doi: 10.3934/dcdsb.2021195
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Global dynamics and bifurcations in a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate

1. 

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

2. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: Jicai Huang

Received  December 2020 Revised  April 2021 Early access August 2021

Fund Project: Research of JH and QP was partially supported by NSFC (No. 11871235) and the Fundamental Research Funds for the Central Universities (CCNU19TS030). Research of QH is partially supported by NSFC (No. 11871060) and the Fundamental Research Funds for the Central university (XDJK2018B031)

In this paper, we analyze a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate. The nonmonotone incidence rate describes the "psychological effect": when the number of the infected individuals (denoted by $ I $) exceeds a certain level, the incidence rate is a decreasing function with respect to $ I $. The piecewise-smooth treatment rate describes the situation where the community has limited medical resources, treatment rises linearly with $ I $ until the treatment capacity is reached, after which constant treatment (i.e., the maximum treatment) is taken.Our analysis indicates that there exists a critical value $ \widetilde{I_0} $ $ ( = \frac{b}{d}) $ for the infective level $ I_0 $ at which the health care system reaches its capacity such that:(i) When $ I_0 \geq \widetilde{I_0} $, the transmission dynamics of the model is determined by the basic reproduction number $ R_0 $: $ R_0 = 1 $ separates disease persistence from disease eradication. (ii) When $ I_0 < \widetilde{I_0} $, the model exhibits very rich dynamics and bifurcations, such as multiple endemic equilibria, periodic orbits, homoclinic orbits, Bogdanov-Takens bifurcations, and subcritical Hopf bifurcation.

Citation: Qin Pan, Jicai Huang, Qihua Huang. Global dynamics and bifurcations in a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021195
References:
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M. E. Alexander and S. M. Moghadas, Bifurcation analysis of an SIRS epidemic model with generalized incidence, SIAM J. Appl. Math., 65 (2005), 1794-1816.  doi: 10.1137/040604947.  Google Scholar

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R. Bogdanov, Bifurcations of a limit cycle of a certain family of vector fields on the plane, (Russian) Trudy Sem. Petrovsk. Vyp., 2 (1976), 23-35.   Google Scholar

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R. I. Bogdanov, Versal deformations of a singular point of a vector field on the plane in the case of zero eigen-values, Functional Analysis and Its Applications, 9 (1975), 144-145.  doi: 10.1007/BF01075453.  Google Scholar

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W. LiuS. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204.  doi: 10.1007/BF00276956.  Google Scholar

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M. LuJ. HuangS. Ruan and P. Yu, Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate, J. Differential Equations, 267 (2019), 1859-1898.  doi: 10.1016/j.jde.2019.03.005.  Google Scholar

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F. Takens, Forced oscillations and bifurcation, Global Analysis of Dynamical Systems, 3 (2001), 1-61.   Google Scholar

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

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W. Wang and S. Ruan, Bifurcation in an epidemic model with constant removal rate of the infectives, J. Math. Anal. Appl., 291 (2004), 775-793.  doi: 10.1016/j.jmaa.2003.11.043.  Google Scholar

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D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429.  doi: 10.1016/j.mbs.2006.09.025.  Google Scholar

[23]

D. Xiao and Y. Zhou, Qualitative analysis of an epidemic model, Can. Appl. Math. Q., 14 (2006), 469-492.   Google Scholar

[24]

Q. YangD. JiangN. Shi and C. Ji, The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence, J. Math. Anal. Appl., 388 (2012), 248-271.  doi: 10.1016/j.jmaa.2011.11.072.  Google Scholar

[25]

Y. Yao, Bifurcations of a Leslie-Gower prey-predator system with ratio-dependent Holling IV functional response and prey harvesting, Math. Meth. Appl. Sci., 43 (2020), 2137-2170.  doi: 10.1002/mma.5944.  Google Scholar

[26]

Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Translated from the Chinese by Anthony Wing Kwok Leung. Translations of Mathematical Monographs, 101. American Mathematical Society, Providence, RI, 1992.  Google Scholar

[27]

X. Zhang and X. Liu, Backward bifurcation of an epidemic model with saturated treatment function, J. Math. Anal. Appl., 348 (2008), 433-443.  doi: 10.1016/j.jmaa.2008.07.042.  Google Scholar

[28]

T. ZhouW. Zhang and Q. Lu, Bifurcation analysis of an SIS epidemic model with saturated incidence rate and saturated treatment function, Applied Mathematics and Computation, 226 (2014), 288-305.  doi: 10.1016/j.amc.2013.10.020.  Google Scholar

show all references

References:
[1]

M. E. Alexander and S. M. Moghadas, Bifurcation analysis of an SIRS epidemic model with generalized incidence, SIAM J. Appl. Math., 65 (2005), 1794-1816.  doi: 10.1137/040604947.  Google Scholar

[2] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1992.   Google Scholar
[3]

