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Revista de Matemática Teoría y Aplicaciones

Print version ISSN 1409-2433

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


A two-patch epidemic model with nonlinear reinfection

Un modelo epidémico de dos poblaciones con reinfección no lineal

Juan G. Calvo1 

Alberto Hernández2 

Mason A. Porter3 

Fabio Sanchez4 

1University of Costa Rica, CIMPA, School of Mathematics, San José, Costa Rica.

2University of Costa Rica, CIMPA, School of Mathematics, San José, Costa Rica.

3University of California Los Angeles, Department of Mathematics, Los Angeles CA, United States of America.

4University of Costa Rica, CIMPA, School of Mathematics, San José, Costa Rica.


The propagation of infectious diseases and its impact on individuals play a major role in disease dynamics, and it is important to incorporate population heterogeneity into efforts to study diseases. As a simplistic but illustrative example, we examine interactions between urban and rural populations on the dynamics of disease spreading. Using a compartmental framework of susceptible-infected-susceptible (SI S ) dynamics with some level of immunity, we formulate a model that allows nonlinear reinfection. We investigate the effects of population movement in a simple scenario: a case with two patches, which allows us to model population movement between urban and rural areas. To study the dynamics of the system, we compute a basic reproduction number for each population (urban and rural). We also compute steady states, determine the local stability of the disease-free steady state, and identify conditions for the existence of endemic steady states. From our analysis and computational experiments, we illustrate that population movement plays an important role in disease dynamics. In some cases, it can be rather beneficial, as it can enlarge the region of stability of a disease-free steady state.

Keywords: dynamical systems; population dynamics; mathematical modeling; biological contagions; population movement.


La propagación de enfermedades infecciosas y su impacto en individuos juega un gran rol en la dinámica de enfermedades, y es importante incorporar heterogeneidad en la población en los esfuerzos por estudiar enfermedades. De manera simplística pero ilustrativa, se examinan interacciones entre una población urbana y una rural en la dinámica de la propagación de una enfermedad. Utilizando un sistema compartimental de dinámicas entre susceptibles-infectados-susceptibles (SIeS) con cierto nivel de inmunidad, se formula un modelo que permite reinfecciones no lineales. Se investiga los efectos de movimiento de poblaciones en un escenario simple: un caso con dos poblaciones, que permite modelar movimiento entre un área urbana y otra rural. Con el fin de estudiar la dinámica del sistema, se calcula el número básico reproductivo para cada comunidad (rural y urbana). Se calculan también puntos de equilibrio, la estabilidad local del estado libre de enfermedad, y se identifican condiciones para la existencia de estados de equilibrio endémicos. Del análisis y experimentos computacionales, se ilustra que el movimiento en la población juega un rol importante en la dinámica del sistema. En algunos casos, puede ser beneficioso, pues incrementa la región de estabilidad del punto de equilibrio del estado libre de infección.

Palabras clave: sistemas dinámicos; dinámica de poblaciones; modelado matemático; contagios biológicos; movimiento de poblaciones.

Mathematics Subject Classification: 92D25, 92D30.

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We thank the Research Center in Pure and Applied Mathematics and the Mathematics Department at Universidad de Costa Rica for their support during the preparation of this manuscript. The authors gratefully acknowledge institutional support for project B8747 from an UCREA grant from the Vice Rectory for Research at Universidad de Costa Rica. We also acknowledge helpful discussions with Profs. Luis Barboza, Carlos Castillo-Chavez, and Esteban Segura.


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Received: July 27, 2019; Revised: September 17, 2019; Accepted: October 31, 2019

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