scholarly journals A General Discrete Time Model of Population Dynamics in the Presence of an Infection

2009 ◽  
Vol 2009 ◽  
pp. 1-15 ◽  
Author(s):  
Giuseppe Izzo ◽  
Yoshiaki Muroya ◽  
Antonia Vecchio
Keyword(s):  
Discrete Model ◽  
Reproduction Number ◽  
Continuous Model ◽  
Host Population ◽  
Threshold Parameter ◽  
Time Model ◽  
Limiting Behavior ◽  
Discrete Time Model ◽  
Discrete Counterpart

We present a set of difference equations which generalizes that proposed in the work of G. Izzo and A. Vecchio (2007) and represents the discrete counterpart of a larger class of continuous model concerning the dynamics of an infection in an organism or in a host population. The limiting behavior of this new discrete model is studied and a threshold parameter playing the role of the basic reproduction number is derived.

scholarly journals Dynamical analysis of a generalized hepatitis B epidemic model and its dynamically consistent discrete model

2021 ◽  
Author(s):  
Manh Tuan Hoang
Keyword(s):  
Hepatitis B ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Discrete Model ◽  
Reproduction Number ◽  
Continuous Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Nsfd Scheme ◽  
The Basic Reproduction Number

The aim of this work is to study qualitative dynamical properties of a generalized hepatitis B epidemic model and its dynamically consistent discrete model. Positivity, boundedness, the basic reproduction number and asymptotic stability properties of the model are analyzed rigorously. By the Lyapunov stability theory and the Poincare-Bendixson theorem in combination with the Bendixson-Dulac criterion, we show that a disease-free equilibrium point is globally asymptotically stable if the basic reproduction number $\mathcal{R}_0 \leq 1$ and a disease-endemic equilibrium point is globally asymptotically stable whenever $\mathcal{R}_0 > 1$. Next, we apply the Mickens’ methodology to propose a dynamically consistent nonstandard finite difference (NSFD) scheme for the continuous model. By rigorously mathematical analyses, it is proved that the constructed NSFD scheme preserves essential mathematical features of the continuous model for all finite step sizes. Finally, numerical experiments are conducted to illustrate the theoretical findings and to demonstrate advantages of the NSFD scheme over standard ones. The obtained results in this work not only improve but also generalize some existing recognized works.

scholarly journals Multiple Periodic Solutions of a Ratio-Dependent Predator-Prey Discrete Model

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Tiejun Zhou ◽  
Xiaolan Zhang ◽  
Min Wang
Keyword(s):  
Periodic Solutions ◽  
Discrete Model ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Time Model ◽  
Predator Prey ◽  
Discrete Time Model ◽  
Multiple Periodic Solutions ◽  
Ratio Dependent

A delayed ratio-dependent predator-prey discrete-time model with nonmonotone functional response is investigated in this paper. By using the continuation theorem of Mawhins coincidence degree theory, some new sufficient conditions are obtained for the existence of multiple positive periodic solutions of the discrete model. An example is given to illustrate the feasibility of the obtained result.

MODELING THE ROLE OF MUTATIONS AND DENSITY INDEPENDENT DISPERSAL IN EVOLUTIONARY RESCUE

2014 ◽  
Vol 22 (01) ◽  
pp. 123-132
Author(s):  
MELKIOR ORNIK
Keyword(s):  
Experimental Research ◽  
Discrete Time ◽  
Environmental Changes ◽  
Discrete Space ◽  
Time Model ◽  
Beneficial Mutations ◽  
Discrete Time Model ◽  
Population Dispersal ◽  
Evolutionary Rescue

Faced with a strong and sudden deterioration of environment, a population encounters two possible options — adapt or perish. In general, it is not known which of those outcomes the environmental changes will lead to. Building on experimental research, we introduce a discrete-space, discrete-time model for environmental rescue based on the influence of population dispersal, as well as, potentially beneficial mutations. Numerical results obtained by the model are shown to correspond well to experimentally obtained data.

