American Institute of Mathematical Sciences

Advanced
July  2010, 14(1): 209-231. doi: 10.3934/dcdsb.2010.14.209

Global asymptotic dynamics of a model for quarantine and isolation

Mohammad A. Safi 1, and  Abba B. Gumel 1,

1. 

Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, R3T 2N2, Canada, Canada

Received  May 2009 Revised  October 2009 Published  April 2010

The paper presents an SEIQHRS model for evaluating the combined impact of quarantine (of asymptomatic cases) and isolation (of individuals with clinical symptoms) on the spread of a communicable disease. Rigorous analysis of the model, which takes the form of a deterministic system of nonlinear differential equations with standard incidence, reveal that it has a globally-asymptotically stable disease-free equilibrium whenever its associated reproduction number is less than unity. Further, the model has a unique endemic equilibrium when the threshold quantity exceeds unity. Using a Krasnoselskii sub-linearity trick, it is shown that the unique endemic equilibrium is locally-asymptotically stable for a special case. A nonlinear Lyapunov function of Volterra type is used, in conjunction with LaSalle Invariance Principle, to show that the endemic equilibrium is globally-asymptotically stable for a special case. Numerical simulations, using a reasonable set of parameter values (consistent with the SARS outbreaks of 2003), show that the level of transmission by individuals isolated in hospitals play an important role in determining the impact of the two control measures (the use of quarantine and isolation could offer a detrimental population-level impact if isolation-related transmission is high enough).
Keywords: reproduction number, Isolation, stability., quarantine, equilibria.
Mathematics Subject Classification: Primary: 92B05, 92C60; Secondary: 92D3.
Citation: Mohammad A. Safi, Abba B. Gumel. Global asymptotic dynamics of a model for quarantine and isolation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 209-231. doi: 10.3934/dcdsb.2010.14.209
[1]

Z. Feng. Final and peak epidemic sizes for SEIR models with quarantine and isolation. Mathematical Biosciences & Engineering, 2007, 4 (4) : 675-686. doi: 10.3934/mbe.2007.4.675
[2]

Tom Burr, Gerardo Chowell. The reproduction number $R_t$ in structured and nonstructured populations. Mathematical Biosciences & Engineering, 2009, 6 (2) : 239-259. doi: 10.3934/mbe.2009.6.239
[3]

Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37
[4]

Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595
[5]

Gerardo Chowell, Catherine E. Ammon, Nicolas W. Hengartner, James M. Hyman. Estimating the reproduction number from the initial phase of the Spanish flu pandemic waves in Geneva, Switzerland. Mathematical Biosciences & Engineering, 2007, 4 (3) : 457-470. doi: 10.3934/mbe.2007.4.457
[6]

Ling Xue, Caterina Scoglio. Network-level reproduction number and extinction threshold for vector-borne diseases. Mathematical Biosciences & Engineering, 2015, 12 (3) : 565-584. doi: 10.3934/mbe.2015.12.565
[7]

Nitu Kumari, Sumit Kumar, Sandeep Sharma, Fateh Singh, Rana Parshad. Basic reproduction number estimation and forecasting of COVID-19: A case study of India, Brazil and Peru. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021170
[8]

PaweŁ Hitczenko, Georgi S. Medvedev. Stability of equilibria of randomly perturbed maps. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 369-381. doi: 10.3934/dcdsb.2017017
[9]

D. J. W. Simpson. On the stability of boundary equilibria in Filippov systems. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3093-3111. doi: 10.3934/cpaa.2021097
[10]

Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455
[11]

Frederic Laurent-Polz, James Montaldi, Mark Roberts. Point vortices on the sphere: Stability of symmetric relative equilibria. Journal of Geometric Mechanics, 2011, 3 (4) : 439-486. doi: 10.3934/jgm.2011.3.439
[12]

Paul Georgescu, Hong Zhang, Daniel Maxin. The global stability of coexisting equilibria for three models of mutualism. Mathematical Biosciences & Engineering, 2016, 13 (1) : 101-118. doi: 10.3934/mbe.2016.13.101
[13]

Anna Ghazaryan, Christopher K. R. T. Jones. On the stability of high Lewis number combustion fronts. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 809-826. doi: 10.3934/dcds.2009.24.809
[14]

Sílvia Cuadrado. Stability of equilibria of a predator-prey model of phenotype evolution. Mathematical Biosciences & Engineering, 2009, 6 (4) : 701-718. doi: 10.3934/mbe.2009.6.701
[15]

Lyudmila Grigoryeva, Juan-Pablo Ortega, Stanislav S. Zub. Stability of Hamiltonian relative equilibria in symmetric magnetically confined rigid bodies. Journal of Geometric Mechanics, 2014, 6 (3) : 373-415. doi: 10.3934/jgm.2014.6.373
[16]

Elbaz I. Abouelmagd, Juan L. G. Guirao, Aatef Hobiny, Faris Alzahrani. Stability of equilibria points for a dumbbell satellite when the central body is oblate spheroid. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1047-1054. doi: 10.3934/dcdss.2015.8.1047
[17]

Shangbing Ai. Global stability of equilibria in a tick-borne disease model. Mathematical Biosciences & Engineering, 2007, 4 (4) : 567-572. doi: 10.3934/mbe.2007.4.567
[18]

William H. Sandholm. Local stability of strict equilibria under evolutionary game dynamics. Journal of Dynamics & Games, 2014, 1 (3) : 485-495. doi: 10.3934/jdg.2014.1.485
[19]

Yacine Chitour, Frédéric Grognard, Georges Bastin. Equilibria and stability analysis of a branched metabolic network with feedback inhibition. Networks & Heterogeneous Media, 2006, 1 (1) : 219-239. doi: 10.3934/nhm.2006.1.219
[20]

Daniela Cárcamo-Díaz, Jesús F. Palacián, Claudio Vidal, Patricia Yanguas. Nonlinear stability of elliptic equilibria in hamiltonian systems with exponential time estimates. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5183-5208. doi: 10.3934/dcds.2021073

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (275)
  • HTML views (0)
  • Cited by (20)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences