Qualitative analysis and optimal control of an SIR model with logistic growth, non-monotonic incidence and saturated treatment

Jayanta Kumar Ghosh¹ ; Prahlad Majumdar² ; Uttam Ghosh²

¹ Boalia Junior High School, Nadia, West Bengal, India.
² Department of Applied Mathematics, University of Calcutta, Kolkata, India.

Mathematical modelling of natural phenomena, Tome 16 (2021), article no. 13.

Résumé

This paper describes an SIR model with logistic growth rate of susceptible population, non-monotonic incidence rate and saturated treatment rate. The existence and stability analysis of equilibria have been investigated. It has been shown that the disease free equilibrium point (DFE) is globally asymptotically stable if the basic reproduction number is less than unity and the transmission rate of infection less than some threshold. The system exhibits the transcritical bifurcation at DFE with respect to the cure rate. We have also found the condition for occurring the backward bifurcation, which implies the value of basic reproduction number less than unity is not enough to eradicate the disease. Stability or instability of different endemic equilibria has been shown analytically. The system also experiences the saddle-node and Hopf bifurcation. The existence of Bogdanov-Takens bifurcation (BT) of co-dimension 2 has been investigated which has also been shown through numerical simulations. Here we have used two control functions, one is vaccination control and other is treatment control. We have solved the optimal control problem both analytically and numerically. Finally, the efficiency analysis has been used to determine the best control strategy among vaccination and treatment.

Détail

DOI : 10.1051/mmnp/2021004

Bibliographie

[1] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology. Springer, New York (2011).

[2] V. Capasso, G. Serio A generalization of the Kermack-Mckendric deterministic epidemic model Math. Biosci 1978 43 61

[3] S.A. Carvalho, S.O. da Silva and I.C. da Cunha, Mathematical modeling of dengue epidemic: control methods and vaccination strategies. Preprint arXiv:1508.00961 (2015).

[4] O. Diekman and J.A.P. Heesterbeek: Mathematical Epidemiology of Infectious Disease. Wiley, New York (2000).

[5] O. Diekmann, J.A.P. Heesterbeek, M.G. Roberts The construction of next-generation matrices for compartmental epidemic models J. R. Soc. Interface 2010 873 885

[6] J.K. Ghosh, U. Ghosh, M.H.A. Biswas, S. Sarkar Qualitative analysis and optimal control strategy of an SIR model with saturated incidence and treatment To appear in: Differ. Equ. Dyn. Syst. 2019

[7] S. Jana, S.K. Nandi, T.K. Kar Complex dynamics of an SIR epidemic model with saturated incidence rate and treatment Acta Biotheor 2016 65 84

[8] Y.A. Kuznetsov: Elements of Applied Bifurcation Theory. Springer, New York (1998).

[9] A.A. Lashari Optimal control of an SIR epidemic model with a saturated treatment Appl. Math. Inf. Sci 2016 185 191

[10] S. Lenhart and J.T. Workman, Optimal control applied to biological model. Mathematical and computational biology series. Chapman and Hall/CRC, Boca Raton (2007).

[11] J. Li, Z. Teng, G. Wang, L. Zhang, C. Hu Stability and bifurcation analysis of an SIR epidemic model with logistic growth and saturated treatment Chaos. Solit. Fract 2017 63 71

[12] M. Lu, J. Huang, S. Ruan Global dynamics of a susceptible-infectious-recovered epidemic model with a generalized nonmonotone incidence rate To appear in: J. Dyn. Differ. Equ. 2020

[13] M. Lu, J. Huang, S. Ruan, P. Yu Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate J. Differ. Equ 2019 1859 1898

[14] M. Martcheva, An Introduction to Mathematical Epidemiology. Springer, New York (2015).

[15] J.D. Murray, Mathematical Biology. Springer, New York (1993).

[16] S.K. Nandi, S. Jana, M. Mandal, T.K. Kar Complex dynamics and optimal treatment of an epidemic model with two infectious diseases Int. J. Appl. Comput. Math 2019 29

[17] L. Perko, Differential Equations and Dynamical Systems, in Vol. 7. Springer, New York (2000).

[18] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, The mathematical theory of optimal processes. Wiley, New Jersey (1962).

[19] O. Sharomi, T. Malik Optimal control in epidemiology Ann. Oper. Res 2017 55 71

[20] R.K. Upadhyay, A.K. Pal, S. Kumari, P. Roy Dynamics of an SEIR epidemic model with nonlinear incidence and treatment rates Nonlinear Dyn 2019 2351 2368

[21] P. Van Den Driessche, J. Watmough Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease trans-mission Math Biosci 2002 29 48

[22] W. Wang Backward bifurcation of an epidemic model with treatment Math. Biosci 2006 58 71

[23] W. Wang, S. Ruan Bifurcation in an epidemic model with constant removal rate of the infectives J. Math. Anal. Appl 2004 775 793

[24] J. Wang, S. Liu, B. Zheng, Y. Takeuchi Qualitative and bifurcation analysis using an SIR model with a saturated treatment function Math. Comput. Model 2012 710 722

[25] D. Xiao, S. Ruan Global analysis of an epidemic model with a nonlinear incidence rate Math. Biosci 2007 419 429

[26] R. Xu, Z. Ma, Z. Wang Global stability of a delayed SIRS epidemic model with saturation incidence and temporary immunity Comput. Math. Appl 2010 3211 3221

[27] X. Zhang, X.N. Liu Backward bifurcation of an epidemic model with saturated treatment function J. Math. Anal. Appl 2008 433 443

[28] J. Zhang, J. Ren, X. Zhang Dynamics of an SLIR model with nonmonotone incidence rate and stochastic perturbation Math. Biosci. Eng 2019 5504 5530

[29] Z. Zhonghua, S. Yaohong Qualitative analysis of a SIR epidemic model with saturated treatment rate J. Appl. Math. Comput 2010 177 194