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June 2018, 15(3): 739-764. doi: 10.3934/mbe.2018033

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

* Corresponding author:Yanni Xiao.

Received  April 18, 2017 Revised  October 08, 2017 Published  December 2017

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.

Citation: Aili Wang, Yanni Xiao, Huaiping Zhu. Dynamics of a Filippov epidemic model with limited hospital beds. Mathematical Biosciences & Engineering, 2018, 15 (3) : 739-764. doi: 10.3934/mbe.2018033
References:
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A. AbdelrazecJ. Bélair and C. Shan, Modeling the spread and control of dengue with limited public health resources, Math. Biosci., 271 (2016), 136-145. doi: 10.1016/j.mbs.2015.11.004.

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M. J. Aman and F. Kashanchi, Zika virus: A new animal model for an arbovirus, PLOS Negl Trop Dis, 10 (2016), e0004702. doi: 10.1371/journal.pntd.0004702.

[3]

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

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F. BizzarriA. Colombo and 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. doi: 10.1137/140959031.

[5]

Y. CaiY. Kang and M. Banerjee, A stochastic SIRS epidemic model with infectious force under intervention strategies, J Differ Equations, 259 (2015), 7463-7502. doi: 10.1016/j.jde.2015.08.024.

[6]

N. S. ChongB. Dionne and 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. doi: 10.1007/s00285-016-0971-y.

[7]

A. Colombo and F. Dercole, Discontinuity induced bifurcations of non-hyperbolic cycles in non-smooth systems, SIAM J Appl. Dyn. Syst., 9 (2010), 62-83. doi: 10.1137/080732377.

[8]

F. Della Rossa and F. Dercole, Generalized boundary equilibria in n-dimensional Filippov systems: The transition between persistence and nonsmooth-fold scenarios, Physica D, 241 (2012), 1903-1910. doi: 10.1016/j.physd.2011.04.012.

[9]

F. Della Rossa and F. Dercole, Generic and generalized boundary operating points in piecewise-linear (discontinuous) control systems, 51st IEEE Conference on Decision and Control, (2012), 7714-7719. doi: 10.1109/CDC.2012.6425950.

[10]

F. Dercole, Border collision bifurcations in the evolution of mutualistic interactions, Int. J. Bifurcat. Chaos, 15 (2005), 2179-2190. doi: 10.1142/S0218127405013241.

[11]

F. DercoleF. Della Rossa and A. Colombo, Two degenerate boundary equilibrium bifurcations in planar Filippov systems, SIAM J Appl. Dyn. Syst., 10 (2011), 1525-1553. doi: 10.1137/100812549.

[12]

F. DercoleR. Ferrière and 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. doi: 10.1098/rspb.2005.3398.

[13]

F. DercoleA. Gragnani and Y. A. Kuznetsov, Numerical sliding bifurcation analysis: An application to a relay control system, IEEE T Circuits-I, 50 (2003), 1058-1063. doi: 10.1109/TCSI.2003.815214.

[14]

F. DercoleA. Gragnani and S. Rinaldi, Bifurcation analysis of piecewise smooth ecological models, Theor. Popul. Biol., 72 (2007), 197-213. doi: 10.1016/j.tpb.2007.06.003.

[15]

F. Dercole and Y. A. Kuznetsov, SlideCont: An Auto97 driver for bifurcation analysis of Filippov systems, ACM Math. Software., 31 (2005), 95-119. doi: 10.1145/1055531.1055536.

[16]

F. Dercole and M. Stefano, Detection and continuation of a border collision bifurcation in a forest fire model, Appl. Math. Comput., 168 (2005), 623-635. doi: 10.1016/j.amc.2004.09.008.

[17]

M. Di BernardoC. J. Budd and A. R. Champneys, Bifurcations in nonsmooth dynamical systems, SIAM Rev., 50 (2008), 629-701. doi: 10.1137/050625060.

[18]

M. Di BernardoP. Kowalczyk and A. Nordmark, Bifurcations of dynamical systems with sliding: Derivation of normal-form mappings, Physica D, 170 (2002), 175-205. doi: 10.1016/S0167-2789(02)00547-X.

[19]

C. A. DonnellyM. C. Fisher and C. Fraser, Epidemiological and genetic analysis of severe acute respiratory syndrome, Lancet Infect. Dis., 4 (2004), 672-683. doi: 10.1016/S1473-3099(04)01173-9.

[20]

S. Echevarría-ZunoJ. M. Mejía-Aranguré and 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. doi: 10.1007/978-94-015-7793-9.

[22]

C. Fraser and C. A. Donnelly S. Cauchemez, Pandemic potential of a strain of influenza a (H1N1): early findings, Science, 324 (2009), 1557-1561. doi: 10.1126/science.1176062.

[23]

J. L. Goodman, Studying "secret serums" toward safe, effective ebola treatments, New Engl. J Med., 371 (2014), 1086-1089. doi: 10.1056/NEJMp1409817.