scholarly journals Stoichiometric knife-edge model on discrete time scale

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Ming Chen ◽  
Lale Asik ◽  
Angela Peace
Keyword(s):  
Discrete Time ◽  
Time Scale ◽  
Ecological Stoichiometry ◽  
Discrete Model ◽  
Nutrient Content ◽  
Continuous Model ◽  
Knife Edge ◽  
Time Model ◽  
Predator Prey ◽  
Predator Prey Model

AbstractEcological stoichiometry is the study of the balance of multiple elements in ecological interactions and processes (Sterner and Elser in Ecological Stoichiometry: The Biology of Elements from Molecules to the Biosphere, 2002). Modeling under this framework enables us to investigate the effect nutrient content on organisms whether the imbalance involves insufficient or excess nutrient content. This phenomenon is called the “stoichiometric knife-edge”. In this paper, a discrete-time predator–prey model that captures this phenomenon is established and qualitatively analyzed. We systematically expound the similarities and differences between our discrete model and the corresponding continuous analog. Theoretical and numerical analyses show that while the discrete and continuous models share many properties, differences also exist. Under certain parameter sets, the models exhibit qualitatively different dynamics. While the continuous model shows limit cycle, Hopf bifurcation, and saddle-node bifurcation, the discrete-time model exhibits richer dynamical behaviors, such as chaos. By comparing the dynamics of the continuous and discrete model, we can conclude that stoichiometric effects of low food quality on predators are robust to the discretization of time. This study can possibly serve as an example for pointing to the importance of time scale in ecological modeling.

Problems with continuous-time malaria models in describing gametocytogenesis

Parasitology ◽  
2008 ◽  
Vol 135 (8) ◽  
pp. 881-896 ◽  
Author(s):  
L. CROOKS
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Continuous Model ◽  
Time Model ◽  
Discrete Time Model ◽  
The Third ◽  
Infection Dynamics ◽  
Constant Rate ◽  
Parasite Multiplication ◽  
The Many

SUMMARYMost mathematical models of malaria infection represent parasites as replicating continuously at a constant rate whereas in reality, malaria parasites replicate at a fixed age. The behaviour of continuous-time models when gametocytogenesis is included, in comparison to a more realistic discrete-time model that incorporates a fixed replication age was evaluated. Both the infection dynamics under gametocytogenesis and implications for predicting the amount parasites should invest into gametocytes (level of investment favoured by natural selection) are considered. It is shown that the many malaria models with constant replication rates can be represented by just 3 basic types. For these 3 types, it is then shown that under gametocytogenesis (i) in 2 cases, parasite multiplication and gametocyte production is mostly much too low, (ii) in the third, parasite multiplication and gametocyte production is mostly much too high, (iii) the effect of gametocyte investment on parasite multiplication is mostly too high, (iv) the effect of gametocyte investment on gametocyte production is nearly always too low and (v) with a simple approximation of fitness, the predicted level of gametocyte investment is mostly much too low. However, a continuous model with 48 age-compartments compares well to the discrete model. These findings are a further argument for modelling malaria infections in discrete time.

scholarly journals Discrete and Continuous Model of Three-Phase Linear Induction Motors “LIMs” Considering Attraction Force

Energies ◽  
2019 ◽  
Vol 12 (4) ◽  
pp. 655 ◽  
Author(s):  
Nicolás Toro-García ◽  
Yeison Garcés-Gómez ◽  
Fredy Hoyos
Keyword(s):  
Induction Motor ◽  
Discrete Time ◽  
Continuous Model ◽  
Attraction Force ◽  
Continuous Time Model ◽  
Time Model ◽  
Linear Induction ◽  
Linear Induction Motor ◽  
Discrete Time Model ◽  
Three Phase

