scholarly journals ROLE OF REPEAT INFECTION IN THE DYNAMICS OF A SIMPLE MODEL OF WANING AND BOOSTING IMMUNITY

2021 ◽  
pp. 1-22
Author(s):  
LAURA F. STRUBE ◽  
MAYA WALTON ◽  
LAUREN M. CHILDS
Keyword(s):  
Simple Model ◽  
Secondary Infection ◽  
Dynamical Behavior ◽  
Stable Limit Cycle ◽  
Epidemiological Data ◽  
Small Region ◽  
Extended Model ◽  
Stable Limit ◽  
Bifurcation Structures ◽  
Repeat Infection

Some infectious diseases produce lifelong immunity while others only produce temporary immunity. In the case of short-lived immunity, the level of protection wanes over time and may be boosted upon re-exposure, via infection or vaccination. Previous work developed a simple model capturing waning and boosting immunity, known as the Susceptible-Infectious-Recovered-Waned-Susceptible (SIRWS) model, which exhibits rich dynamical behavior including supercritical and subcritical Hopf bifurcations among other structures. Here, we extend the bifurcation analyses of the SIRWS model to examine the influence of all parameters on these bifurcation structures. We show that the bistable region, involving both a stable fixed point and a stable limit cycle, exists only for a small region of biologically realistic parameter space. Furthermore, we contrast the SIRWS model with a modified version, where immune boosting may involve the occurrence of a secondary infection. Analysis of this extended model shows that oscillations and bistability, as found in the SIRWS model, depend on strong assumptions about infectivity and recovery rate from secondary infection. Understanding the dynamics of models of waning and boosting immunity is important for accurately assessing epidemiological data.

scholarly journals Trapping Phenomenon Attenuates the Consequences of Tipping Points for Limit Cycles

2017 ◽  
Vol 7 (1) ◽  
Author(s):  
Everton S. Medeiros ◽  
Iberê L. Caldas ◽  
Murilo S. Baptista ◽  
Ulrike Feudel
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Nonlinear Dynamical Systems ◽  
Dynamical Behavior ◽  
Stable Limit Cycle ◽  
Tipping Points ◽  
Stable Limit ◽  
Nonlinear Dynamical ◽  
Natural Systems ◽  
Fold Bifurcation

Abstract Nonlinear dynamical systems may be exposed to tipping points, critical thresholds at which small changes in the external inputs or in the system’s parameters abruptly shift the system to an alternative state with a contrasting dynamical behavior. While tipping in a fold bifurcation of an equilibrium is well understood, much less is known about tipping of oscillations (limit cycles) though this dynamics are the typical response of many natural systems to a periodic external forcing, like e.g. seasonal forcing in ecology and climate sciences. We provide a detailed analysis of tipping phenomena in periodically forced systems and show that, when limit cycles are considered, a transient structure, so-called channel, plays a fundamental role in the transition. Specifically, we demonstrate that trajectories crossing such channel conserve, for a characteristic time, the twisting behavior of the stable limit cycle destroyed in the fold bifurcation of cycles. As a consequence, this channel acts like a “ghost” of the limit cycle destroyed in the critical transition and instead of the expected abrupt transition we find a smooth one. This smoothness is also the reason that it is difficult to precisely determine the transition point employing the usual indicators of tipping points, like critical slowing down and flickering.

Dynamics of a delayed competitive system affected by toxic substances with imprecise biological parameters

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5271-5293
Author(s):  
A.K. Pal ◽  
P. Dolai ◽  
G.P. Samanta
Keyword(s):  
Equilibrium Points ◽  
Stable Limit Cycle ◽  
Limit Cycle Oscillation ◽  
Toxic Substances ◽  
Stable Limit ◽  
Biological Parameters ◽  
Delay Model ◽  
Competitive System ◽  
Interval Numbers ◽  
Cycle Oscillation

In this paper we have studied the dynamical behaviours of a delayed two-species competitive system affected by toxicant with imprecise biological parameters. We have proposed a method to handle these imprecise parameters by using parametric form of interval numbers. We have discussed the existence of various equilibrium points and stability of the system at these equilibrium points. In case of toxic stimulatory system, the delay model exhibits a stable limit cycle oscillation. Computer simulations are carried out to illustrate our analytical findings.

scholarly journals STUDYING THE STABILITY BY USING LOCAL LINEARIZATION METHOD

2020 ◽  
Vol 2 (1) ◽  
pp. 27-31
Author(s):  
Abdulghafoor Jasim Salim ◽  
Kais Ismail Ebrahem ◽  
Suhirman
Keyword(s):  
Singular Point ◽  
Limit Cycle ◽  
Autoregressive Model ◽  
Stable Limit Cycle ◽  
Linearization Method ◽  
Stable Limit ◽  
Local Linearization ◽  
Non Linear ◽  
The Stability ◽  
Local Linearization Method

