Coronavirus epidemic model with isolation and nonlinear incidence rate

Document Type : Original Article

Authors

Department of Mathematics, Faculty of Science (Girls), Al-Azhar University, Cairo, Egypt

Abstract

In our article we proposed an epidemiological model consisting of five compartments that describe corona virus disease with isolation. We developed an SEIR system to produce the dynamical behaviour of infection by adding isolation compartment F(t) (that is because of the isolation of the infected individuals will reduce the spread of the disease). We formulated our model with nonlinear incidence rate. First, we discussed the positivity and boundedness of the model. Then we calculated the basic reproduction number of the model under certain conditions. By analysing the local and the global stability of our model, it is noticed that this stability depends on the basic reproductive number. We found that the system has no endemic equilibrium state if R0 < 1. Finally, we discussed the global stability by using Lyapunov function of Goh-volterra type with LaSalle’s invariance principle. It was shown that the system has a unique equilibrium point which is global asymptotically stable.

Keywords

Main Subjects

References

[1] Chen N, Zhou M, Dong X, Qu J, Gong F, Han Y, Qiu Y, Wang J, Liu Y, Wei Y, et al. Epidemiological and clinical characteristics of 99 cases of 2019 novel coronavirus pneumonia in Wuhan, China: a descriptive study. Lancet. 2020; 395(10223):507–13.
[2] WHO. Coronavirus disease 2019 (COVID-19): situation report. 2020; 51.
[3] Cook K, Driessche P. Analysis of an SEIRS epidemic model with two delays. J. Math. Biol. 1996; 35: 240-260.
[4] Abta A, Kaddar A, Alaoui H T. Global stability for delay SIR and SEIR epidemic models with saturated incidence rates. Election. J. Differential Equations. 2012; (386): 956-965.
[5] Han S, Lei C. Global stability of equilibria of a diffusive SEIR epidemic model with nonlinear incidence. Appl. Math. Lett. 2019; (98):  114-120.
[6] Liu Q et al. Stationary distribution and extinction of a stochastic SEIR epidemic model with standard incidence. Physics A.2017; (476): 58-69.
[7] Tian B, Yuan R. Traveling waves for a diffusive SEIR epidemic model with non-local reaction. Appl. Math. Model. 2017; (50): 432-449.
[8] He SB, Peng YX, Sun KH. SEIR Modeling of the COVID-19 and its dynamics. Nonlinear Dynam. 2020; 101(3): 1667- 1680.
[9] Hsieh Y H. Middle East respiratory syndrome coronavirus (MERS-CoV) nosocomial outbreak in south korea: insights from modeling. PeerJ. 2015; 3:1505.
[10] Kim Y, Lee S, Chu C, Choe S, Hong S, Shin Y. The characteristics of middle eastern respiratory syndrome coronavirus transmission dynamics in South Korea. Osong Public Health Res Perspect. 2016; 7(1): 49–55.
[11] Chen T M, Rui J, Wang Q P, Zhao Z Y, Cui J A, Yin L. A mathematical model for simulating the phase-based transmissibility of a novel coronavirus. Infect Dis Poverty. 2020; 9(1): 1–8.
[12] Sookaromdee P, Wiwanitkit V et al. Imported cases of 2019-novel coronavirus (2019-ncov) infections in Thailand: mathematical modelling of the outbreak. Asian Pac J Trop Med. 2020; 13(3): 139–40.
[13] Mwalili S, Kimathi M, Ojiambo V et al. SEIR model for COVID-19 dynamics incorporating the environment and social distancing. BMC Res Notes. 2020; 13, 352.
[14] Safi M A, Garba S M. Global stability analysis of SEIR model with Holling type Π incidence function; Comp and Math. Methods in medicine. 2012; ID 826052, 8 pages.
[15] Zeb A, Alzahrani E, Erturk V S, Zaman G. Mathematical model for Coronavirus disease 2019 (COVID-19) containing isolation class. Hindawi, BioMed Research International. 2020; Article ID 3452402. https://dot.org/10.1155/2020/3452402
[16] Li M Y, Smith H L, Wang L, “Global dynamics of an SEIR epidemic model with vertical transmission,” SIAM Journal on Applied Mathematics. 2001; 62 (1): 58–69.
[17] Li G, Jin Z. “Global stability of a SEIR epidemic model with infectious force in latent, infected and immune period,” Chaos, Solitons and Fractals.2005; 25 (5): 1177–1184.
[18] Zhang J, Ma Z. “Global dynamics of an SEIR epidemic model with saturating contact rate”. Mathematical Biosciences. 2003; 185 (1): 15–32.
[19] Smith H L, Waltman P. The Theory of the Chemostat, Cambridge University Press, 1995.
[20] Hethcote H W, Thieme H R. “Stability of the endemic equilibrium in epidemic models with subpopulations”. Mathematical Biosciences. 1985; 75 (2): 205–227.
[21] Diekmann O, Heesterbeek J A, Metz J A. “On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations”. Journal of Mathematical Biology. 1990; 28 (4): 365–382.
[22] Van Den Driessche P, Watmough J. “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission”. Mathematical Biosciences. 2002; 180: 29–48.
[23] Hethcote H W. “Mathematics of infectious diseases”. SIAM Review. 2000; 42(4): 599–653.
[24] Anderson R M, Verlag R M. Population Biology of Infectious Diseases, Springer. New York, NY, USA, 1982.
[25] Hale J K. Ordinary Differential Equations. John Wiley & Sons, New York, NY, USA, 1969.
[26] LaSalle J P. The Stability of Dynamical Systems, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1976.
Al-Azhar Bulletin of Science
Volume 32, Issue 2-B
December 2021
Pages 11-22

Files

Share

How to cite

Statistics