[24]

L. V. Green, How many hospital beds, Inquiry: J. Health Car., 39 (2002), 400-412. doi: 10.5034/inquiryjrnl_39.4.400.

[25]

M. GuardiaS. J. Hogan and 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. doi: 10.1137/090766826.

[26]

M. GuardiaT. M. Seara and M. A. Teixeira, Generic bifurcations of low codimension of planar filippov systems, J. Differ. Equations, 250 (2011), 1967-2023. doi: 10.1016/j.jde.2010.11.016.

[27]

A. B. GumelS. Ruan and T. Day, Modelling strategies for controlling SARS outbreaks, P. Roy. Soc. B-Biol. Sci., 271 (2004), 2223-2232. doi: 10.1098/rspb.2004.2800.

[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. doi: 10.1016/j.jtbi.2011.01.016.

[29]

Y. A. KuznetsovS. Rinaldi and A. Gragnani, One-parameter bifurcations in planar filippov systems, Int. J. Bifurcat. Chaos, 13 (2003), 2157-2188. doi: 10.1142/S0218127403007874.

[30]

I. M. LonginiA. Nizam and S. Xu, Containing pandemic influenza at the source, Science, 309 (2005), 1083-1087. doi: 10.1126/science.1115717.

[31]

M. E. M. MezaA. Bhaya and E. Kaszkurewicz, Threshold policies control for predator-prey systems using a control liapunov function approach, Theor. Popul. Biol., 67 (2005), 273-284. doi: 10.1016/j.tpb.2005.01.005.

[32]

W. QinS. Tang and C. Xiang, Effects of limited medical resource on a Filippov infectious disease model induced by selection pressure, Appl. Math. Comput., 283 (2016), 339-354. doi: 10.1016/j.amc.2016.02.042.

[33]

Z. SadiqueB. Lopman and 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. doi: 10.1093/infdis/jiv410.

[34]

C. ShanY. Yi and H. Zhu, Nilpotent singularities and dynamics in an SIR type of compartmental model with hospital resources, J Differ Equations, 260 (2016), 4339-4365. doi: 10.1016/j.jde.2015.11.009.

[35]

C. Shan and 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. doi: 10.1016/j.jde.2014.05.030.

[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. doi: 10.1371/journal.pone.0150513.

[37]

S. TangJ. Liang and Y. Xiao, Sliding bifurcations of filippov two stage pest control models with economic thresholds, SIAM J Appl. Dyn. Syst., 72 (2012), 1061-1080. doi: 10.1137/110847020.

[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. doi: 10.1371/journal.pone.0010911.

[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. doi: 10.1142/S0218127413501447.

[41]

W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58-71. doi: 10.1016/j.mbs.2005.12.022.

[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). doi: 10.1038/srep07838.

[45]

X. Zhang and 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. doi: 10.1016/j.nonrwa.2007.10.011.

show all references

References:
[1]

A. AbdelrazecJ. Bélair and C. Shan, Modeling the spread and control of dengue with limited public health resources, Math. Biosci., 271 (2016), 136-145. doi: 10.1016/j.mbs.2015.11.004.

[2]

M. J. Aman and F. Kashanchi, Zika virus: A new animal model for an arbovirus, PLOS Negl Trop Dis, 10 (2016), e0004702. doi: 10.1371/journal.pntd.0004702.

[3]

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

[4]

F. BizzarriA. Colombo and 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. doi: 10.1137/140959031.

[5]

Y. CaiY. Kang and M. Banerjee, A stochastic SIRS epidemic model with infectious force under intervention strategies, J Differ Equations, 259 (2015), 7463-7502. doi: 10.1016/j.jde.2015.08.024.

[6]

N. S. ChongB. Dionne and 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. doi: 10.1007/s00285-016-0971-y.

[7]

A. Colombo and F. Dercole, Discontinuity induced bifurcations of non-hyperbolic cycles in non-smooth systems, SIAM J Appl. Dyn. Syst., 9 (2010), 62-83. doi: 10.1137/080732377.

[8]

F. Della Rossa and F. Dercole, Generalized boundary equilibria in n-dimensional Filippov systems: The transition between persistence and nonsmooth-fold scenarios, Physica D, 241 (2012), 1903-1910. doi: 10.1016/j.physd.2011.04.012.

[9]

F. Della Rossa and F. Dercole, Generic and generalized boundary operating points in piecewise-linear (discontinuous) control systems, 51st IEEE Conference on Decision and Control, (2012), 7714-7719. doi: 10.1109/CDC.2012.6425950.

[10]

F. Dercole, Border collision bifurcations in the evolution of mutualistic interactions, Int. J. Bifurcat. Chaos, 15 (2005), 2179-2190. doi: 10.1142/S0218127405013241.

[11]

F. DercoleF. Della Rossa and A. Colombo, Two degenerate boundary equilibrium bifurcations in planar Filippov systems, SIAM J Appl. Dyn. Syst., 10 (2011), 1525-1553. doi: 10.1137/100812549.