Abstract: In this paper we study the stability of one of a non linear autoregressive model with trigonometric term  by using local linearization method proposed by Tuhro Ozaki .We find the singular point ,the stability of the singular point and the limit cycle. We conclude  that the proposed model under certain conditions have a non-zero singular point which is  a asymptotically salable ( when  0 ) and have an  orbitaly stable limit cycle . Also we give some examples in order to explain the method. Key Words : Non-linear Autoregressive model; Limit cycle; singular point; Stability.

scholarly journals Méthode d'agrégation des variables appliquée à la dynamique des populations

2006 ◽  
Vol Volume 5, Special Issue TAM... ◽  
Author(s):  
Pierre Auger ◽  
Abderrahim El Abdllaoui ◽  
Rachid Mchich
Keyword(s):  
Stable Limit Cycle ◽  
Stable Limit ◽  
Dynamique Des Populations ◽  
Globally Asymptotically Stable ◽  
Density Dependent ◽  
Migration Rates ◽  
First Case ◽  
International Audience ◽  
Predator Model ◽  
Predator System

International audience We present the method of aggregation of variables in the case of ordinary differential equations. We apply the method to a prey - predator model in a multi - patchy environment. In this model, preys can go to a refuge and therefore escape to predation. The predator must return regularly to his terrier to feed his progeny. We study the effect of density-dependent migration on the global stability of the prey-predator system. We consider constant migration rates, but also density-dependent migration rates. We prove that the positif equilibrium is globally asymptotically stable in the first case, and that its stability changes in the second case. The fact that we consider density-dependent migration rates leads to the existence of a stable limit cycle via a Hopf bifurcation. Nous présentons les grandes lignes de laméthode d'agrégation des variables dans les systèmes d'équations différentielles ordinaires. Nous appliquons laméthode à un modèle proie-prédateur spatialisé. Dans ce modèle, les proies peuvent échapper à la prédation en se réfugiant sur un site. Le prédateur doit aussi retourner régulièrement dans son terrier pour nourrir sa progéniture. Nous étudions les effets de migration dépendant de la densité des populations sur la stabilité globale du système proie-prédateur. Nous considérons des taux de migration constants, puis densité-dépendants. Dans le cas de taux constants il existe un équilibre positif toujours stable alors que dans le cas de taux de migration densité-dépendants, il existe un cycle limite stable via une bifurcation de Hopf.

Bifurcation of Codimension 3 in a Predator–Prey System of Leslie Type with Simplified Holling Type IV Functional Response

2016 ◽  
Vol 26 (02) ◽  
pp. 1650034 ◽  
Author(s):  
Jicai Huang ◽  
Xiaojing Xia ◽  
Xinan Zhang ◽  
Shigui Ruan
Keyword(s):  
Functional Response ◽  
Limit Cycle ◽  
Stable Limit Cycle ◽  
Positive Equilibrium ◽  
Stable Limit ◽  
Homoclinic Loop ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Stable Equilibria

It was shown in [Li & Xiao, 2007] that in a predator–prey model of Leslie type with simplified Holling type IV functional response some complex bifurcations can occur simultaneously for some values of parameters, such as codimension 1 subcritical Hopf bifurcation and codimension 2 Bogdanov–Takens bifurcation. In this paper, we show that for the same model there exists a unique degenerate positive equilibrium which is a degenerate Bogdanov–Takens singularity (focus case) of codimension 3 for other values of parameters. We prove that the model exhibits degenerate focus type Bogdanov–Takens bifurcation of codimension 3 around the unique degenerate positive equilibrium. Numerical simulations, including the coexistence of three hyperbolic positive equilibria, two limit cycles, bistability states (one stable equilibrium and one stable limit cycle, or two stable equilibria), tristability states (two stable equilibria and one stable limit cycle), a stable limit cycle enclosing a homoclinic loop, a homoclinic loop enclosing an unstable limit cycle, or a stable limit cycle enclosing three unstable hyperbolic positive equilibria for various parameter values, confirm the theoretical results.

