Global dynamics of a delay differential system of a two-patch SIS-model with transport-related infections

Yukihiko Nakata; Gergely Röst

Mathematica Bohemica (2015)

  • Volume: 140, Issue: 2, page 171-193
  • ISSN: 0862-7959

Abstract

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We describe the global dynamics of a disease transmission model between two regions which are connected via bidirectional or unidirectional transportation, where infection occurs during the travel as well as within the regions. We define the regional reproduction numbers and the basic reproduction number by constructing a next generation matrix. If the two regions are connected via bidirectional transportation, the basic reproduction number R 0 characterizes the existence of equilibria as well as the global dynamics. The disease free equilibrium always exists and is globally asymptotically stable if R 0 < 1 , while for R 0 > 1 an endemic equilibrium occurs which is globally asymptotically stable. If the two regions are connected via unidirectional transportation, the disease free equilibrium always exists, but for R 0 > 1 two endemic equilibria can appear. In this case, the regional reproduction numbers determine which one of the two is globally asymptotically stable. We describe how the time delay influences the dynamics of the system.

How to cite

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Nakata, Yukihiko, and Röst, Gergely. "Global dynamics of a delay differential system of a two-patch SIS-model with transport-related infections." Mathematica Bohemica 140.2 (2015): 171-193. <http://eudml.org/doc/271571>.

@article{Nakata2015,
abstract = {We describe the global dynamics of a disease transmission model between two regions which are connected via bidirectional or unidirectional transportation, where infection occurs during the travel as well as within the regions. We define the regional reproduction numbers and the basic reproduction number by constructing a next generation matrix. If the two regions are connected via bidirectional transportation, the basic reproduction number $R_\{0\}$ characterizes the existence of equilibria as well as the global dynamics. The disease free equilibrium always exists and is globally asymptotically stable if $R_\{0\}<1$, while for $R_\{0\}>1$ an endemic equilibrium occurs which is globally asymptotically stable. If the two regions are connected via unidirectional transportation, the disease free equilibrium always exists, but for $R_\{0\}>1$ two endemic equilibria can appear. In this case, the regional reproduction numbers determine which one of the two is globally asymptotically stable. We describe how the time delay influences the dynamics of the system.},
author = {Nakata, Yukihiko, Röst, Gergely},
journal = {Mathematica Bohemica},
keywords = {SIS model; asymptotically autonomous system; global asymptotic stability; Lyapunov functional; transport-related infection},
language = {eng},
number = {2},
pages = {171-193},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Global dynamics of a delay differential system of a two-patch SIS-model with transport-related infections},
url = {http://eudml.org/doc/271571},
volume = {140},
year = {2015},
}

TY - JOUR
AU - Nakata, Yukihiko
AU - Röst, Gergely
TI - Global dynamics of a delay differential system of a two-patch SIS-model with transport-related infections
JO - Mathematica Bohemica
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 140
IS - 2
SP - 171
EP - 193
AB - We describe the global dynamics of a disease transmission model between two regions which are connected via bidirectional or unidirectional transportation, where infection occurs during the travel as well as within the regions. We define the regional reproduction numbers and the basic reproduction number by constructing a next generation matrix. If the two regions are connected via bidirectional transportation, the basic reproduction number $R_{0}$ characterizes the existence of equilibria as well as the global dynamics. The disease free equilibrium always exists and is globally asymptotically stable if $R_{0}<1$, while for $R_{0}>1$ an endemic equilibrium occurs which is globally asymptotically stable. If the two regions are connected via unidirectional transportation, the disease free equilibrium always exists, but for $R_{0}>1$ two endemic equilibria can appear. In this case, the regional reproduction numbers determine which one of the two is globally asymptotically stable. We describe how the time delay influences the dynamics of the system.
LA - eng
KW - SIS model; asymptotically autonomous system; global asymptotic stability; Lyapunov functional; transport-related infection
UR - http://eudml.org/doc/271571
ER -

References

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