SciELO - Scientific Electronic Library Online

 
vol.27 issue1A two-patch epidemic model with nonlinear reinfectionAssessing the invasion speed of triatomine populations, chagas disease vectors author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand

Journal

Article

Indicators

Related links

  • Have no similar articlesSimilars in SciELO

Share


Revista de Matemática Teoría y Aplicaciones

Print version ISSN 1409-2433

Rev. Mat vol.27 n.1 San José Jan./Jun. 2020

http://dx.doi.org/10.15517/rmta.v27i1.39948 

Artículo

A delay differential equations model for disease transmission dynamics

Un modelo de ecuaciones diferenciales con retraso para la dinámica de transmisión de enfermedades

Mustafa Erdem1 

Muntaser Safan2 

Carlos Castillo-Chavez3 

1Arizona State University, Simon A. Levin Mathematical, Computational and Modeling Sciences Center, Tempe AZ, United States. mustafa.erdem@asu.edu

2Arizona State University, Simon A. Levin Mathematical, Computational and Modeling Sciences Center, Tempe, United States. Mansoura University, Mathematics Department, Faculty of Science, Mansoura, Egypt. Umm Al-Qura University, Department of Mathematical Sciences, Faculty of Applied Sciences, Makkah, Saudi Arabia. muntaser_safan@yahoo.com

3Arizona State University, Simon A. Levin Mathematical, Computational and Modeling Sciences Center, Tempe AZ, United States; Brown University, Visiting Provost Professor of Applied Mathematics. carlos_castillo-chavez@brown.edu

Abstract

A delay differential equations epidemic model of SIQR (Susceptible-Infective-Quarantined-Recovered) type, with arbitrarily distributed periods in the isolation or quarantine class, is proposed. Its essential mathematical features are analyzed. In addition, conditions that support the existence of periodic solutions via Hopf bifurcation are identified. Nonexponential waiting times in the quarantine/isolation class lead not only to oscillations but can also support stability switches.

Keywords: delay differential equation; integro-differential equation; epidemic model; quarantine; stability switch; oscillations; stage structure.

Resumen

Se propone un modelo epidémico de ecuaciones diferenciales con retraso del tipo SIQR (por sus siglas en inglés) (Susceptible-Infeccioso-En cuarentena-Recuperado), con períodos arbitrariamente distribuidos en la clase de aislamiento o cuarentena. Se analizan sus características matemáticas esenciales. Además, se identifican las condiciones que respaldan la existencia de soluciones periódicas a través de la bifurcación de Hopf. Los tiempos de espera no exponenciales en la clase de cuarentena/aislamiento conducen no solo a oscilaciones sino que también pueden soportar cambios de estabilidad.

Palabras clave: ecuación diferencial con retraso; ecuación integro-diferencial; modelo epidémico; cuarentena; cambio de estabilidad; oscilaciones; estructura por etapas.

Mathematics Subject Classification: 92D25, 92D30, 92B99.

Ver contenido complete en PDF.

Acknowledgements

Authors thank the reviewers as well as the editors for their effort.

References

E, Beretta; Y, Kuang. Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal. 33 (2002), no. 5, 1144-1165. [ Links ]

F, G, Boese. Stability with respect to the delay: On a paper of K.L. Cooke and P. van den Driessche, J. Math. Anal. and Applications 228 (1998), no. 2, 293-321. [ Links ]

F, Brauer; C, Castillo-Chávez. Mathematical models in population biology and epidemiology, Texts in Applied Mathematics 40, Springer-Verlag, New York, 2001. [ Links ]

C, Castillo-Chavez; H,W, Hethcote; V, Andreasen; S,A, Levin; W,M, Liu. Cross-immunity in the dynamics of homogeneous and heterogeneous populations, in: T,G, Hallam; L,G, Gross; S,A, Levin. (Eds.) Mathematical Ecology, Proceedings of the Autumn Course Research Seminars, (Trieste, 1986), World Sci. Publishing, Teaneck, NJ (1988), 303-316. [ Links ]

C, Castillo-Chavez; H,W, Hethcote; V, Andreasen; S,A, Levin; W,M, Liu. Epidemiological models with age structure, proportionate mixing, and cross-immunity, J. Math. Biol. 27 (1989), no. 3, 233-258. [ Links ]

G, Chowell; P,W, Fenimore; M,A, Castillo-Garsow; C, Castillo-Chavez. SARS outbreaks in Ontario, Hong Kong and Singapore: the role of diagnosis and isolation as a control mechanism, J. Theor. Biol. 224 (2003), no. 1, 1-8. [ Links ]

R, Curtiss III. The impact of vaccines and vaccinations: challenges and opportunities for modelers, Math. Biosc. Eng. 8 (2011), no. 1, 77-93. [ Links ]

