---
_id: '1555'
abstract:
- lang: eng
  text: 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.
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.
article_processing_charge: No
article_type: original
author:
- first_name: Diána
  full_name: Knipl, Diána
  last_name: Knipl
- first_name: Pawel
  full_name: Pilarczyk, Pawel
  id: 3768D56A-F248-11E8-B48F-1D18A9856A87
  last_name: Pilarczyk
- first_name: Gergely
  full_name: Röst, Gergely
  last_name: Röst
citation:
  ama: Knipl D, Pilarczyk P, Röst G. Rich bifurcation structure in a two patch vaccination
    model. <i>SIAM Journal on Applied Dynamical Systems</i>. 2015;14(2):980-1017.
    doi:<a href="https://doi.org/10.1137/140993934">10.1137/140993934</a>
  apa: Knipl, D., Pilarczyk, P., &#38; Röst, G. (2015). Rich bifurcation structure
    in a two patch vaccination model. <i>SIAM Journal on Applied Dynamical Systems</i>.
    Society for Industrial and Applied Mathematics . <a href="https://doi.org/10.1137/140993934">https://doi.org/10.1137/140993934</a>
  chicago: Knipl, Diána, Pawel Pilarczyk, and Gergely Röst. “Rich Bifurcation Structure
    in a Two Patch Vaccination Model.” <i>SIAM Journal on Applied Dynamical Systems</i>.
    Society for Industrial and Applied Mathematics , 2015. <a href="https://doi.org/10.1137/140993934">https://doi.org/10.1137/140993934</a>.
  ieee: D. Knipl, P. Pilarczyk, and G. Röst, “Rich bifurcation structure in a two
    patch vaccination model,” <i>SIAM Journal on Applied Dynamical Systems</i>, vol.
    14, no. 2. Society for Industrial and Applied Mathematics , pp. 980–1017, 2015.
  ista: 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.
  mla: Knipl, Diána, et al. “Rich Bifurcation Structure in a Two Patch Vaccination
    Model.” <i>SIAM Journal on Applied Dynamical Systems</i>, vol. 14, no. 2, Society
    for Industrial and Applied Mathematics , 2015, pp. 980–1017, doi:<a href="https://doi.org/10.1137/140993934">10.1137/140993934</a>.
  short: D. Knipl, P. Pilarczyk, G. Röst, SIAM Journal on Applied Dynamical Systems
    14 (2015) 980–1017.
date_created: 2018-12-11T11:52:42Z
date_published: 2015-01-01T00:00:00Z
date_updated: 2021-01-12T06:51:34Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1137/140993934
ec_funded: 1
intvolume: '        14'
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: http://discovery.ucl.ac.uk/1473750/1/99393.pdf
month: '01'
oa: 1
oa_version: Published Version
page: 980 - 1017
project:
- _id: 255F06BE-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '622033'
  name: Persistent Homology - Images, Data and Maps
publication: SIAM Journal on Applied Dynamical Systems
publication_identifier:
  eissn:
  - 1536-0040
publication_status: published
publisher: 'Society for Industrial and Applied Mathematics '
publist_id: '5616'
quality_controlled: '1'
scopus_import: 1
status: public
title: Rich bifurcation structure in a two patch vaccination model
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 14
year: '2015'
...