Rich bifurcation structure in a two patch vaccination model

D. Knipl, P. Pilarczyk, G. Röst, SIAM Journal on Applied Dynamical Systems 14 (2015) 980–1017.

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Journal Article | Published | English
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Abstract
We show that incorporating spatial dispersal of individuals into a simple vaccination epidemic model may give rise to a model that exhibits rich dynamical behavior. Using an SIVS (susceptible-infected-vaccinated-susceptible) model as a basis, we describe the spread of an infectious disease in a population split into two regions. In each subpopulation, both forward and backward bifurcations can occur. This implies that for disconnected regions the two-patch system may admit several steady states. We consider traveling between the regions and investigate the impact of spatial dispersal of individuals on the model dynamics. We establish conditions for the existence of multiple nontrivial steady states in the system, and we study the structure of the equilibria. The mathematical analysis reveals an unusually rich dynamical behavior, not normally found in the simple epidemic models. In addition to the disease-free equilibrium, eight endemic equilibria emerge from backward transcritical and saddle-node bifurcation points, forming an interesting bifurcation diagram. Stability of steady states, their bifurcations, and the global dynamics are investigated with analytical tools, numerical simulations, and rigorous set-oriented numerical computations.
Publishing Year
Date Published
2015-01-01
Journal Title
SIAM Journal on Applied Dynamical Systems
Acknowledgement
Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria (pawel.pilarczyk@ist.ac.at). This author’s work was partially supported by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement 622033, by Fundo Europeu de Desenvolvimento Regional (FEDER) through COMPETE—Programa Operacional Factores de Competitividade (POFC), by the Portuguese national funds through Funda ̧caoparaaCiˆencia e a Tecnologia (FCT) in the framework of the research project FCOMP-01-0124-FEDER-010645 (ref. FCT PTDC/MAT/098871/2008), and by European Research Council through StG 259559 in the framework of the EPIDELAY project.
Volume
14
Issue
2
Page
980 - 1017
IST-REx-ID

Cite this

Knipl D, Pilarczyk P, Röst G. Rich bifurcation structure in a two patch vaccination model. SIAM Journal on Applied Dynamical Systems. 2015;14(2):980-1017. doi:10.1137/140993934
Knipl, D., Pilarczyk, P., & Röst, G. (2015). Rich bifurcation structure in a two patch vaccination model. SIAM Journal on Applied Dynamical Systems, 14(2), 980–1017. https://doi.org/10.1137/140993934
Knipl, Diána, Pawel Pilarczyk, and Gergely Röst. “Rich Bifurcation Structure in a Two Patch Vaccination Model.” SIAM Journal on Applied Dynamical Systems 14, no. 2 (2015): 980–1017. https://doi.org/10.1137/140993934.
D. Knipl, P. Pilarczyk, and G. Röst, “Rich bifurcation structure in a two patch vaccination model,” SIAM Journal on Applied Dynamical Systems, vol. 14, no. 2, pp. 980–1017, 2015.
Knipl D, Pilarczyk P, Röst G. 2015. Rich bifurcation structure in a two patch vaccination model. SIAM Journal on Applied Dynamical Systems. 14(2), 980–1017.
Knipl, Diána, et al. “Rich Bifurcation Structure in a Two Patch Vaccination Model.” SIAM Journal on Applied Dynamical Systems, vol. 14, no. 2, Society for Industrial and Applied Mathematics , 2015, pp. 980–1017, doi:10.1137/140993934.

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