Analysis of chemical kinetic systems over the entire parameter space - I. The Sal’nikov thermokinetic oscillator

1988 ◽  
Vol 416 (1851) ◽  
pp. 391-402 ◽  
Keyword(s):  
Steady State ◽  
Parameter Space ◽  
Limit Cycles ◽  
Stable Limit Cycle ◽  
Chemical Kinetic ◽  
Phase Portraits ◽  
Stable Steady State ◽  
Stable Limit ◽  
Unstable Steady State ◽  
Undamped Oscillations

In this series of papers we re-examine, using recently developed techniques, some chemical kinetic models that have appeared in the literature with a view to obtaining a complete description of all the qualitatively distinct behaviour that the system can exhibit. Each of the schemes is describable by two coupled ordinary differential equations and contain at most three independent parameters. We find that even with these relatively simple chemical schemes there are regions of parameter space in which the systems display behaviour not previously found. Quite often these regions are small and it seems unlikely that they would be found via classical methods. In part I of the series we consider one of the thermally coupled kinetic oscillator models studied by Sal’nikov. He showed that there is a region in parameter space in which the system would be in a state of undamped oscillations because the relevant phase portrait consists of an unstable steady state surrounded by a stable limit cycle. Our analysis has revealed two further regions in which the phase portraits contain, respectively, two limit cycles of opposite stability enclosing a stable steady state and three limit cycles of alternating stability surrounding an unstable steady state. This latter region is extremely small, so much so that it could be reasonably neglected in any predictions made from the model.

Self-Excited Vibration Model of Dragonfly’s Wing Based on the Concept of Bionic Design for Small- or Micro-Sized Actuators

1997 ◽  
Author(s):  
Sagiri Ishimoto ◽  
Hiromu Hashimoto
Keyword(s):  
Nonlinear Equation ◽  
Driving Forces ◽  
Stable Limit Cycle ◽  
Equation Of Motion ◽  
Stable Limit ◽  
Bionic Design ◽  
Wing Vibration ◽  
Vibration Model ◽  
Excited Vibration ◽  
Sympetrum Frequens

Abstract This paper describes a self-excited vibration model of dragonfly’s wing based on the concept of bionic design, which is expected as a technological hint to solve the scale effect problems in developing the small- or micro-sized actuators. From a morphological consideration of flight muscle of dragonfly, the nonlinear equation of motion for the wing considering the air drag force due to flapping of wing is formulated. In the model, the dry friction-type and Van der Pol-type driving forces are employed to power the flight muscles and to generate the stable self-excited wing vibration. Two typical Japanese dragonflies, “Anotogaster sieboldii Selys” and “Sympetrum frequens Selys”, are selected as examples, and the self-excited vibration analyses for these dragonfly’s wings are demonstrated. The linearized solutions for the nonlinear equation of motion are compared with the nonlinear solutions, and the vibration system parameters to generate the stable limit cycle of self-excited wing vibration are determined.

scholarly journals Height control for a two-mass hopping robot based on stable limit cycle

2016 ◽  
Vol 13 (6) ◽  
pp. 172988141665774
Author(s):  
Taihui Zhang ◽  
Honglei An ◽  
Qing Wei ◽  
Wenqi Hou ◽  
Hongxu Ma
Keyword(s):  
Limit Cycle ◽  
Control Algorithm ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
Inverted Pendulum Model ◽  
Hopping Robot ◽  
Control Objective ◽  
Upper Level ◽  
Mass Spring ◽  
And Control

Differing from the commonly used spring loaded inverted pendulum model, this paper makes use of a two-mass spring model considering impact between the foot and ground which is closer to the real hopping robot. The height of upper mass which includes the upper leg and body is the main control objective. Then we develop a new kind of control algorithm acting on two levels: The upper level aims to achieve the desired velocity of the upper mass based on a stable limit cycle, where three different controllers are used to regulate the limit cycle; the target of the lower level is to drive the system to converge to the desired state and control the contact force between the foot and ground within an appropriate range based on the inner force control at the same time. Simulation results presented in this paper confirm the efficiency of this control algorithm.

Effect of joint stiffness on the dynamic stability of a one-link force-controlled manipulator

Robotica ◽  
1989 ◽  
Vol 7 (4) ◽  
pp. 339-342
Author(s):  
Bing C. Chiou ◽  
M. Shahinpoor
Keyword(s):  
Dynamic Stability ◽  
Closed Loop ◽  
Force Feedback ◽  
Joint Stiffness ◽  
Stable Limit Cycle ◽  
Critical Value ◽  
Interaction Process ◽  
Feedback Gain ◽  
Stable Limit ◽  
Damping Force

SUMMARYStudies are the effects of joint flexibility on the dynamic stability of a one-link force-controlled manipulator. The closed-loop dynamic equation with the explicit force controller and the damping force controller are first derived. Stability analysis is then carried out by computing the system eigenvalues. Results indicate a conditionally stable system. Due to the presence of discontinuous contacts with the environment during the interaction process, the system exhibits a stable limit cycle when the force feedback gain goes beyond the critical value.

Sign in / Sign up

Export Citation Format

Share Document