Philadelphia, PA - "The impact of human mobility on disease dynamics has been the focus of mathematical epidemiology for many years, especially since the 2002-03 SARS outbreak, which showed that an infectious agent can spread across the globe very rapidly via transportation networks," says mathematician Gergely Röst. Röst is co-author of a paper to be published this week in the SIAM Journal on Applied Dynamical Systems that presents a mathematical model to study the effects of individual movement on infectious disease spread.
The basic reproduction number--referred to as R nought or R0--the central quantity in epidemiology, determines the average number of secondary infections caused by a typical infected individual in a susceptible population. In most cases, when the reproduction number is less than 1, the system has only the "disease-free" equilibrium, and the disease is expected to die out. As the number increases through 1, a stable "endemic" equilibrium emerges, that is, the disease is maintained in the population without the need for external inputs.
When a backward bifurcation occurs, stable endemic equilibria can co-exist with a stable disease-free equilibrium even when the reproduction number is less than one. This makes it insufficient to simply reduce R0 to below 1 in order to eliminate disease. R0 must be further reduced to avoid endemic states and ensure eradication. Hence, a backward bifurcation has important public health implications.
"In our research, we investigated the interplay of backward bifurcation and spatial dispersal. As no such study has been done before, we started with a minimal system that includes only two patches and three compartments in each location such that the local dynamics shows backward bifurcation," explains Knipl. "It was fascinating to see how rich dynamics were generated by our model: instead of the usual disease free and single endemic equilibria, we found that up to nine equilibria can exist."
The rich dynamics resulting from spatial dispersal is not normally seen in simple epidemic models. The stability of steady states and their bifurcations and dynamics are investigated with analytical tools and numerical simulations.
While the model was not constructed to study a specific epidemic, the authors say that it can be considered as a prototype system that unites two phenomena that have been studied only separately before. They believe that similar bifurcation diagrams will be found in the future when studying more realistic models concerning a wide variety of diseases. They hope their paper will motivate such future work whenever spatial mobility and backward bifurcation are intertwined.
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Source Article:
Rich Bifurcation Structure in a Two-patch Vaccination Model
SIAM Journal on Applied Dynamical Systems
http://epubs.siam.org/journal/sjaday
Paper will be published online on June 11, 2015.
About the authors:
Diana Knipl is a postdoctoral researcher at the Agent-Based Modelling Laboratory in York University, Toronto. Pawe? Pilarczyk is a Marie Curie postdoctoral fellow in the Edelsbrunner Group at the Institute of Science and Technology Austria. Gergely Röst is a research associate professor at Bolyai Institute at the University of Szeged, Hungary.
If reporters would like to arrange an interview with the paper's authors, please contact karthika@siam.org
SIAM Journal on Applied Dynamical Systems