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 $
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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 |
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.
Keywords:
SIRS epidemic model,
nonmonotone incidence rate,
piecewise-smooth treatment,
global stability,
Bogdanov-Takens bifurcation,
Hopf bifurcation.
Mathematics Subject Classification:
Primary: 58F15, 58F17; Secondary: 53C35.
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
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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. |
| [2] | R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1992. 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.
|
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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. |
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V. Capasso, E. 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 |
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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. |
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J. C. Eckalbar and W. L. Eckalbar,
Dynamics of an epidemic model with quadratic treatment, Nonlinear Anal. RWA, 12 (2011), 320-332.
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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. |
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H. W. Hethcote and P. van den Driessche,
Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271-287.
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Y. A. Kuzenetsov, Elements of Applied Bifurcation Theory, Springer: New York, 1995.
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W. Liu, H. 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. |
| [13] |
W. Liu, S. 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. |
| [14] |
M. Lizana and J. Rivero,
Multiparametric bifurcations for a model in epidemiology, J. Math. Biol., 35 (1996), 21-36.
doi: 10.1007/s002850050040. |
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M. Lu, J. Huang, S. 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. |
| [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. |
| [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. |
| [18] |
Y. Tang, D. Huang, S. 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. |
| [19] |
F. Takens,
Forced oscillations and bifurcation, Global Analysis of Dynamical Systems, 3 (2001), 1-61.
|
| [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. |
| [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. |
| [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. |
| [23] |
D. Xiao and Y. Zhou,
Qualitative analysis of an epidemic model, Can. Appl. Math. Q., 14 (2006), 469-492.
|
| [24] |
Q. Yang, D. Jiang, N. 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. |
| [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. |
| [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. |
| [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. |
| [28] |
T. Zhou, W. 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. |
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. |
| [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.
|
| [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. |
| [5] |
V. Capasso, E. 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. |
| [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. |
| [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. |
| [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. |
| [10] |
Y. A. Kuzenetsov, Elements of Applied Bifurcation Theory, Springer: New York, 1995.
doi: 10.1007/978-1-4757-2421-9. |
| [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. Liu, H. 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. |
| [13] |
W. Liu, S. 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. |
| [14] |
M. Lizana and J. Rivero,
Multiparametric bifurcations for a model in epidemiology, J. Math. Biol., 35 (1996), 21-36.
doi: 10.1007/s002850050040. |
| [15] |
M. Lu, J. Huang, S. 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. |
| [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. |
| [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. |
| [18] |
Y. Tang, D. Huang, S. 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. |
| [19] |
F. Takens,
Forced oscillations and bifurcation, Global Analysis of Dynamical Systems, 3 (2001), 1-61.
|
| [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. |
| [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. |
| [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. |
| [23] |
D. Xiao and Y. Zhou,
Qualitative analysis of an epidemic model, Can. Appl. Math. Q., 14 (2006), 469-492.
|
| [24] |
Q. Yang, D. Jiang, N. 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. |
| [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. |
| [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. |
| [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. |
| [28] |
T. Zhou, W. 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. |
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 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 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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