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

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

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

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)

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  • 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.

    Mathematics Subject Classification: Primary: 34C23, 34C25; Secondary: 92D30.


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  • 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

    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. $

    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

    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

    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

    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

    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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