The threshold value of the number of hospital beds in a SEIHR epidemic model

Figure 1.  The distribution of equilibria on the $ (K,A) $ plane. There is a unique endemic equilibrium $ E_1^* $ in region $ \Omega_1 $. There are two equilibria $ E_1^* $ and $ E_2^* $ in region $ \Omega_2 $ where $ \mathbb{\hat{R}}_0<\mathbb{R}_0<1 $, and the two equilibria coalesce when the parameters approach the green curve $ A = A_4(K) $ where $ \mathbb{R}_0 = \mathbb{\hat{R}}_0 $. In other regions of the first quadrant, there is no endemic equilibrium

Figure 2.  The phase portraits and the corresponding bifurcation diagrams near $ E_0 $. Figure (a) and (b) correspond to $ 0<K<K^* $, (c) and (d) correspond to $ K\geq K^* $

Figure 3.  The backward bifurcation diagram of system (3) when $ K = 100 $. When the value of parameter $ A $ passes through Hopf point $ H $, the stability of the endemic equilibria will change. Parameter values are listed in Table 1

Figure 4.  Bifurcation curves in $ (K,A) $ plane. The blue solid curve represents saddle-node bifurcation $ (SN) $, the red dotted curve represents Hopf bifurcation $ (Hopf) $, the black dashed curve represents Homoclinic bifurcation $ (Hom) $ and the purple solid line, $ L $, represents $ \mathbb{R}_0 = 1 $. These three bifurcation curves and straight-line $ L $ divide the first quadrant of the $ (K,A) $ plane into seven regions, where $ D_3 $ represents the region between $ Hopf $ and $ Hom $ when $ \mathbb{R}_0<1 $, $ D_6 $ represents the region where the limit cycle branched by $ Hopf $ exists when $ \mathbb{R}_0>1 $, and others are shown in the figure. The existence and stability of equilibria in each region are different. Each region gives a schematic diagram of the phase diagram near endemic equilibria except $ D_1 $. Parameter values are listed in Table 1, $ K^* = 126.6059 $

Figure 5.  The local phase portraits near the positive equilibrium $ E_1 $ in regions $ D_2 $, $ D_3 $ and $ D_4 $ of Fig. 4. Parameter values are listed in Table 1

Figure 6.  The local phase portrait near the positive equilibrium $ E_1 $ in region $ D_5 $ of Fig. 4. $ E_1 $ is stable. $ K = 80 $, $ A = 16 $. Parameter values are listed in Table 1

Figure 7.  The phase portrait near the positive equilibrium $ E_1 $ when the value of parameter (K, A) is in region $ D_6 $ of Fig. 4. $ E_1 $ is stable. Parameter values are listed in Table 1

Figure 8.  The phase portrait in the $ S-I-H $ space when the value of parameter $ (K,A) $ is in region $ D_7 $ of Fig. 4. The unique endemic equilibrium is unstable and a stable periodic solution exists. Parameter values are listed in Table 1. $ K = 140 $, $ A = 16.5 $

Figure 9.  The bifurcation diagram in plane $ (\mathbb{R}_0,I) $ when $ K = 60 $, $ A = 8.4 $, $ \mathbb{R}_0 = 0.6593 $ in region $ D_3 $. $ H $ is the Hopf bifurcation point and $ Hom $ is the Homoclinic orbit. The system has Hopf bifurcation at $ H $ and branches out an unstable limit cycle. Parameter values are listed in Table 1

Figure 10.  The bifurcation diagram in $ (A,I) $ plane with different $ K $, where $ K^* = 126.6059 $. When $ 0<K<K^* $, the system will undergo a backward bifurcation. When $ K\geq K^* $, the system will undergo a forward bifurcation. Parameter values are listed in Table 1

Figure 11.  The contour plot of $ \mathbb{R}_0 $, where $ \mathbb{R}_0 $ is 0.1, 0.5, 1, 1.5. The shorter the duration of exposed period ($ 1/\delta $) is, the greater the number of basic reproduction number ($ \mathbb{R}_0 $) is. This effect is more obvious with the increase of transmission rate $ \beta $. Parameter values are listed in Table 1