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Dynamics of a Filippov epidemic model with limited hospital beds

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

b. Department of Applied Mathematics, Xi'an Jiaotong University, Xi'an 710049, China

c. Laboratory of Mathematical Parallel Systems, Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada

Received: , Published:

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.

# References

[1] A. Abdelrazec,J. Bélair,C. Shan, Modeling the spread and control of dengue with limited public health resources, Math. Biosci., 271 (2016): 136-145.

[2] M. J. Aman and F. Kashanchi, Zika virus: A new animal model for an arbovirus, *PLOS Negl Trop Dis*, **10** (2016), e0004702.

[3] M. Bernardo, C. Budd, A. R. Champneys and et al.,
*Piecewise-smooth Dynamical Systems: Theory and Applications*, Springer, 2008.

[4] F. Bizzarri,A. Colombo,F. Dercole, Necessary and sufficient conditions for the noninvertibility of fundamental solution matrices of a discontinuous system, SIAM J Appl. Dyn. Syst., 15 (2016): 84-105.

[5] Y. Cai,Y. Kang,M. Banerjee, A stochastic SIRS epidemic model with infectious force under intervention strategies, J Differ Equations, 259 (2015): 7463-7502.

[6] N. S. Chong,B. Dionne,R. Smith, An avian-only Filippov model incorporating culling of both susceptible and infected birds in combating avian influenza, J Math. Biol., 73 (2016): 751-784.

[7] A. Colombo,F. Dercole, Discontinuity induced bifurcations of non-hyperbolic cycles in non-smooth systems, SIAM J Appl. Dyn. Syst., 9 (2010): 62-83.

[8] F. Della Rossa,F. Dercole, Generalized boundary equilibria in n-dimensional Filippov systems: The transition between persistence and nonsmooth-fold scenarios, Physica D, 241 (2012): 1903-1910.

[9] F. Della Rossa,F. Dercole, Generic and generalized boundary operating points in piecewise-linear (discontinuous) control systems, 51st IEEE Conference on Decision and Control, null (2012): 7714-7719.

[10] F. Dercole, Border collision bifurcations in the evolution of mutualistic interactions, Int. J. Bifurcat. Chaos, 15 (2005): 2179-2190.

[11] F. Dercole,F. Della Rossa,A. Colombo, Two degenerate boundary equilibrium bifurcations in planar Filippov systems, SIAM J Appl. Dyn. Syst., 10 (2011): 1525-1553.

[12] F. Dercole,R. Ferrière,A. Gragnani, Coevolution of slow-fast populations: Evolutionary sliding, evolutionary pseudo-equilibria and complex Red Queen dynamics, P. Roy. Soc. B-Biol. Sci., 273 (2006): 983-990.

[13] F. Dercole,A. Gragnani,Y. A. Kuznetsov, Numerical sliding bifurcation analysis: An application to a relay control system, IEEE T Circuits-I, 50 (2003): 1058-1063.

[14] F. Dercole,A. Gragnani,S. Rinaldi, Bifurcation analysis of piecewise smooth ecological models, Theor. Popul. Biol., 72 (2007): 197-213.

[15] F. Dercole,Y. A. Kuznetsov, SlideCont: An Auto97 driver for bifurcation analysis of Filippov systems, ACM Math. Software., 31 (2005): 95-119.

[16] F. Dercole,M. Stefano, Detection and continuation of a border collision bifurcation in a forest fire model, Appl. Math. Comput., 168 (2005): 623-635.

[17] M. Di Bernardo,C. J. Budd,A. R. Champneys, Bifurcations in nonsmooth dynamical systems, SIAM Rev., 50 (2008): 629-701.

[18] M. Di Bernardo,P. Kowalczyk,A. Nordmark, Bifurcations of dynamical systems with sliding: Derivation of normal-form mappings, Physica D, 170 (2002): 175-205.

[19] C. A. Donnelly,M. C. Fisher,C. Fraser, Epidemiological and genetic analysis of severe acute respiratory syndrome, Lancet Infect. Dis., 4 (2004): 672-683.

[20] S. Echevarría-Zuno,J. M. Mejía-Aranguré,A. J. Mar-Obeso, Infection and death from influenza a H1N1 virus in mexico: a retrospective analysis, Lancet, 374 (2010): 2072-2079.

