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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 


Backward bifurcation in neutrophil-pathogen interaction

Bifurcación hacia atrás en interacción neutrófilo-patógeno

Christopher M. Kribs1 

Omomayowa Olawoyin2 

1University of Texas at Arlington, Department of Mathematics, Arlington TX 76012, United States;

2University of Texas at Arlington, Department of Mathematics, Arlington TX 76012, United States;


Bacterial infections elicit immune responses including neutrophils, whose recruitment is stimulated by the bacteria’s presence but which die after eliminating those bacteria. This dual interaction between bacteria and neutrophil concentrations, more complicated than the simple predator-prey relationship that describes macrophage-bacteria interactions, creates an environment in which neutrophils may only be able to clear sufficiently small infections. This study describes this relationship using a simple nonlinear dynamical system which exhibits bistability behavior known as a backward bifurcation. Bacterial growth is assumed limited by a key nutrient. In contrast to a previous study which held neutrophil and nutrient levels constant and required saturation terms to produce bistability, our model shows that simple bilinear terms support bistability when nutrient and neutrophil densities are allowed to vary in response to bacterial density. An example application involving Borrelia burgdorferi, which feeds on manganese, illustrates why neutrophils’ rapid response is key to their ability to contain bacterial infections.

Keywords: bistability; resource limitations; phagocytosis; hysteresis.


Las infecciones bacterianas provocan respuestas inmunitarias, incluyendo neutrófilos, cuyo reclutamiento es estimulado por la presencia de la bacteria pero que muere después de eliminar esas bacterias. Esta doble interacción entre las concentraciones de bacterias y neutrófilos, más complicada que la simple relación depredador-presa que describe las interacciones entre bacterias y macrófagos, crea un ambiente en el que los neutrófilos tal vez sólo puedan despejar infecciones suficientemente pequeñas. Este estudio describe esta relación utilizando un sistema dinámico no lineal sencillo que exhibe un comportamiento de biestabilidad conocido como una bifurcación hacia atrás. El crecimiento bacteriano se supone limitado por un nutriente clave. En contraste con un estudio anterior que mantuvo los niveles de neutrófilos y nutrientes constantes y requería términos de saturación para producir la biestabilidad, nuestro modelo muestra que los términos bilineales sencillos fomentan la biestabilidad cuando las densidades de nutrientes y neutrófilos pueden variar en respuesta a la densidad bacteriana. Un ejemplo aplicado a la bacteria Borrelia burgdorferi, que se alimenta de manganeso, ilustra por qué la respuesta rápida de los neutrófilos es clave para su capacidad de contener las infecciones bacterianas.

Palabras clave: biestabilidad; recursos limitados; fagocitosis; histéresis.

Mathematics Subject Classification: 92C37.

Ver contenido complete en PDF.


This project was inspired by 4, for which the authors thank their colleagues. CK dedicates this project to the memory of Fred and Carolyn Kribs.


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Received: May 12, 2019; Revised: August 31, 2019; Accepted: September 17, 2019

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