# American Institute of Mathematical Sciences

July  2022, 27(7): 3533-3561. doi: 10.3934/dcdsb.2021195

## 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 Published  July 2022 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 and Continuous Dynamical Systems - B, 2022, 27 (7) : 3533-3561. doi: 10.3934/dcdsb.2021195
##### References:
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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. [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.

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##### 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. [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. [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. [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.
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 $x_1$, $x_2$
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 $A = \frac{235}{32},\, n = 1,\, m = \frac{20}{8},\,p = \frac{1}{4},\,q = \frac{17}{8}$
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 $IV$
An unstable limit cycle bifurcates from $E_2(x_2,y_2)$ for system (9)
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 $R_0>1$
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 $R_0> R_0^*$
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 $(q,A) = (2.3,7.433)$
$(a)-(d)$ are the local enlarged view of $(b)-(e)$ in Figure 7, respectively
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 $(q,A) = (2.3,7.433)$
$(a)-(d)$ are the local enlarged view of $(b)-(e)$ in Figure 9, respectively
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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