[21] A. F. Filippov and F. M. Arscott,
*Differential Equations with Discontinuous Righthand Sides: Control Systems*, Springer, 1988.

[22] C. Fraser,C. A. Donnelly S. Cauchemez, Pandemic potential of a strain of influenza a (H1N1): early findings, Science, 324 (2009): 1557-1561.

[23] J. L. Goodman, Studying "secret serums" toward safe, effective ebola treatments, New Engl. J Med., 371 (2014): 1086-1089.

[24] L. V. Green, How many hospital beds, Inquiry: J. Health Car., 39 (2002): 400-412.

[25] M. Guardia,S. J. Hogan,T. M. Seara, An analytical approach to codimension-2 sliding bifurcations in the dry-friction oscillator, SIAM J Appl. Dyn. Syst., 9 (2010): 769-798.

[26] M. Guardia,T. M. Seara,M. A. Teixeira, Generic bifurcations of low codimension of planar filippov systems, J. Differ. Equations, 250 (2011): 1967-2023.

[27] A. B. Gumel,S. Ruan,T. Day, Modelling strategies for controlling SARS outbreaks, P. Roy. Soc. B-Biol. Sci., 271 (2004): 2223-2232.

[28] V. Křivan, On the gause predator-prey model with a refuge: A fresh look at the history, J. of Theor. Biol., 274 (2011): 67-73.

[29] Y. A. Kuznetsov,S. Rinaldi,A. Gragnani, One-parameter bifurcations in planar filippov systems, Int. J. Bifurcat. Chaos, 13 (2003): 2157-2188.

[30] I. M. Longini,A. Nizam,S. Xu, Containing pandemic influenza at the source, Science, 309 (2005): 1083-1087.

[31] M. E. M. Meza,A. Bhaya,E. Kaszkurewicz, Threshold policies control for predator-prey systems using a control liapunov function approach, Theor. Popul. Biol., 67 (2005): 273-284.

[32] W. Qin,S. Tang,C. Xiang, Effects of limited medical resource on a Filippov infectious disease model induced by selection pressure, Appl. Math. Comput., 283 (2016): 339-354.

[33] Z. Sadique,B. Lopman,B. S. Cooper, Cost-effectiveness of ward closure to control outbreaks of norovirus infection in United Kingdom National Health Service Hospitals, J. Infect. Dis., 213 (2016): S19-S26.

[34] C. Shan,Y. Yi,H. Zhu, Nilpotent singularities and dynamics in an SIR type of compartmental model with hospital resources, J Differ Equations, 260 (2016): 4339-4365.

[35] C. Shan,H. Zhu, Bifurcations and complex dynamics of an sir model with the impact of the number of hospital beds, J Differ Equations, 257 (2014): 1662-1688.

[36] X. Sun, Y. Xiao, S. Tang and et al., Early HAART initiation may not reduce actual reproduction number and prevalence of MSM infection: Perspectives from coupled within-and between-host modelling studies of Chinese MSM populations, *PloS one*, **11** (2016), e0150513.

[37] S. Tang,J. Liang,Y. Xiao, Sliding bifurcations of filippov two stage pest control models with economic thresholds, SIAM J Appl. Dyn. Syst., 72 (2012): 1061-1080.

[38] S. Tang, Y. Xiao, Y. Yang and et al., Community-based measures for mitigating the 2009 H1N1 pandemic in china, *PloS One*, **5** (2010), e10911.

[39] V. I. Utkin,
*Sliding Modes and Their Applications in Variable Structure Systems*, Mir, Moscow, 1978.

[40] A. Wang and Y. Xiao, Sliding bifurcation and global dynamics of a Filippov epidemic model with vaccination, *Int. J. Bifurcat. Chaos*, **23** (2013), 1350144, 32pp.

[41] W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006): 58-71.

[42] WHO Ebola Response Team, Ebola virus disease in west africa--the first 9 months of the epidemic and forward projections, New Engl. J Med., 371 (2014): 1481-1495.

[43] World health organization, World health statistics, 2005-2015.

[44] Y. Xiao, S. Tang and J. Wu, Media impact switching surface during an infectious disease outbreak, *Sci. Rep. -UK*, **5** (2015).

[45] X. Zhang,X. Liu, Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment, Nonlinear Anal-Real, 10 (2009): 565-575.

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