# American Institute of Mathematical Sciences

December  2021, 26(12): 6229-6252. doi: 10.3934/dcdsb.2021016

## Complex dynamics of a SIRS epidemic model with the influence of hospital bed number

 1 Department of Mathematics, Hangzhou Normal University, Hangzhou, 310036, China 2 School of Mathematical Sciences, East China Normal University, Shanghai 200241, China

* Corresponding author: Yancong Xu, Email: Yancongx@hznu.edu.cn

Received  May 2020 Revised  September 2020 Published  December 2021 Early access  January 2021

Fund Project: The first author was supported by the National NSF of China (No. 11671114, 11871022) and NSF of Zhejiang (LY20A010002); the second author was supported by the National NSF of China (No. 11901144) and NSF of Zhejiang (Y201840020)

In this paper, the nonlinear dynamics of a SIRS epidemic model with vertical transmission rate of neonates, nonlinear incidence rate and nonlinear recovery rate are investigated. We focus on the influence of public available resources (especially the number of hospital beds) on disease control and transmission. The existence and stability of equilibria are analyzed with the basic reproduction number as the threshold value. The conditions for the existence of transcritical bifurcation, Hopf bifurcation, saddle-node bifurcation, backward bifurcation and the normal form of Bogdanov-Takens bifurcation are obtained. In particular, the coexistence of limit cycle and homoclinic cycle, and the coexistence of stable limit cycle and unstable limit cycle are also obtained. This study indicates that maintaining enough number of hospital beds is very crucial to the control of the infectious diseases no matter whether the immunity loss population are involved or not. Finally, numerical simulations are also given to illustrate the theoretical results.

Citation: Yancong Xu, Lijun Wei, Xiaoyu Jiang, Zirui Zhu. Complex dynamics of a SIRS epidemic model with the influence of hospital bed number. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6229-6252. doi: 10.3934/dcdsb.2021016
##### References:
 [1] Y. Bai and X. Mu, Global asymptotic stability of a generalized SIRS epidemic model with transfer from infectious to susceptible, J. of Applied Analysis and Computation, 8 (2018), 402-412.  doi: 10.11948/2018.402. [2] 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. [3] O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts, The construction of next-generation matrices for compartmental epidemic models, J. R. Soc., Interface, 7 (2010), 873-885.  doi: 10.1098/rsif.2009.0386. [4] E. J. Doedel, T. F. Fairgrieve, B. Sandstede, A. R. Champneys, Y. A. Kuznetsov and X. Wang, Auto07p, continuation and bifurcation software for ordinary differential equations, (2007). [5] J. Guckenheimer, P. Holmes and M. Slemrod, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, 1983. doi: 10.1007/978-1-4612-1140-2. [6] Z. X. Hu, W. B. Ma and S. Ruan, Analysis of SIR epidemic models with nonlinear incidence rate and treatment, Math. Biosci., 238 (2012), 12-20.  doi: 10.1016/j.mbs.2012.03.010. [7] Z. Hu, P. Bi, W. Ma and S. Ruan, Bifurcations of an SIRS epidemic model with nonlinear incidence rate, Discrete Contin. Dynam. Syst. Ser. B, 15 (2011), 93-112.  doi: 10.3934/dcdsb.2011.15.93. [8] W. M. Liu, H. W. Hetchote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380.  doi: 10.1007/BF00277162. [9] Y. G. Lin, D. Q. Jiang and M. L. Jin, Stationary distribution of a stochastic SIR model with saturated incidence and its asymptotic stability, Acta Mathematica Scientia, 35 (2015), 619-629.  doi: 10.1016/S0252-9602(15)30008-4. [10] W. M. Liu, H. W. Hetchote and S. A. Levin, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological model, J. Math. Biol., 23 (1986), 187-204.  doi: 10.1007/BF00276956. [11] Q. Liu, D. Jiang, T. Hayat and B. Ahmad, Analysis of a delayed vaccinated SIR epidemic model with temporary immunity and Levy jumps, Nonlinear Analysis: Hybrid Systems, 27 (2018), 29-43.  doi: 10.1016/j.nahs.2017.08.002. [12] G. H. Li and Y. X. Zhang, Dynamic behaviors of a modified SIR model in epidemic diseases using nonlinear incidence and recovery rates, Plos One, 2017 (2017), 1-28.  doi: 10.1371/journal.pone.0175789. [13] 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. Diff. Eqs., 267 (2019), 1859-1898.  doi: 10.1016/j.jde.2019.03.005. [14] 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. Diff. Eq., (2020). doi: 10.1007/s10884-020-09862-3. [15] K. M. O'Reilly, R. Lowe, W. J. Edmunds, et al., Projecting the end of the Zika virus epidemic in Latin America: A modelling analysis, BMC medicine, 180 (2018). doi: 10.1186/s12916-018-1158-8. [16] Y. Pan, D. Pi, S. Yao and H. Meng, The global stability of two epidemic models with nonlinear recovery incidence rate, Modern Physics Letters B, 32 (2018), 1850357, 9 pp. doi: 10.1142/S0217984918503578. [17] L. Perko, Differential Equations and Dynamical Systems, Springer, 2001. doi: 10.1007/978-1-4613-0003-8. [18] S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Diff. Eqs., 188 (2003), 135-163.  doi: 10.1016/S0022-0396(02)00089-X. [19] E. Rivero-Esquivel, E. Avila-Vales and G. García-Almeida, Stability and bifurcation analysis of a SIR model with saturated incidence rate and saturated treatment, Mathematics and Computers in Simulation, 121 (2016), 109-132.  doi: 10.1016/j.matcom.2015.09.005. [20] C. Shan and H. Zhu, Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds, J. Diff. Eqs., 257 (2014), 1662-1688.  doi: 10.1016/j.jde.2014.05.030. [21] B. Sandstede and Y. C. Xu, Snakes and isolas in non-reversible conservative systems, Dynamical Systems, 27 (2012), 317-329.  doi: 10.1080/14689367.2012.691961. [22] 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. [23] P. van den Dreessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [24] W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58-71.  doi: 10.1016/j.mbs.2005.12.022. [25] S. Wiggins, Introduction to Applied Nonlinear Dynamics Systems and Chaos, second edition, Texts in Applied Mathematics, vol. 2, Spring-Verlag, 2003. [26] Y. C. Xu, Y. Yang, F. W. Meng and P. Yu, Modelling and analysis of recurrent autoimmune disease, Nonlinear Analysis, Real World Applications, 54 (2020), 103109, 28 pp. doi: 10.1016/j.nonrwa.2020.103109. [27] X. Yuan, F. Wang, Y. Xue and L. Yakui, Global stability of an SIR model with differential infectivity on complex networks, Physica A: Statistical Mechanics and its Applications, 499 (2018), 443-456.  doi: 10.1016/j.physa.2018.02.065.

