Article Contents
Article Contents

# The threshold value of the number of hospital beds in a SEIHR epidemic model

• To investigate the impact of the number of hospital beds on the control of infectious diseases and help allocate the limited medical resources in a region, a SEIHR epidemic model including exposed and hospitalized classes is established. Different from available models, the hospitalization rate is expressed as a function of the number of empty beds. The existence and stability of the equilibria are analyzed, and it is proved that the system undergoes backward bifurcation, Hopf bifurcation, and Bogdanov-Takens bifurcation of codimension $2$ under certain conditions by using the center manifold theory and normal form theory. In particular, our results show that there is a threshold value for the capacity of hospital beds in a region. If the capacity of hospital beds is lower than this threshold value, there will be a backward bifurcation, which means that even if the basic reproduction number, $\mathbb{R}_0$, is less than unity, it is not enough to prevent the outbreaks. Before taking disease control measures, one should compare the number of beds with the threshold value to avoid misjudgment and try to increase the capacity of hospital beds above this threshold value. The method to estimate the threshold value is also given. In addition, the impacts of the duration of the exposed period on the basic reproduction number and disease transmission are investigated.

Mathematics Subject Classification: Primary: 34C23, 34C60; Secondary: 92D30.

 Citation:

• Figure 1.  The distribution of equilibria on the $(K,A)$ plane. There is a unique endemic equilibrium $E_1^*$ in region $\Omega_1$. There are two equilibria $E_1^*$ and $E_2^*$ in region $\Omega_2$ where $\mathbb{\hat{R}}_0<\mathbb{R}_0<1$, and the two equilibria coalesce when the parameters approach the green curve $A = A_4(K)$ where $\mathbb{R}_0 = \mathbb{\hat{R}}_0$. In other regions of the first quadrant, there is no endemic equilibrium

Figure 2.  The phase portraits and the corresponding bifurcation diagrams near $E_0$. Figure (a) and (b) correspond to $0<K<K^*$, (c) and (d) correspond to $K\geq K^*$

Figure 3.  The backward bifurcation diagram of system (3) when $K = 100$. When the value of parameter $A$ passes through Hopf point $H$, the stability of the endemic equilibria will change. Parameter values are listed in Table 1

Figure 4.  Bifurcation curves in $(K,A)$ plane. The blue solid curve represents saddle-node bifurcation $(SN)$, the red dotted curve represents Hopf bifurcation $(Hopf)$, the black dashed curve represents Homoclinic bifurcation $(Hom)$ and the purple solid line, $L$, represents $\mathbb{R}_0 = 1$. These three bifurcation curves and straight-line $L$ divide the first quadrant of the $(K,A)$ plane into seven regions, where $D_3$ represents the region between $Hopf$ and $Hom$ when $\mathbb{R}_0<1$, $D_6$ represents the region where the limit cycle branched by $Hopf$ exists when $\mathbb{R}_0>1$, and others are shown in the figure. The existence and stability of equilibria in each region are different. Each region gives a schematic diagram of the phase diagram near endemic equilibria except $D_1$. Parameter values are listed in Table 1, $K^* = 126.6059$

Figure 5.  The local phase portraits near the positive equilibrium $E_1$ in regions $D_2$, $D_3$ and $D_4$ of Fig. 4. Parameter values are listed in Table 1

Figure 6.  The local phase portrait near the positive equilibrium $E_1$ in region $D_5$ of Fig. 4. $E_1$ is stable. $K = 80$, $A = 16$. Parameter values are listed in Table 1

Figure 7.  The phase portrait near the positive equilibrium $E_1$ when the value of parameter (K, A) is in region $D_6$ of Fig. 4. $E_1$ is stable. Parameter values are listed in Table 1

Figure 8.  The phase portrait in the $S-I-H$ space when the value of parameter $(K,A)$ is in region $D_7$ of Fig. 4. The unique endemic equilibrium is unstable and a stable periodic solution exists. Parameter values are listed in Table 1. $K = 140$, $A = 16.5$

Figure 9.  The bifurcation diagram in plane $(\mathbb{R}_0,I)$ when $K = 60$, $A = 8.4$, $\mathbb{R}_0 = 0.6593$ in region $D_3$. $H$ is the Hopf bifurcation point and $Hom$ is the Homoclinic orbit. The system has Hopf bifurcation at $H$ and branches out an unstable limit cycle. Parameter values are listed in Table 1

Figure 10.  The bifurcation diagram in $(A,I)$ plane with different $K$, where $K^* = 126.6059$. When $0<K<K^*$, the system will undergo a backward bifurcation. When $K\geq K^*$, the system will undergo a forward bifurcation. Parameter values are listed in Table 1

Figure 11.  The contour plot of $\mathbb{R}_0$, where $\mathbb{R}_0$ is 0.1, 0.5, 1, 1.5. The shorter the duration of exposed period ($1/\delta$) is, the greater the number of basic reproduction number ($\mathbb{R}_0$) is. This effect is more obvious with the increase of transmission rate $\beta$. Parameter values are listed in Table 1