A fifth-order dynamic continuous model of a linear induction motor (LIM), without considering “end effects” and considering attraction force, was developed. The attraction force is necessary in considering the dynamic analysis of the mechanically loaded linear induction motor. To obtain the circuit parameters of the LIM, a physical system was implemented in the laboratory with a Rapid Prototype System. The model was created by modifying the traditional three-phase model of a Y-connected rotary induction motor in a d–q stationary reference frame. The discrete-time LIM model was obtained through the continuous time model solution for its application in simulations or computational solutions in order to analyze nonlinear behaviors and for use in discrete time control systems. To obtain the solution, the continuous time model was divided into a current-fed linear induction motor third-order model, where the current inputs were considered as pseudo-inputs, and a second-order subsystem that only models the currents of the primary with voltages as inputs. For the discrete time model, the current-fed model is discretized by solving a set of differential equations, and the subsystem is discretized by a first-order Taylor series. Finally, a comparison of the continuous and discrete time model behaviors was shown graphically in order to validate the discrete time model.

A Response Time Model on the Role of Spectral and Duration Cues in Vowel Perception

2014 ◽  
Author(s):  
Gabriel Tillman ◽  
Don van Ravenzwaaij ◽  
Scott Brown ◽  
Titia Benders
Keyword(s):  
Response Time ◽  
Time Model ◽  
Vowel Perception ◽  
Response Time Model

scholarly journals Model-based evaluation of school- and non-school-related measures to control the COVID-19 pandemic

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Ganna Rozhnova ◽  
Christiaan H. van Dorp ◽  
Patricia Bruijning-Verhagen ◽  
Martin C. J. Bootsma ◽  
Janneke H. H. M. van de Wijgert ◽  
...  
Keyword(s):  
Reproduction Number ◽  
Additional Benefit ◽  
Transmission Model ◽  
Time Points ◽  
School Based ◽  
Admission Data ◽  
Age Structured ◽  
The Impact ◽  
Pandemic Wave

AbstractThe role of school-based contacts in the epidemiology of SARS-CoV-2 is incompletely understood. We use an age-structured transmission model fitted to age-specific seroprevalence and hospital admission data to assess the effects of school-based measures at different time points during the COVID-19 pandemic in the Netherlands. Our analyses suggest that the impact of measures reducing school-based contacts depends on the remaining opportunities to reduce non-school-based contacts. If opportunities to reduce the effective reproduction number (Re) with non-school-based measures are exhausted or undesired and Re is still close to 1, the additional benefit of school-based measures may be considerable, particularly among older school children. As two examples, we demonstrate that keeping schools closed after the summer holidays in 2020, in the absence of other measures, would not have prevented the second pandemic wave in autumn 2020 but closing schools in November 2020 could have reduced Re below 1, with unchanged non-school-based contacts.

scholarly journals Threshold Dynamics in a Model for Zika Virus Disease with Seasonality

2021 ◽  
Vol 83 (4) ◽  
Author(s):  
Mahmoud A. Ibrahim ◽  
Attila Dénes
Keyword(s):  
Zika Virus ◽  
Reproduction Number ◽  
Virus Disease ◽  
Threshold Dynamics ◽  
Global Dynamics ◽  
Threshold Parameter ◽  
Globally Asymptotically Stable ◽  
Linear Integral ◽  
The Impact ◽  
Disease Free

AbstractWe present a compartmental population model for the spread of Zika virus disease including sexual and vectorial transmission as well as asymptomatic carriers. We apply a non-autonomous model with time-dependent mosquito birth, death and biting rates to integrate the impact of the periodicity of weather on the spread of Zika. We define the basic reproduction number $${\mathscr {R}}_{0}$$ R 0 as the spectral radius of a linear integral operator and show that the global dynamics is determined by this threshold parameter: If $${\mathscr {R}}_0 < 1,$$ R 0 < 1 , then the disease-free periodic solution is globally asymptotically stable, while if $${\mathscr {R}}_0 > 1,$$ R 0 > 1 , then the disease persists. We show numerical examples to study what kind of parameter changes might lead to a periodic recurrence of Zika.

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