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##### References:
 [1] Y. Bai and X. Mu, Global asymptotic stability of a generalized SIRS epidemic model with transfer from infectious to susceptible, J. of Applied Analysis and Computation, 8 (2018), 402-412.  doi: 10.11948/2018.402. [2] 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. [3] O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts, The construction of next-generation matrices for compartmental epidemic models, J. R. Soc., Interface, 7 (2010), 873-885.  doi: 10.1098/rsif.2009.0386. [4] E. J. Doedel, T. F. Fairgrieve, B. Sandstede, A. R. Champneys, Y. A. Kuznetsov and X. Wang, Auto07p, continuation and bifurcation software for ordinary differential equations, (2007). [5] J. Guckenheimer, P. Holmes and M. Slemrod, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, 1983. doi: 10.1007/978-1-4612-1140-2. [6] Z. X. Hu, W. B. Ma and S. Ruan, Analysis of SIR epidemic models with nonlinear incidence rate and treatment, Math. Biosci., 238 (2012), 12-20.  doi: 10.1016/j.mbs.2012.03.010. [7] Z. Hu, P. Bi, W. Ma and S. Ruan, Bifurcations of an SIRS epidemic model with nonlinear incidence rate, Discrete Contin. Dynam. Syst. Ser. B, 15 (2011), 93-112.  doi: 10.3934/dcdsb.2011.15.93. [8] W. M. Liu, H. W. Hetchote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380.  doi: 10.1007/BF00277162. [9] Y. G. Lin, D. Q. Jiang and M. L. Jin, Stationary distribution of a stochastic SIR model with saturated incidence and its asymptotic stability, Acta Mathematica Scientia, 35 (2015), 619-629.  doi: 10.1016/S0252-9602(15)30008-4. [10] W. M. Liu, H. W. Hetchote and S. A. Levin, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological model, J. Math. Biol., 23 (1986), 187-204.  doi: 10.1007/BF00276956. [11] Q. Liu, D. Jiang, T. Hayat and B. Ahmad, Analysis of a delayed vaccinated SIR epidemic model with temporary immunity and Levy jumps, Nonlinear Analysis: Hybrid Systems, 27 (2018), 29-43.  doi: 10.1016/j.nahs.2017.08.002. [12] G. H. Li and Y. X. Zhang, Dynamic behaviors of a modified SIR model in epidemic diseases using nonlinear incidence and recovery rates, Plos One, 2017 (2017), 1-28.  doi: 10.1371/journal.pone.0175789. [13] 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. Diff. Eqs., 267 (2019), 1859-1898.  doi: 10.1016/j.jde.2019.03.005. [14] 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. Diff. Eq., (2020). doi: 10.1007/s10884-020-09862-3. [15] K. M. O'Reilly, R. Lowe, W. J. Edmunds, et al., Projecting the end of the Zika virus epidemic in Latin America: A modelling analysis, BMC medicine, 180 (2018). doi: 10.1186/s12916-018-1158-8. [16] Y. Pan, D. Pi, S. Yao and H. Meng, The global stability of two epidemic models with nonlinear recovery incidence rate, Modern Physics Letters B, 32 (2018), 1850357, 9 pp. doi: 10.1142/S0217984918503578. [17] L. Perko, Differential Equations and Dynamical Systems, Springer, 2001. doi: 10.1007/978-1-4613-0003-8. [18] S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Diff. Eqs., 188 (2003), 135-163.  doi: 10.1016/S0022-0396(02)00089-X. [19] E. Rivero-Esquivel, E. Avila-Vales and G. García-Almeida, Stability and bifurcation analysis of a SIR model with saturated incidence rate and saturated treatment, Mathematics and Computers in Simulation, 121 (2016), 109-132.  doi: 10.1016/j.matcom.2015.09.005. [20] C. Shan and H. Zhu, Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds, J. Diff. Eqs., 257 (2014), 1662-1688.  doi: 10.1016/j.jde.2014.05.030. [21] B. Sandstede and Y. C. Xu, Snakes and isolas in non-reversible conservative systems, Dynamical Systems, 27 (2012), 317-329.  