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# Dynamics of a Filippov epidemic model with limited hospital beds

• * Corresponding author:Yanni Xiao.
• A Filippov epidemic model is proposed to explore the impact of capacity and limited resources of public health system on the control of epidemic diseases. The number of infected cases is chosen as an index to represent a threshold policy, that is, the capacity dependent treatment policy is implemented when the case number exceeds a critical level, and constant treatment rate is adopted otherwise. The proposed Filippov model exhibits various local sliding bifurcations, including boundary focus or node bifurcation, boundary saddle bifurcation and boundary saddle-node bifurcation, and global sliding bifurcations, including grazing bifurcation and sliding homoclinic bifurcation to pseudo-saddle. The impact of some key parameters including the threshold level on disease control is examined by numerical analysis. Our results suggest that strengthening the basic medical conditions, i.e. increasing the minimum treatment ratio, or enlarging the input of medical resources, i.e. increasing HBPR (i.e. hospital bed-population ratio) as well as the possibility and level of maximum treatment ratio, can help to contain the case number at a relatively low level when the basic reproduction number $R_0>1$. If $R_0<1$, implementing these strategies can help in eradicating the disease although the disease cannot always be eradicated due to the occurring of backward bifurcation in the system.

Mathematics Subject Classification: Primary: 92D30, 92B05; Secondary: 34C05.

 Citation: • • Figure 1.  Boundary node bifurcation for Filippov system (4).

Figure 2.  Boundary saddle bifurcation for Filippov system (4). Here we choose $I_c$ as a bifurcation parameter and fix all other parameters as follows: $\Lambda = 5, \mu = 0.08, \beta = 1.4, h_0 = 0.3, h_1 = 0.7, b = 3, \nu = 0.7, I_c = 2 \mbox{(a)}, I_c =1.1463\mbox{(b)}, I_c = 1\mbox{(c)}.$

Figure 3.  Boundary saddle node bifurcation for Filippov system (4). Here we choose $I_c$ as a bifurcation parameter and fix all other parameters as follows: $\Lambda = 6, \mu = 0.1, \beta = 1.4, h_0 = 0.3, h_1 = 0.7834, \nu = 0.6, I_c = 2 \mbox{(a)}, I_c =2.5661\mbox{(b)}, I_c = 3\mbox{(c)}.$

Figure 4.  Local and global sliding bifurcations for Filippov system (4). We select $I_c$ as a bifurcation parameter and fix all other parameters as follows: $\Lambda = 8, \mu = 0.1, \beta = 1.8, h_0 = 0.2, h_1 = 2, b = 3.28, \nu = 0.6, I_c = 4 \mbox{(a)}, I_c = 4.7844 \mbox{(b)}, I_c = 4.95 \mbox{(c)}, I_c = 5.03 \mbox{(d)}, I_c = 5.2 \mbox{(e)}, I_c = 6.009782 \mbox{(f)}, I_c = 6.3 \mbox{(g)}, I_c = 6.8556 \mbox{(h)}.$ Here the black thick solid line represents a periodic cycle while the blue thick solid line stands for the homoclinic cycle.

Figure 5.  Evolution of the sliding modes and pseudo-equilibria for Filippov system (4) with respect to the threshold level $I_c$. Here we fix all other parameters as follows: $\Lambda = 8, \mu = 0.1, \beta = 1.8, h_0 = 0.2, \nu = 0.6$ and $h_1 = 0.8, b = 5\mbox{(a)}; h_1 = 2, b = 3.28\mbox{(b)}.$

Figure 6.  Evolution of the sliding modes (grey thick solid lines), the regular endemic equilibria (circle points and square points) and pseudo-equilibria (diamond points) for Filippov system (4) with respect to the parameter $b$. Here we fix all other parameters as follows: $\Lambda = 8, \mu = 0.1, \beta = 1.8, h_0 = 0.2, \nu = 0.6$ and $h_1 = 0.8, I_c = 8.6 \mbox{(a)}; h_1 = 2, I_c = 8 \mbox{(b)}.$

Figure 7.  Evolution of the infected cases with respect to the maximum per capita treatment rate $h_1$. Here we fix all other parameters as follows: $\Lambda = 8, \mu = 0.1, \beta = 1.8, \nu = 0.6, b = 5, h_0 = 0.2.$

Figure 8.  Evolution of the equilibria with respect to the minimum per capita treatment rate $h_0$. Here we fix all other parameters as follows: $\Lambda = 8, \mu = 0.1, \beta = 1.8, \nu = 0.6$ and $b = 5, h_1 = 1.05 \mbox{(a)}; b = 3.28, h_1 = 2 \mbox{(b)}.$

Table 1.  Existence of endemic equilibria for system $S_{G_2}$

 Range of parameter values Existence of endemic equilibria $R_0>1$ $\frac{-a_1+\sqrt{C_0}}{2a_0}<\frac{\Lambda}{\mu+\nu}$ $E_2$ $R_0<1$ $a_0>0, a_1<0, C_0>0, \frac{-a_1+\sqrt{C_0}}{2a_0}<\frac{\Lambda}{\mu+\nu}$ $E_2, E_3$ $a_0>0, a_1<0, C_0=0, \frac{-a_1}{2a_0}<\frac{\Lambda}{\mu+\nu}$ $E_*$ $a_0=0, a_1<0, \frac{-a_2}{a_1}<\frac{\Lambda}{\mu+\nu}$ $E_4$ $C_0<0$ Nonexistence $a_0>0, a_1>0, C_0\geq 0$ Nonexistence $a_0=0, a_1\geq 0$ Nonexistence $R_0=1$ $R_1>1, \frac{-a_1}{2a_0}<\frac{\Lambda}{\mu+\nu}$ $E_2$ $R_1\leq 1$ Nonexistence
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Tables(1)

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