TY - JOUR
T1 - Backward bifurcations, turning points and rich dynamics in simple disease models
AU - Zhang, Wenjing
AU - Wahl, Lindi M.
AU - Yu, Pei
N1 - Publisher Copyright:
© 2016, Springer-Verlag Berlin Heidelberg.
Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.
PY - 2016/10/1
Y1 - 2016/10/1
N2 - In this paper, dynamical systems theory and bifurcation theory are applied to investigate the rich dynamical behaviours observed in three simple disease models. The 2- and 3-dimensional models we investigate have arisen in previous investigations of epidemiology, in-host disease, and autoimmunity. These closely related models display interesting dynamical behaviors including bistability, recurrence, and regular oscillations, each of which has possible clinical or public health implications. In this contribution we elucidate the key role of backward bifurcations in the parameter regimes leading to the behaviors of interest. We demonstrate that backward bifurcations with varied positions of turning points facilitate the appearance of Hopf bifurcations, and the varied dynamical behaviors are then determined by the properties of the Hopf bifurcation(s), including their location and direction. A Maple program developed earlier is implemented to determine the stability of limit cycles bifurcating from the Hopf bifurcation. Numerical simulations are presented to illustrate phenomena of interest such as bistability, recurrence and oscillation. We also discuss the physical motivations for the models and the clinical implications of the resulting dynamics.
AB - In this paper, dynamical systems theory and bifurcation theory are applied to investigate the rich dynamical behaviours observed in three simple disease models. The 2- and 3-dimensional models we investigate have arisen in previous investigations of epidemiology, in-host disease, and autoimmunity. These closely related models display interesting dynamical behaviors including bistability, recurrence, and regular oscillations, each of which has possible clinical or public health implications. In this contribution we elucidate the key role of backward bifurcations in the parameter regimes leading to the behaviors of interest. We demonstrate that backward bifurcations with varied positions of turning points facilitate the appearance of Hopf bifurcations, and the varied dynamical behaviors are then determined by the properties of the Hopf bifurcation(s), including their location and direction. A Maple program developed earlier is implemented to determine the stability of limit cycles bifurcating from the Hopf bifurcation. Numerical simulations are presented to illustrate phenomena of interest such as bistability, recurrence and oscillation. We also discuss the physical motivations for the models and the clinical implications of the resulting dynamics.
KW - Backward bifurcation
KW - Bistability
KW - Concave incidence rate
KW - Convex incidence rate
KW - Disease model
KW - Forward bifurcation
KW - Hopf bifurcation
KW - Negative backward bifurcation
KW - Positive backward bifurcation
KW - Recurrence
KW - Turning point
UR - http://www.scopus.com/inward/record.url?scp=84975706869&partnerID=8YFLogxK
U2 - 10.1007/s00285-016-0976-6
DO - 10.1007/s00285-016-0976-6
M3 - Article
C2 - 26921202
AN - SCOPUS:84975706869
VL - 73
SP - 947
EP - 976
JO - Journal of Mathematical Biology
JF - Journal of Mathematical Biology
SN - 0303-6812
IS - 4
ER -