R. Bogdanov, Bifurcations of a limit cycle of a certain family of vector fields on the plane, (Russian) Trudy Sem. Petrovsk. Vyp., 2 (1976), 23-35.   Google Scholar

[4]

R. I. Bogdanov, Versal deformations of a singular point of a vector field on the plane in the case of zero eigen-values, Functional Analysis and Its Applications, 9 (1975), 144-145.  doi: 10.1007/BF01075453.  Google Scholar

[5]

V. CapassoE. Crosso and G. Serio, Mathematical models in epidemiological studies. I. Application to the epidemic of cholera verified in Bari in 1973, Annali Sclavo, 19 (1977), 193-208.   Google Scholar

[6]

V. Capasso and G. Serio, A generalization of the Kermack-Mckendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.  Google Scholar

[7]

J. C. Eckalbar and W. L. Eckalbar, Dynamics of an epidemic model with quadratic treatment, Nonlinear Anal. RWA, 12 (2011), 320-332.  doi: 10.1016/j.nonrwa.2010.06.018.  Google Scholar

[8]

P. Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, John A. Jacquez Memorial Volume. Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[9]

H. W. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271-287.  doi: 10.1007/BF00160539.  Google Scholar

[10]

Y. A. Kuzenetsov, Elements of Applied Bifurcation Theory, Springer: New York, 1995. doi: 10.1007/978-1-4757-2421-9.  Google Scholar

[11]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roal Soc. Lond., 115 (1927), 700-721.   Google Scholar

[12]

W. LiuH. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380.  doi: 10.1007/BF00277162.  Google Scholar

[13]

W. LiuS. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204.  doi: 10.1007/BF00276956.  Google Scholar

[14]

M. Lizana and J. Rivero, Multiparametric bifurcations for a model in epidemiology, J. Math. Biol., 35 (1996), 21-36.  doi: 10.1007/s002850050040.  Google Scholar

[15]

M. LuJ. HuangS. Ruan and P. Yu, Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate, J. Differential Equations, 267 (2019), 1859-1898.  doi: 10.1016/j.jde.2019.03.005.  Google Scholar

[16]

M. Lu, J. Huang, S. Ruan and P. Yu, Global dynamics of a susceptible-infectious-recovered epidemic model with a generalized nonmonotone incidence rate, J. Dyn. Differ. Equ., (2020). https://doi.org/10.1007/s10884-020-09862-3 doi: 10.1007/s10884-020-09862-3.  Google Scholar

[17]

X. Liu and L. Yang, Stability analysis of an SEIQV epidemic model with saturated incidence rate, Nonlinear Anal. RWA, 13 (2012), 2671-2679.  doi: 10.1016/j.nonrwa.2012.03.010.  Google Scholar

[18]

Y. TangD. HuangS. Ruan and W. Zhang, Coexistence of limit cycles and homoclinic loops in a SIRS model with a nonlinear incidence rate, SIAM J. Appl. Math., 69 (2008), 621-639.  doi: 10.1137/070700966.  Google Scholar

[19]

F. Takens, Forced oscillations and bifurcation, Global Analysis of Dynamical Systems, 3 (2001), 1-61.   Google Scholar

[20]

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

[21]

W. Wang and S. Ruan, Bifurcation in an epidemic model with constant removal rate of the infectives, J. Math. Anal. Appl., 291 (2004), 775-793.  doi: 10.1016/j.jmaa.2003.11.043.  Google Scholar

[22]

D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429.  doi: 10.1016/j.mbs.2006.09.025.  Google Scholar

[23]

D. Xiao and Y. Zhou, Qualitative analysis of an epidemic model, Can. Appl. Math. Q., 14 (2006), 469-492.   Google Scholar

[24]

Q. YangD. JiangN. Shi and C. Ji, The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence, J. Math. Anal. Appl., 388 (2012), 248-271.  doi: 10.1016/j.jmaa.2011.11.072.  Google Scholar

[25]

Y. Yao, Bifurcations of a Leslie-Gower prey-predator system with ratio-dependent Holling IV functional response and prey harvesting, Math. Meth. Appl. Sci., 43 (2020), 2137-2170.  doi: 10.1002/mma.5944.  Google Scholar

[26]

Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Translated from the Chinese by Anthony Wing Kwok Leung. Translations of Mathematical Monographs, 101. American Mathematical Society, Providence, RI, 1992.  Google Scholar

[27]

X. Zhang and X. Liu, Backward bifurcation of an epidemic model with saturated treatment function, J. Math. Anal. Appl., 348 (2008), 433-443.  doi: 10.1016/j.jmaa.2008.07.042.  Google Scholar

[28]

T. ZhouW. Zhang and Q. Lu, Bifurcation analysis of an SIS epidemic model with saturated incidence rate and saturated treatment function, Applied Mathematics and Computation, 226 (2014), 288-305.  doi: 10.1016/j.amc.2013.10.020.  Google Scholar