[12]

F. DercoleR. Ferrière and 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. doi: 10.1098/rspb.2005.3398.

[13]

F. DercoleA. Gragnani and Y. A. Kuznetsov, Numerical sliding bifurcation analysis: An application to a relay control system, IEEE T Circuits-I, 50 (2003), 1058-1063. doi: 10.1109/TCSI.2003.815214.

[14]

F. DercoleA. Gragnani and S. Rinaldi, Bifurcation analysis of piecewise smooth ecological models, Theor. Popul. Biol., 72 (2007), 197-213. doi: 10.1016/j.tpb.2007.06.003.

[15]

F. Dercole and Y. A. Kuznetsov, SlideCont: An Auto97 driver for bifurcation analysis of Filippov systems, ACM Math. Software., 31 (2005), 95-119. doi: 10.1145/1055531.1055536.

[16]

F. Dercole and M. Stefano, Detection and continuation of a border collision bifurcation in a forest fire model, Appl. Math. Comput., 168 (2005), 623-635. doi: 10.1016/j.amc.2004.09.008.

[17]

M. Di BernardoC. J. Budd and A. R. Champneys, Bifurcations in nonsmooth dynamical systems, SIAM Rev., 50 (2008), 629-701. doi: 10.1137/050625060.

[18]

M. Di BernardoP. Kowalczyk and A. Nordmark, Bifurcations of dynamical systems with sliding: Derivation of normal-form mappings, Physica D, 170 (2002), 175-205. doi: 10.1016/S0167-2789(02)00547-X.

[19]

C. A. DonnellyM. C. Fisher and C. Fraser, Epidemiological and genetic analysis of severe acute respiratory syndrome, Lancet Infect. Dis., 4 (2004), 672-683. doi: 10.1016/S1473-3099(04)01173-9.

[20]

S. Echevarría-ZunoJ. M. Mejía-Aranguré and 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. doi: 10.1007/978-94-015-7793-9.

[22]

C. Fraser and C. A. Donnelly S. Cauchemez, Pandemic potential of a strain of influenza a (H1N1): early findings, Science, 324 (2009), 1557-1561. doi: 10.1126/science.1176062.

[23]

J. L. Goodman, Studying "secret serums" toward safe, effective ebola treatments, New Engl. J Med., 371 (2014), 1086-1089. doi: 10.1056/NEJMp1409817.

[24]

L. V. Green, How many hospital beds, Inquiry: J. Health Car., 39 (2002), 400-412. doi: 10.5034/inquiryjrnl_39.4.400.

[25]

M. GuardiaS. J. Hogan and 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. doi: 10.1137/090766826.

[26]

M. GuardiaT. M. Seara and M. A. Teixeira, Generic bifurcations of low codimension of planar filippov systems, J. Differ. Equations, 250 (2011), 1967-2023. doi: 10.1016/j.jde.2010.11.016.

[27]

A. B. GumelS. Ruan and T. Day, Modelling strategies for controlling SARS outbreaks, P. Roy. Soc. B-Biol. Sci., 271 (2004), 2223-2232. doi: 10.1098/rspb.2004.2800.

[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. doi: 10.1016/j.jtbi.2011.01.016.

[29]

Y. A. KuznetsovS. Rinaldi and A. Gragnani, One-parameter bifurcations in planar filippov systems, Int. J. Bifurcat. Chaos, 13 (2003), 2157-2188. doi: 10.1142/S0218127403007874.

[30]

I. M. LonginiA. Nizam and S. Xu, Containing pandemic influenza at the source, Science, 309 (2005), 1083-1087. doi: 10.1126/science.1115717.

[31]

M. E. M. MezaA. Bhaya and E. Kaszkurewicz, Threshold policies control for predator-prey systems using a control liapunov function approach, Theor. Popul. Biol., 67 (2005), 273-284. doi: 10.1016/j.tpb.2005.01.005.

[32]

W. QinS. Tang and C. Xiang, Effects of limited medical resource on a Filippov infectious disease model induced by selection pressure, Appl. Math. Comput., 283 (2016), 339-354. doi: 10.1016/j.amc.2016.02.042.

[33]

Z. SadiqueB. Lopman and 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. doi: 10.1093/infdis/jiv410.

[34]

C. ShanY. Yi and H. Zhu, Nilpotent singularities and dynamics in an SIR type of compartmental model with hospital resources, J Differ Equations, 260 (2016), 4339-4365. doi: 10.1016/j.jde.2015.11.009.

[35]

C. Shan and 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. doi: 10.1016/j.jde.2014.05.030.

[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. doi: 10.1371/journal.pone.0150513.

[37]

S. TangJ. Liang and Y. Xiao, Sliding bifurcations of filippov two stage pest control models with economic thresholds, SIAM J Appl. Dyn. Syst., 72 (2012), 1061-1080. doi: 10.1137/110847020.

[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. doi: 10.1371/journal.pone.0010911.

[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. doi: 10.1142/S0218127413501447.

[41]

W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58-71. doi: 10.1016/j.mbs.2005.12.022.

[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). doi: 10.1038/srep07838.

[45]

X. Zhang and 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. doi: 10.1016/j.nonrwa.2007.10.011.

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
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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