Bifurcation analysis and optimal control of an epidemic model with limited number of hospital beds. (English) Zbl 1519.92287
Summary: This paper deals with a three-dimensional nonlinear mathematical model to analyze an epidemic’s future course when the public healthcare facilities, specifically the number of hospital beds, are limited. The feasibility and stability of the obtained equilibria are analyzed, and the basic reproduction number \((R_0)\) is obtained. We show that the system exhibits transcritical bifurcation. To show the existence of Bogdanov-Takens bifurcation, we have derived the normal form. We have also discussed a generalized Hopf (or Bautin) bifurcation at which the first Lyapunov coefficient evanescences. To show the existence of saddle-node bifurcation, we used Sotomayor’s theorem. Furthermore, we have identified an optimal layout of hospital beds in order to control the disease with minimum possible expenditure. An optimal control setting is studied analytically using optimal control theory, and numerical simulations of the optimal regimen are presented as well.
MSC:
| 92D30 | Epidemiology |
| 34A34 | Nonlinear ordinary differential equations and systems |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 49J15 | Existence theories for optimal control problems involving ordinary differential equations |
Keywords:
hospital beds; Hopf bifurcation; saddle-node bifurcation; transcritical bifurcation; Bogdanov-Takens bifurcation; optimal controlSoftware:
SlideContReferences:
| [1] | Abdelrazec, A., Bélair, J., Shan, C. and Zhu, H., Modeling the spread and control of dengue with limited public health resources,Math. Biosci.271 (2016) 136-145. · Zbl 1364.92036 |
| [2] | Boaden, R., Proudlove, N. and Wilson, M., An exploratory study of bed management,J. Manag. Med.13(4) (1999) 234-250. |
| [3] | Bogdanov, R. I., Bifurcation of the limitcycle of a family of plane vector fields/versal deformations of a singularity of a vector field on the plane in the case of zero eigenvalues,Sel. Math. Sov.1 (1984) 373-387. · Zbl 0518.58029 |
| [4] | Bogdanov, R. I., Versal deformations of a singular point of a vector field on the plane in the case of zero eigenvalues,Funktsional. Anal. Prilozhen.9 (1975) 63-63. |
| [5] | Castillo-Chavez, C. and Song, B., Dynamical models of tuberculosis and their applications,Math. Biosci. Eng.1(2) (2004) 361. · Zbl 1060.92041 |
| [6] | Dercole, F. and Kuznetsov, Y. A., SlideCont: An Auto97 driver for bifurcation analysis of Filippov systems,ACM Trans. Math. Software31 (2005) 95-119. · Zbl 1073.65070 |
| [7] | Djiomba Njankou, S. D. and Nyabadza, F., Modelling the potential impact of limited hospital beds on Ebola virus disease dynamics,Math. Methods Appl. Sci.41 (2018) 8528-8544. · Zbl 1406.92361 |
| [8] | Dubey, B., Patra, A., Srivastava, P. K. and Dubey, U. S., Modeling and analysis of an SEIR model with different types of nonlinear treatment rates,J. Biol. Syst.21 (2013) 1350023. · Zbl 1342.92235 |
| [9] | Echevarría-Zuno, S., Mejía-Aranguré, J. M., Mar-Obeso, A. J., Grajales-Muñiz, C., Robles-Pérez, E., González-León, M., Ortega-Alvarez, M. C., Gonzalez-Bonilla, C., Rascón-Pacheco, R. A., Borja-Aburto, V. H., Infection and death from influenza A H1N1 virus in Mexico: A retrospective analysis, Lancet374 (2009) 2072-2079. |
| [10] | Imran, M., Khan, A., Ansari, A. R. and Shah, S. T. H., Modeling transmission dynamics of Ebola virus disease,Int. J. Biomath.10(04) (2017) 1750057. · Zbl 1373.92123 |
| [11] | Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Vol. 42 (Springer, 2013). · Zbl 0515.34001 |
| [12] | Kar, T. K. and Jana, S., A theoretical study on mathematical modelling of an infectious disease with application of optimal control,Biosystems111(1) (2013) 37-50. |
| [13] | Lata, K., Mishra, S. N. and Misra, A. K., An optimal control problem for carrier dependent diseases,Biosystems187 (2020) 104039. |
