illustrate the theoretical results.

Keywords: SIRS epidemic model, nonlinear recovery rate, nonlinear incidence rate, Hopf bifurcation, Bogdanov-Takens bifurcation.
Mathematics Subject Classification: Primary: 34C23, 34C25; Secondary: 92D30.
Citation: Yancong Xu, Lijun Wei, Xiaoyu Jiang, Zirui Zhu. Complex dynamics of a SIRS epidemic model with the influence of hospital bed number. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021016
References:
[1]

Y. Bai and X. Mu, Global asymptotic stability of a generalized SIRS epidemic model with transfer from infectious to susceptible, J. of Applied Analysis and Computation, 8 (2018), 402-412.  doi: 10.11948/2018.402.  Google Scholar

[2]

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Z. Hu, P. Bi, W. Ma and S. Ruan, Bifurcations of an SIRS epidemic model with nonlinear incidence rate, Discrete Contin. Dynam. Syst. Ser. B, 15 (2011), 93-112.  doi: 10.3934/dcdsb.2011.15.93.  Google Scholar

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Y. G. Lin, D. Q. Jiang and M. L. Jin, Stationary distribution of a stochastic SIR model with saturated incidence and its asymptotic stability, Acta Mathematica Scientia, 35 (2015), 619-629.  doi: 10.1016/S0252-9602(15)30008-4.  Google Scholar

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W. M. Liu, H. W. Hetchote and S. A. Levin, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological model, J. Math. Biol., 23 (1986), 187-204.  doi: 10.1007/BF00276956.  Google Scholar

[11]

Q. Liu, D. Jiang, T. Hayat and B. Ahmad, Analysis of a delayed vaccinated SIR epidemic model with temporary immunity and Levy jumps, Nonlinear Analysis: Hybrid Systems, 27 (2018), 29-43.  doi: 10.1016/j.nahs.2017.08.002.  Google Scholar

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G. H. Li and Y. X. Zhang, Dynamic behaviors of a modified SIR model in epidemic diseases using nonlinear incidence and recovery rates, Plos One, 2017 (2017), 1-28.  doi: 10.1371/journal.pone.0175789.  Google Scholar

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M. Lu, J. Huang, S. Ruan and P. Yu, Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate, J. Diff. Eqs., 267 (2019), 1859-1898.  doi: 10.1016/j.jde.2019.03.005.  Google Scholar

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M. Lu, J. Huang, S. Ruan and P. Yu, Global dynamics of a susceptible-infectious-recovered epidemic model with a generalized nonmonotone incidence rate, J. Dyn. Diff. Eq., (2020). doi: 10.1007/s10884-020-09862-3.  Google Scholar

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S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Diff. Eqs., 188 (2003), 135-163.  doi: 10.1016/S0022-0396(02)00089-X.  Google Scholar

[19]

E. Rivero-Esquivel, E. Avila-Vales and G. García-Almeida, Stability and bifurcation analysis of a SIR model with saturated incidence rate and saturated treatment, Mathematics and Computers in Simulation, 121 (2016), 109-132.  doi: 10.1016/j.matcom.2015.09.005.  Google Scholar

[20]

C. Shan and H. Zhu, Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds, J. Diff. Eqs., 257 (2014), 1662-1688.  doi: 10.1016/j.jde.2014.05.030.  Google Scholar

[21]

B. Sandstede and Y. C. Xu, Snakes and isolas in non-reversible conservative systems, Dynamical Systems, 27 (2012), 317-329.  doi: 10.1080/14689367.2012.691961.  Google Scholar

[22]

Y. Tang, D. Huang, S. Ruan and W. Zhang, Coexistence of limit cycles and homoclinic loops in a SIRS model with a nonlinear incidence rate, SIAM J. Appl. Math., 69 (2008), 621-639.  doi: 10.1137/070700966.  Google Scholar

[23]

P. van den Dreessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[24]

W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58-71.  doi: 10.1016/j.mbs.2005.12.022.  Google Scholar

[25]

S. Wiggins, Introduction to Applied Nonlinear Dynamics Systems and Chaos, second edition, Texts in Applied Mathematics, vol. 2, Spring-Verlag, 2003.  Google Scholar

[26]

Y. C. Xu, Y. Yang, F. W. Meng and P. Yu, Modelling and analysis of recurrent autoimmune disease, Nonlinear Analysis, Real World Applications, 54 (2020), 103109, 28 pp. doi: 10.1016/j.nonrwa.2020.103109.  Google Scholar

[27]

X. Yuan, F. Wang, Y. Xue and L. Yakui, Global stability of an SIR model with differential infectivity on complex networks, Physica A: Statistical Mechanics and its Applications, 499 (2018), 443-456.  doi: 10.1016/j.physa.2018.02.065.  Google Scholar

show all references

References:
[1]

