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

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)