| [14] | Lenhart, S. and Workman, J. T., Optimal Control Applied to Biological Models (Chapman and Hall/CRC, 2007). · Zbl 1291.92010 |
| [15] | Longini, I. M., Nizam, A., Xu, S., Ungchusak, K., Hanshaoworakul, W., Cummings, D. A. and Halloran, M. E., Containing pandemic influenza at the source,Science309 (2005) 1083-1087. |
| [16] | Lukes, D. L., Differential Equations: Classical to Controlled (Academic Press, Edinburgh, 1982). · Zbl 0509.34003 |
| [17] | Misra, A. K. and Maurya, J., Modeling the importance of temporary hospital beds on the dynamics of emerged infectious disease,Chaos31(10) (2021) 103125. · Zbl 07867384 |
| [18] | Oliveira, J. F., Jorge, D. C. P., Veiga, R. V., Rodrigues, M. S., Torquato, M. F., da Silva, N. B., Fiaccone, R. L., Cardim, L. L., Pereira, F. A. C., de Castro, C. P., Paiva, A. S. S., Amad, A. A. S., Lima, E. A. B. F., Souza, D. S., Pinho, S. T. R., Ramos, P. I. P. and Andrade, R. F. S., Mathematical modeling of COVID-19 in 14.8 million individuals in Bahia, Brazil, Nat. Commun.12(1) (2021) 1-13. |
| [19] | Pontijagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V. and Mishchenko, E. F., The Mathematical Theory of Optimal Process (John Wiley & Sons, 1962). · Zbl 0102.32001 |
| [20] | Qin, W., Tang, S., Xiang, C. and Yang, Y., Effects of limited medical resource on a Filippov infectious disease model induced by selection pressure,Appl. Math. Comput.283 (2016) 339-354. · Zbl 1410.92061 |
| [21] | Sadique, Z., Lopman, B., Cooper, B. S. and Edmunds, W. J., Cost-effectiveness of ward closure to control outbreaks of norovirus infection in United Kingdom National Health Service Hospitals, J. Infect. Dis.213 (2016) S19-S26. |
| [22] | Shan, C. and Zhu, H., Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds,J. Differential Equations257 (2014) 1662-1688. · Zbl 1300.34113 |
| [23] | Shan, C., Yi, Y. and Zhu, H., Nilpotent singularities and dynamics in an SIR type of compartmental model with hospital resources,J. Differential Equations260 (2016) 4339-4365. · Zbl 1339.34057 |
| [24] | Tiwari, P. K., Rai, R. K., Misra, A. K. and Chattopadhyay, J., Dynamics of infectious diseases: Local versus global awareness,Internat. J. Bifur. Chaos Appl. Sci. Engrg.31 (2021) 2150102. · Zbl 1471.34102 |
| [25] | Van den Driessche, P. and Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,Math. Biosci.180 (2002) 29-48. · Zbl 1015.92036 |
| [26] | Wang, A., Xiao, Y. and Zhu, H., Dynamics of a Filippov epidemic model with limited hospital beds,Math. Biosci. Eng.15 (2018) 739. · Zbl 1406.92625 |
| [27] | Wang, W., Backward bifurcation of an epidemic model with treatment,Math. Biosci.201(1-2) (2006) 58-71. · Zbl 1093.92054 |
| [28] | Weissman, G. E., Crane-Droesch, A., Chivers, C., Luong, T., Hanish, A., Levy, M. Z., Lubken, J., Becker, M., Draugelis, M. E., Anesi, G. L., Brennan, P. J., Christie, J. D., William Hanson III, C., Mikkelsen, M. E. and Halpern, S. D., Locally informed simulation to predict hospital capacity needs during the COVID-19 pandemic,Ann. Intern. Med.173(1) (2020) 21-28. |
| [29] | Zhang, M., Ge, J. and Lin, Z., The impact of the number of hospital beds and spatial heterogeneity on an SIS epidemic model,Acta Appl. Math.167(1) (2020) 59-73. · Zbl 1440.35335 |
| [30] | Zhang, X. and Liu, X., Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment,Nonlinear Anal. Real World Appl.10(2) (2009) 565-575. · Zbl 1167.34338 |
| [31] | Zhao, H., Wang, L., Oliva, S. M. and Zhu, H., Modeling and dynamics analysis of Zika transmission with limited medical resources, Bull. Math. Biol.82(8) (2020) 1-50. · Zbl 1508.92154 |
| [32] | World Health Organization, World Health Statistics 2022, https://www.who.int/data/gho/publications/world-health-statistics. |
| [33] | World Health Organization, WHO Coronavirus (COVID-19) Dashboard, https://covid19.who.int/. |
| [34] | Hospital Bed Population Ratio (HBPR), OECD Data, https://data.oecd.org/healtheqt/hospital-beds.htm. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