Y. Bai and X. Mu, Global asymptotic stability of a generalized SIRS epidemic model with transfer from infectious to susceptible, J. of Applied Analysis and Computation, 8 (2018), 402-412.  doi: 10.11948/2018.402.  Google Scholar

[2]

V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.  Google Scholar

[3]

O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts, The construction of next-generation matrices for compartmental epidemic models, J. R. Soc., Interface, 7 (2010), 873-885.  doi: 10.1098/rsif.2009.0386.  Google Scholar

[4]

E. J. Doedel, T. F. Fairgrieve, B. Sandstede, A. R. Champneys, Y. A. Kuznetsov and X. Wang, Auto07p, continuation and bifurcation software for ordinary differential equations, (2007). Google Scholar

[5]

J. Guckenheimer, P. Holmes and M. Slemrod, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar

[6]

Z. X. Hu, W. B. Ma and S. Ruan, Analysis of SIR epidemic models with nonlinear incidence rate and treatment, Math. Biosci., 238 (2012), 12-20.  doi: 10.1016/j.mbs.2012.03.010.  Google Scholar

[7]

Z. Hu, P. Bi, W. Ma and S. Ruan, Bifurcations of an SIRS epidemic model with nonlinear incidence rate, Discrete Contin. Dynam. Syst. Ser. B, 15 (2011), 93-112.  doi: 10.3934/dcdsb.2011.15.93.  Google Scholar

[8]

W. M. Liu, H. W. Hetchote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380.  doi: 10.1007/BF00277162.  Google Scholar

[9]

Y. G. Lin, D. Q. Jiang and M. L. Jin, Stationary distribution of a stochastic SIR model with saturated incidence and its asymptotic stability, Acta Mathematica Scientia, 35 (2015), 619-629.  doi: 10.1016/S0252-9602(15)30008-4.  Google Scholar

[10]

W. M. Liu, H. W. Hetchote and S. A. Levin, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological model, J. Math. Biol., 23 (1986), 187-204.  doi: 10.1007/BF00276956.  Google Scholar

[11]

Q. Liu, D. Jiang, T. Hayat and B. Ahmad, Analysis of a delayed vaccinated SIR epidemic model with temporary immunity and Levy jumps, Nonlinear Analysis: Hybrid Systems, 27 (2018), 29-43.  doi: 10.1016/j.nahs.2017.08.002.  Google Scholar

[12]

G. H. Li and Y. X. Zhang, Dynamic behaviors of a modified SIR model in epidemic diseases using nonlinear incidence and recovery rates, Plos One, 2017 (2017), 1-28.  doi: 10.1371/journal.pone.0175789.  Google Scholar

[13]

M. Lu, J. Huang, S. Ruan and P. Yu, Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate, J. Diff. Eqs., 267 (2019), 1859-1898.  doi: 10.1016/j.jde.2019.03.005.  Google Scholar

[14]

M. Lu, J. Huang, S. Ruan and P. Yu, Global dynamics of a susceptible-infectious-recovered epidemic model with a generalized nonmonotone incidence rate, J. Dyn. Diff. Eq., (2020). doi: 10.1007/s10884-020-09862-3.  Google Scholar

[15]

K. M. O'Reilly, R. Lowe, W. J. Edmunds, et al., Projecting the end of the Zika virus epidemic in Latin America: A modelling analysis, BMC medicine, 180 (2018). doi: 10.1186/s12916-018-1158-8.  Google Scholar

[16]

Y. Pan, D. Pi, S. Yao and H. Meng, The global stability of two epidemic models with nonlinear recovery incidence rate, Modern Physics Letters B, 32 (2018), 1850357, 9 pp. doi: 10.1142/S0217984918503578.  Google Scholar

[17]

L. Perko, Differential Equations and Dynamical Systems, Springer, 2001. doi: 10.1007/978-1-4613-0003-8.  Google Scholar

[18]

S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Diff. Eqs., 188 (2003), 135-163.  doi: 10.1016/S0022-0396(02)00089-X.  Google Scholar

[19]

E. Rivero-Esquivel, E. Avila-Vales and G. García-Almeida, Stability and bifurcation analysis of a SIR model with saturated incidence rate and saturated treatment, Mathematics and Computers in Simulation, 121 (2016), 109-132.  doi: 10.1016/j.matcom.2015.09.005.  Google Scholar

[20]

C. Shan and H. Zhu, Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds, J. Diff. Eqs., 257 (2014), 1662-1688.  doi: 10.1016/j.jde.2014.05.030.  Google Scholar

[21]

B. Sandstede and Y. C. Xu, Snakes and isolas in non-reversible conservative systems, Dynamical Systems, 27 (2012), 317-329.  doi: 10.1080/14689367.2012.691961.  Google Scholar