Table 1.  Parameters description

 Parameter Value Reference $d$ $3.5\times10^{-5}$ [21] $\beta$ $0.001$ [21] $\delta$ $0.19$ [21] $\mu$ $5.5889\times10^{-4}$ [21] $\gamma$ $0.03$ [21] $\gamma_h$ $0.1$ [21] $c$ $363.91$ Assume
•  [1] C. Castillochavez, Mathematical Models in Population Biology and Epidemiology, Springer, 2001. [2] T. Chen, J. Rui, Q. Wang, Z. Zhao, J. Cui and L. Yin, A mathematical model for simulating the phase-based transmissibility of a novel coronavirus, Infectious Diseases of Poverty, 9 24, (2020). [3] L. Ferrari, G. Gerardi, G. Manzi, A. Micheletti, F. Nicolussi, E. Biganzoli and S. Salini, Modeling provincial COVID-19 epidemic data using an adjusted time-dependent SIRD model, Int. J. Environ. Res. Public Health, 18 (2021), 6563.  doi: 10.3390/ijerph18126563. [4] W. H. Hamer,  Epidemic Disease in England: The Evidence of Variability and of Persistency of Type, Bedford Press, 1906. [5] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics, Proceedings of the Royal Society of London, 115 (1927), 700-721. [6] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics, Proceedings of the Royal Society of London, 138 (1932), 55-83. [7] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Scienses, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4757-2421-9. [8] G. Lan, S. Yuan and B. Song, The impact of hospital resources and environmental perturbations to the dynamics of SIRS model, Journal of the Franklin Institute, 358 (2021), 2405-2433.  doi: 10.1016/j.jfranklin.2021.01.015. [9] M. Lavielle, M. Faron, J. H. Lefevre and J.-D. Zeitoun, Predicting the propagation of COVID-19 at an international scale: Extension of an SIR model, BMJ Open, 11 (2021), e041472. doi: 10.1136/bmjopen-2020-041472. [10] J. Li, P. Yuan, J. Heffernan, T. Zheng, N. Ogden, B. Sander, J. Li, Q. Li, J. Bélair, J. D. Kong, et al., Observation wards and control of the transmission of COVID-19 in Wuhan, Bull World Health Organ, 98 (2020). [11] M. Y. Li and J. S. Muldowney, Global stability for the SEIR model in epidemiology, Mathematical Biosciences, 125 (1995), 155-164.  doi: 10.1016/0025-5564(95)92756-5. [12] L. Perko, Differential Equations and Dynamical Systems, Springer, 1991. doi: 10.1007/978-1-4684-0392-3. [13] R. Ross, The Prevention of Malaria, John Murray, 1911. [14] D. Saikia, K. Bora and M. P. Bora, Covid-19 outbreak in india: An SEIR model-based analysis, Nonlinear Dynamics, 104 (2021), 4727-4751.  doi: 10.1007/s11071-021-06536-7. [15] C. Shan, Y. Yi and H. Zhu, Nilpotent singularities and dynamics in an SIR type of compartmental model with hospital resources, J. Differential Equations, 260 (2016), 4339-4365.  doi: 10.1016/j.jde.2015.11.009. [16] C. Shan and H. Zhu, Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds, J. Differential Equations, 257 (2014), 1662-1688.  doi: 10.1016/j.jde.2014.05.030. [17] B. Tang, N. L. Bragazzi, Q. Li, S. Tang, Y. Xiao and J. Wu, An updated estimation of the risk of transmission of the novel coronavirus (2019-nCov), Infectious Disease Modelling, 5 (2020), 248-255.  doi: 10.1016/j.idm.2020.02.001. [18] B. Tang, X. Wang, Q. Li, N. L. Bragazzi, S. Tang, Y. Xiao and J. Wu, Estimation of the transmission risk of the 2019-nCov and its implication for public health interventions, Journal of Clinical Medicine, 9 (2020), 462. [19] P. Van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [20] H. Wan and J. Cui, Rich dynamics of an epidemic model with saturation recovery, J. Appl. Math., 2013 (2013), Art. ID 314958, 9 pp. doi: 10.1155/2013/314958. [21] H. Wan, J. Cui and G. Yang, Risk estimation and prediction of the transmission of coronavirus disease-2019 (COVID-19) in the mainland of China excluding Hubei province, Infectious Diseases of Poverty, 9 (2020), 1-9. [22] W. Wang, Backward bifurcation of an epidemic model with treatment, Mathematical Biosciences, 201 (2006), 58-71.  doi: 10.1016/j.mbs.2005.12.022. [23] W. Wang and S. Ruan, Bifurcations 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. [24] X. Wang, Q. Li, X. Sun, S. He, F. Xia, P. Song, Y. Shao, J. Wu, R. A. Cheke, S. Tang and Y. Xiao, Effects of medical resource capacities and intensities of public mitigation measures on outcomes of COVID-19 outbreaks, BMC Public Health, 21 (2021), Article number: 605, 11 pp. doi: 10.1186/s12889-021-10657-4. [25] M. Yin, Q. Zhu and X. Lü, Parameter estimation of the incubation period of COVID-19 based on the doubly interval-censored data model, Nonlinear Dynamics, 106 (2021), 1347-1358.  doi: 10.1007/s11071-021-06587-w. [26] J. Zhang, J. Lou, Z. Ma and J. Wu, A compartmental model for the analysis of SARS transmission patterns and outbreak control measures in China, Applied Mathematics and Computation, 162 (2005), 909-924.  doi: 10.1016/j.amc.2003.12.131. [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] H. Zhao, L. Wang, S. M. Oliva and H. Zhu, Modeling and dynamics analysis of Zika transmission with limited medical resources, Bull. Math. Biol., 82 (2020), Paper No. 99, 50 pp. doi: 10.1007/s11538-020-00776-1.

Figures(11)

Tables(1)