Dynamics of a Filippov epidemic model with limited hospital beds

Aili Wang; Yanni Xiao; Huaiping Zhu; Aili Wang; Yanni Xiao; Huaiping Zhu

doi:10.3934/mbe.2018033

Mathematical Biosciences and Engineering

2018, Volume 15, Issue 3: 739-764. doi: 10.3934/mbe.2018033

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Dynamics of a Filippov epidemic model with limited hospital beds

. Institute of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji 721013, China

Received: 18 April 2017 Revised: 08 October 2017 Published: 01 June 2018
MSC : Primary: 92D30, 92B05; Secondary: 34C05

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 $R_0 \gt 1$. If $R_0 \lt 1$, 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.
- Filippov system,
- limited hospital beds,
- sliding mode dynamics,
- pseudo-equilibrium,
- threshold policy,
- sliding bifurcation
Citation: Aili Wang, Yanni Xiao, Huaiping Zhu. Dynamics of a Filippov epidemic model with limited hospital beds[J]. Mathematical Biosciences and Engineering, 2018, 15(3): 739-764. doi: 10.3934/mbe.2018033

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Abstract

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 $R_0 \gt 1$. If $R_0 \lt 1$, 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.

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