[22]

Y. Tang, D. Huang, S. Ruan and W. Zhang, Coexistence of limit cycles and homoclinic loops in a SIRS model with a nonlinear incidence rate, SIAM J. Appl. Math., 69 (2008), 621-639.  doi: 10.1137/070700966.  Google Scholar

[23]

P. van den Dreessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[24]

W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58-71.  doi: 10.1016/j.mbs.2005.12.022.  Google Scholar

[25]

S. Wiggins, Introduction to Applied Nonlinear Dynamics Systems and Chaos, second edition, Texts in Applied Mathematics, vol. 2, Spring-Verlag, 2003.  Google Scholar

[26]

Y. C. Xu, Y. Yang, F. W. Meng and P. Yu, Modelling and analysis of recurrent autoimmune disease, Nonlinear Analysis, Real World Applications, 54 (2020), 103109, 28 pp. doi: 10.1016/j.nonrwa.2020.103109.  Google Scholar

[27]

X. Yuan, F. Wang, Y. Xue and L. Yakui, Global stability of an SIR model with differential infectivity on complex networks, Physica A: Statistical Mechanics and its Applications, 499 (2018), 443-456.  doi: 10.1016/j.physa.2018.02.065.  Google Scholar

Figure 1.  Bifurcation diagram of system (4) with respect to parameters $ \mu_{1} $ and $ d $ if $ k(bm+\beta)>(b+\beta)(p\delta+\mu_{0}) $, there exist one and two endemic equilibria in $ D_{1} $ and $ D_{0}, $ respectively. Two endemic equilibria coalesce and a saddle-node bifurcation occurs on $ C_{\Delta}^{-} $; the forward bifurcation occurs on $ C_{\Delta}^{+} $ and the backward bifurcation occurs on $ C_{\Delta}^{-} $, respectively. There is no endemic equilibrium in other regions
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Figure 2.  $ (a) $ Phase diagram of system (4) with no endemic equilibrium for $ R_{0}<1, k(bm+\beta)<(b+\beta)(p\delta+\mu_{0}) $ $ (b) $ Phase diagram of system (4) with one endemic equilibrium $ E_{*} $ for $ R_{0} = 1, d = d_{3}, s_{1} = 0. $
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Figure 3.  Bifurcation diagram of system (4) as $ \mu_{1} = 0.184 $, $ R_{0} = 1, d_{2} = 0.0797434. $ (a) system (4) undergoes the forward bifurcation for $ d_{2}<d = 0.1 $. (b) system (3) undergoes the backward bifurcation for $ d_{2}>d = 0.05 $, where $ \rm{BP} $ and $ \rm{HB} $ denote the transcritical bifurcation point and the subcritical Hopf bifurcation point
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Figure 4.  (a) Bifurcation diagram of system (4) as $ \mu_{1} = 0.184 $, $ R_{0} = 1, d_{2} = 0.0797434, d = d_{2}, $ system (4) undergoes the pitchfork bifurcation, where $ \rm{PB} $ denotes the pitchfork bifurcation point, $ HB_1 $ and $ HB_2 $ are supercritical Hopf bifurcation points. (b) One-parameter bifurcation diagram of system (4) with $ I $ and $ d $, where $ HB_1 $, $ HB_2 $ and $ SN $ denote, respectively, the subcritical Hopf bifurcation point, the supercritical bifurcation point and the saddle-node bifurcation point of limit cycles
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Figure 5.  (a) One-parameter bifurcation diagram of system (4) with $ I $ and $ \beta $ where $ HB_1 $ and $ HB_2 $ denote the supercritical bifurcation points. (b) Two-parameter Hopf bifurcation diagram with $ \beta $ and $ d $, where $ H $ and $ GH $ denote, respectively, the Hopf bifurcation curve and the generalized Hopf bifurcation point
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Figure 6.  $ (a) $ One-parameter bifurcation diagram of system (4) with the parameter $ d $ as a free parameter, where $ HB_1 $ and $ HB_2 $ denote two supercritical Hopf bifurcation points; $ (b) $ Two-parameter bifurcation diagram of system (4) with respect to parameters $ d $ and $ \mu_{1} $, where $ Hom, H, SN $ and $ SN_{lc} $ denote the homoclinic orbit bifurcation curve (black), Hopf bifurcation curve (green) and saddle-node bifurcation curve (red), the saddle-node bifurcation curve of limit cycles (the dotted blue line from $ GH $ to $ C_2 $), respectively. GH denotes the generalized Hopf bifurcation point
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Figure 7.  $ (a) $ Zoomed two-parameter bifurcation diagram of system (4) in Figure 6 (b); $ (b) $ Phase portraits in different regions of parameters in Figure 7 (a)
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