O, Diekmann; S,A, van Gils; S,M, Verduyn Lunel; H,-O, Walther. Delay equations: Functional-, Complex- and Nonlinear Analysis, Applied Mathematical Sciences 110, Springer-Verlag, New York , 1995. [ Links ]

M, Erdem; M, Safan; C, Castillo-Chavez. Mathematical models of influenza with imperfect quarantine, Bull. Math. Biol. 79 (2017), no. 7, 1612-1636. [ Links ]

Z, Feng; W, Huang; C, Castillo-Chavez. On the role of variable latent periods in mathematical models for tuberculosis, J. Dynamics and Differential Equations 13 (2001), no. 2, 425-452. [ Links ]

Z, Feng; H,R, Thieme. Recurrent outbreaks of childhood diseases revisited: the impact of isolation, Math. Biosci. 128 (1995), no. 1-2, 93-130. [ Links ]

Z, Feng; H,R, Thieme. Endemic models with arbitrarily distributed periods of infection II: fast disease dynamics and permanent recovery, SIAM Journal on Applied Mathematics 61 (2000), no. 3, 983-1012. [ Links ]

Z, Feng; H,R, Thieme. Endemic models with arbitrarily distributed periods of infection I: fundamental properties of the Model, SIAM Journal on Applied Mathematics 61 (2000), no. 3, 803-833. [ Links ]

Z, Feng; Y, Yang; D, Xu; P, Zhang; M,M, McCauley; J,W, Glasser. Timely identification of optimal control strategies for emerging infectious diseases, J. Theor. Biol. 259 (2009), no. 1, 165-171. [ Links ]

L, Gao; H, Hethcote. Simulations of rubella vaccination strategies in China, Math Biosc 202 (2006), no. 2, 371-385. [ Links ]

G,F, Gensini; M,H, Yacoub; A,A, Conti. The concept of quarantine in history: from plague to SARS, Journal of Infection 49 (2004), no. 4, 257-261. [ Links ]

J,K, Hale; S,M, Verduyn-Lunel. Introduction to Functional Differential Equations, Applied Mathematical Sciences 99, Springer-Verlag, New York , 1993. [ Links ]

M,A, Herrera-Valdez; M, Cruz-Aponte; C, Castillo-Chavez. Multiple outbreaks for the same pandemic: local transportation and social distancing explain the different “waves” of A-H1N1pdm cases observed in Mexico during 2009, Math. Biosc. Eng. 8 (2011), no. 1, 21-48. [ Links ]

H, W, Hethcote. The mathematics of infectious diseases, SIAM Rev. 42 (2000), no. 4, 599-653. [ Links ]

H,W, Hethcote; M, Zhien; L, Shengbing. Effects of quarantine in six endemic models for infectious diseases, Math. Biosci. 180 (2002), no. 1-2, 141-160. [ Links ]

N, W, McLachlan. Modern operational calculus with applications in technical mathematics, Revised edition, Dover Publications, Inc., New York, 1962. [ Links ]

A, Mubayi; C, Kribs-Zaleta; M, Martcheva; C, Castillo-Chavez. A cost-based comparison of quarantine strategies for new emerging diseases, Math. Biosc. Eng. 7 (2010), no. 3, 687-717. [ Links ]

M, Nuño. A mathematical model for the dynamics of influenza at the population and host level, Ph.D. Thesis, Cornell University, 2005. [ Links ]

M, Safan. Mathematical analysis of an SIR respiratory infection model with sex and gender disparity: special reference to influenza A, Mathematical Biosciences and Engineering 16 (2019), no. 4, 2613-2649. [ Links ]

M, Safan; M, Kretzschmar; K,P, Hadeler. Vaccination based control of infections in SIRS models with reinfection: special reference to pertussis, Journal of Mathematical Biology 67 (2013), no. 5, 1083-1110. [ Links ]

E, Shim. Prioritization of delayed vaccination for pandemic influenza, Math. Biosc. Eng. 8 (2011), no. 1, 95-112. [ Links ]

H, R, Thieme. Mathematics in population biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, New Jersey, 2003. [ Links ]

H, R, Thieme. The transition through stages with arbitrary length distributions, and applications in epidemics, in: C, Castillo-Chavez; S, Blower; P, van den Driessche; D, Kirschner; A,A, Yakubu. (Eds.) Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory. (Minneapolis, MN, 1999), The IMA Volumes in Mathematics and its Applications, 126, Springer, New York, 2002, pp. 45-84. [ Links ]

Y, Yang; Z, Feng; D, Xu. Analysis of a model with multiple infectious stages and arbitrarily distributed stage durations, Math. Model. Nat. Phenom. 3 (2008), no. 7, 180-193. [ Links ]

Received: May 18, 2019; Revised: June 20, 2019; Accepted: September 17, 2019

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License