doi: 10.1080/14689367.2012.691961. [22] 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. [23] P. van den Dreessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [24] W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58-71.  doi: 10.1016/j.mbs.2005.12.022. [25] S. Wiggins, Introduction to Applied Nonlinear Dynamics Systems and Chaos, second edition, Texts in Applied Mathematics, vol. 2, Spring-Verlag, 2003. [26] Y. C. Xu, Y. Yang, F. W. Meng and P. Yu, Modelling and analysis of recurrent autoimmune disease, Nonlinear Analysis, Real World Applications, 54 (2020), 103109, 28 pp. doi: 10.1016/j.nonrwa.2020.103109. [27] X. Yuan, F. Wang, Y. Xue and L. Yakui, Global stability of an SIR model with differential infectivity on complex networks, Physica A: Statistical Mechanics and its Applications, 499 (2018), 443-456.  doi: 10.1016/j.physa.2018.02.065.
Bifurcation diagram of system (4) with respect to parameters $\mu_{1}$ and $d$ if $k(bm+\beta)>(b+\beta)(p\delta+\mu_{0})$, there exist one and two endemic equilibria in $D_{1}$ and $D_{0},$ respectively. Two endemic equilibria coalesce and a saddle-node bifurcation occurs on $C_{\Delta}^{-}$; the forward bifurcation occurs on $C_{\Delta}^{+}$ and the backward bifurcation occurs on $C_{\Delta}^{-}$, respectively. There is no endemic equilibrium in other regions
$(a)$ Phase diagram of system (4) with no endemic equilibrium for $R_{0}<1, k(bm+\beta)<(b+\beta)(p\delta+\mu_{0})$ $(b)$ Phase diagram of system (4) with one endemic equilibrium $E_{*}$ for $R_{0} = 1, d = d_{3}, s_{1} = 0.$
Bifurcation diagram of system (4) as $\mu_{1} = 0.184$, $R_{0} = 1, d_{2} = 0.0797434.$ (a) system (4) undergoes the forward bifurcation for $d_{2}<d = 0.1$. (b) system (3) undergoes the backward bifurcation for $d_{2}>d = 0.05$, where $\rm{BP}$ and $\rm{HB}$ denote the transcritical bifurcation point and the subcritical Hopf bifurcation point
(a) Bifurcation diagram of system (4) as $\mu_{1} = 0.184$, $R_{0} = 1, d_{2} = 0.0797434, d = d_{2},$ system (4) undergoes the pitchfork bifurcation, where $\rm{PB}$ denotes the pitchfork bifurcation point, $HB_1$ and $HB_2$ are supercritical Hopf bifurcation points. (b) One-parameter bifurcation diagram of system (4) with $I$ and $d$, where $HB_1$, $HB_2$ and $SN$ denote, respectively, the subcritical Hopf bifurcation point, the supercritical bifurcation point and the saddle-node bifurcation point of limit cycles
(a) One-parameter bifurcation diagram of system (4) with $I$ and $\beta$ where $HB_1$ and $HB_2$ denote the supercritical bifurcation points. (b) Two-parameter Hopf bifurcation diagram with $\beta$ and $d$, where $H$ and $GH$ denote, respectively, the Hopf bifurcation curve and the generalized Hopf bifurcation point
$(a)$ One-parameter bifurcation diagram of system (4) with the parameter $d$ as a free parameter, where $HB_1$ and $HB_2$ denote two supercritical Hopf bifurcation points; $(b)$ Two-parameter bifurcation diagram of system (4) with respect to parameters $d$ and $\mu_{1}$, where $Hom, H, SN$ and $SN_{lc}$ denote the homoclinic orbit bifurcation curve (black), Hopf bifurcation curve (green) and saddle-node bifurcation curve (red), the saddle-node bifurcation curve of limit cycles (the dotted blue line from $GH$ to $C_2$), respectively. GH denotes the generalized Hopf bifurcation point
$(a)$ Zoomed two-parameter bifurcation diagram of system (4) in Figure 6 (b); $(b)$ Phase portraits in different regions of parameters in Figure 7 (a)
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