Figure 1.  The positive real roots of $ f(x) = 0 $ when $ m+n-A<0 $: (a) no positive root. (b) a positive double root $ x_* $ (i.e., $ \overline{x}_2 $). (c) two different positive single roots $ x_1 $, $ x_2 $
Figure 2.  A unique positive equilibrium $ E_{*} $ of system (9): (a) a saddle-node with a stable parabolic sector for $ A = \frac{219}{32},\, n = \frac{9}{10},\, m = \frac{20}{8},\,p = \frac{1}{4},\,q = \frac{7}{4} $; (b) a saddle-node with an unstable parabolic sector for $ A = \frac{737}{96},\, n = \frac{16}{15},\, m = \frac{20}{8},\,p = \frac{1}{4},\,q = \frac{19}{8} $; (c) a cusp of codimension two for $ A = \frac{235}{32},\, n = 1,\, m = \frac{20}{8},\,p = \frac{1}{4},\,q = \frac{17}{8} $
Figure 3.  Bogdanov-Takens bifurcation diagram and corresponding phase portraits of system (21) for $ n = 1,\, m = \frac{20}{8},\,p = \frac{1}{4} $. (a) Bifurcation diagram; (b) No positive equilibria when $ (q,A) = (2.3,7.429) $ lies in the region $I $; (c) An unstable focus $ E_2 $ and a saddle $ E_1 $ occur when $ (q,A) = (2.3,7.431) $ lies in the region $ II $; (d) An unstable limit cycle occurs when $ (q,A) = (2.3,7.4315) $ lies in the region $ III $; (e) An unstable homoclinic loop occurs when $ (q,A) = (2.3,7.4320743) $ lies on the curve $ \mathcal{H L} $; (f) A stable focus $ E_2 $ when $ (q,A) = (2.3,7.433) $ lies in the region $ IV $
Figure 4.  An unstable limit cycle bifurcates from $ E_2(x_2,y_2) $ for system (9)
Figure 5.  The phase portraits of system (5) when $ A\leq x_0 $. (a) The disease-free equilibrium $ E_0 $ is globally asymptotically stable when $ R_0\leq1 $; (b) The endemic equilibrium $ E^*(x^*,y^*) $ is globally asymptotically stable when $ R_0>1 $
Figure 6.  The phase portraits of system (5) when $ R_0\geq R_0^* $. (a) Two positive equilibria $ E^*(x^*,y^*) $ and $ E_2 $ when $ R_0 = R_0^* $. (b) A unique positive equilibrium $ E^*(x^*,y^*) $ when $ R_0 = R_0^* $; (c) A unique positive equilibrium $ E_2 $ when $ R_0> R_0^* $
Figure 7.  Phase portraits for Bogdanov-Takens bifurcation in system (5) when $ 0<R_0<1 $, where $ n = 1,\, m = \frac{20}{8},\,p = \frac{1}{4},\,x_0 = 0.2 $. (a) No positive equilibria when $ (q,A) = (2.3,7.429) $; (b) Two positive equilibria occur in the region $ \{(x,y)|x>x_0 = 0.2,\, y>0\} $ when $ (q,A) = (2.3,7.431) $: $ E_1 $ is always a saddle, $ E_2 $ is an unstable focus; (c) An unstable limit cycle occurs when $ (q,A) = (2.3,7.4315) $; (d) An unstable homoclinic loop occurs when $ (q,A) = (2.3,7.4320743) $; (e) $ E_2 $ becomes as a stable focus when $ (q,A) = (2.3,7.433) $
Figure 7, respectively">Figure 8.  $ (a)-(d) $ are the local enlarged view of $ (b)-(e) $ in Figure 7, respectively
Figure 9.  Phase portraits for Bogdanov-Takens bifurcation in system (5) when $ 1<R_0<R_0^* $, where $ n = 1,\, m = \frac{20}{8},\,p = \frac{1}{4},\,x_0 = 0.4 $. (a) $ E^* $ always exists in the region $ \{(x,y)|x<x_0 = 0.4,\, y>0\} $, and no positive equilibria lie in the region $ \{(x,y)|x>x_0 = 0.4,\, y>0\} $ when $ (q,A) = (2.3,7.429) $; (b) Two positive equilibria occur in the region $ \{(x,y)|x>x_0 = 0.4,\, y>0\} $ when $ (q,A) = (2.3,7.431) $: $ E_1 $ is always a saddle, $ E_2 $ is an unstable focus; (c) An unstable limit cycle occurs when $ (q,A) = (2.3,7.4315) $; (d) An unstable homoclinic loop occurs when $ (q,A) = (2.3,7.4320743) $; (e) $ E_2 $ becomes as a stable focus when $ (q,A) = (2.3,7.433) $
Figure 9, respectively">Figure 10.  $ (a)-(d) $ are the local enlarged view of $ (b)-(e) $ in Figure 9, respectively
Figure 11.  An unstable limit cycle bifurcates from $ E_2(x_2,y_2) $ in system (5). (a) $ 0<R_0<1 $. (b) $ 1<R_0<R_0^* $
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