SciELO - Scientific Electronic Library Online

vol.26 issue1Recurrence patterns in the k-mino game author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand




Related links

  • Have no similar articlesSimilars in SciELO


Revista de Matemática Teoría y Aplicaciones

Print version ISSN 1409-2433

Rev. Mat vol.26 n.1 San José Jan./Jun. 2019 


The role of short-term immigration on disease dynamics: an sir model with age-structure

El rol de la immigración en periodos cortos sobre la dinámica de enfermedades: un modelo sir con estructura de edad

Fabio Sánchez1 

Juan G. Calvo2 

1Centro de Investigación en Matemática Pura y Aplicada (CIMPA) - Escuela de Matemática, Universidad de Costa Rica, San José, Costa Rica. E-Mail:

2Misma dirección que/Same address as: F. Sánchez. Universidad de Costa Rica, San José, Costa RicaE-Mail:



We formulate an age-structured nonlinear partial differential equation model that features short-term immigration effects in a population. In- dividuals can immigrate into the population as any of the three stages in the model: susceptible, infected or recovered. Global stability of the immigration-free and infection-free equilibria is discussed. A generalized numerical framework is established and specific short-term immigration scenarios are explored.

Keywords: age-structured model; epidemic models; immigration; migratory effects; finite difference methods



Se formula un modelo de ecuaciones diferenciales parciales no lineal y con estructura de edad, que incluye inmigración en un periodo breve de tiempo y sus efectos en la población. Los inmigrantes pueden ser de cualquier clase en el modelo: susceptibles, infectados o recuperados. Se discute además la estabilidad global del equilibrio correspondiente al mo- delo libre de infección e inmigración. Un método numérico general es discutido, y se exploran escenarios específicos con inmigración en perio- dos cortos de tiempo.

Palabras clave: modelo con estructura de edad; modelos epidémicos; inmi- gración; efectos migratorios; métodos de diferencias finitas

Mathematics Subject Classification: 92D25, 92D30, 35Q92, 65M06.


Ver contenido completo en pdf


The authors would like to thank the Research Center in Pure and Applied Math- ematics and the Mathematics Department at Universidad de Costa Rica for their support during the preparation of this manuscript. We would also like to thank the research vice-chancellor for the support of project 821-B8-747.


Allen, L. J.; Brauer, F.; Van den Driessche, P.; Wu, J. (2008) Mathematical Epidemiology. Springer. [ Links ]

Alvarado, L. (2018) “Costa Rica once again un- der malaria alert”, in: costa-rica-once-again-under-malaria-alert/73681, accessed 12/Nov/2018, 10:00am. [ Links ]

American Community Survey (ACS). In: https://www.census. gov/programs-surveys/acs/, accessed 13/Nov/2018, 10:00am. [ Links ]

BBC News. (2018) “Migrant caravan: what is it and why does it matter?”, in:, accessed 9/Nov/2018, 12:00md. [ Links ]

BBC News. (2018) “Puerto Rico increases hurricane Maria death toll to 2,975”, in: world-us-canada-45338080, accessed 11/Nov/2018, 3:00pm. [ Links ]

BBC News. (2018) “Venezuela’s migration crisis: is enough being done?,” in:, accessed 9/Nov/2018, 11:10am. [ Links ]

Center for Disease Control and Prevention, in: https://www.cdc. gov/cpr/partnerships/state-pages/puerto-rico.htm, accessed 13/Nov/2018, 11:00am. [ Links ]

Center for Disease Control and Prevention. “Entomology & ecology”, in: index.html, accessed 13/Nov/2018, 10:30am. [ Links ]

Center for Disease Control and Prevention. “Immi- grant and refugee health”, in: immigrantrefugeehealth/exams/medical-examination. html, accessed 7/Dec/2018, 7:44am. [ Links ]

Franceschetti, A.; Pugliese, A. (2008) “Threshold behaviour of a SIR epi- demic model with age structure and immigration”, Journal of Mathemati- cal Biology 57(1): 1-27. [ Links ]

Hanson, R. (2018) “4 charts show Venezuela’s worsen- ing migrant crisis,”, in: 4-charts-show-venezuelas-worsening-migrant-crisis-102464, accessed 9/Nov/2018, 11:00am. [ Links ]

Hirschman, C. (2013) “The contributions of immigrants to american cul- ture”, Daedalus 142(3): 26-47. [ Links ]

Holmes, E.E.; Lewis, M.A.; Banks, J.E.; Veit, R.R. (1994) “Partial differ- ential equations in ecology: spatial interactions and population dynamics”, Ecology 75(1): 17-29. [ Links ]

Kribs-Zaleta, C.M.; Martcheva, M. (2002) “Vaccination strategies and backward bifurcation in an age-since-infection structured model”, Mathe- matical Biosciences 177-178: 317-332. [ Links ]

Martcheva, M.; Thieme, H.R. (2003) “Progression age enhanced back- ward bifurcation in an epidemic model with super-infection”, Journal of Mathematical Biology 46(5): 385-424. [ Links ]

McKitterick, M. (2018) “Number of criminals in migrant cara- van - does it matter?”, in: number-of-criminals-in-migrant-caravan-does-it-matter-/4687576.html, accessed 5/Dec/2018, 10:00am. [ Links ]

Meléndez, E.; Hinojosa, J. (2017) “Estimates of post-hurricane Maria exodus from Puerto Rico”, in:https://centropr.hunter. estimates-post-hurricane-maria-exodus-puerto-rico, accessed 11/Nov/2018, 10:00am. [ Links ]

Migration Policy Institute. “Profile of the unauthorized population: United States”, in: unauthorized-immigrant-population/state/US, accessed 13/Nov/2018, 11:30am. [ Links ]

O’Brien, M. (2018) “The migrant caravan and public health”, in: the-migrant-caravan-and-public-health/, accessed 5/Dec/2018, 10:20am. [ Links ]

Sanchez, F.; Calvo, J.G.; Segura, E.; Feng, Z. (2018) “A partial differ- ential equation model with age-structure and nonlinear recidivism: condi- tions for a backward bifurcation and a general numerical implementation”, arXiv:1712.09105v2, submitted. [ Links ]

Sands, G. (2018) “This year saw the most people in immigration deten- tion since 2001,” in: politics/ice-detention/index.html, accessed 13/Nov/2018, 08:00am. [ Links ]

Semple, K. (2018) “Nicaraguan migrants fleeing turmoil test Costa Rica’s good will”, in: americas/nicaragua-migrants-costa-rica.html, accessed 9/Nov/2018, 11:20am. [ Links ]

Shim, E.; Feng, Z.; Martcheva, M.; Castillo-Chavez, C. (2006) “An age- structured epidemic model of rotavirus with vaccination”, Journal of Math- ematical Biology 53(4): 719-746. [ Links ]

Schofield, H. (2016) “What next after the jungle?”, in: https://www., accessed 9/Nov/2018, 11:00am [ Links ]

Received: November 16, 2018; Accepted: December 12, 2018

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