Mathematical analysis of local and global dynamics of a new epidemic model

2021 ◽  
Keyword(s):  
Mathematical Analysis ◽  
Epidemic Model ◽  
Global Dynamics ◽  
Local And Global Dynamics

scholarly journals Global dynamics of a novel deterministic and stochastic SIR epidemic model with vertical transmission and media coverage

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaodong Wang ◽  
Chunxia Wang ◽  
Kai Wang
Keyword(s):  
Vertical Transmission ◽  
Epidemic Model ◽  
Media Coverage ◽  
Existence And Uniqueness ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Global Dynamics ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Stochastic Sir

AbstractIn this paper, we study a novel deterministic and stochastic SIR epidemic model with vertical transmission and media coverage. For the deterministic model, we give the basic reproduction number $R_{0}$ R 0 which determines the extinction or prevalence of the disease. In addition, for the stochastic model, we prove existence and uniqueness of the positive solution, and extinction and persistence in mean. Furthermore, we give numerical simulations to verify our results.

scholarly journals Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yakui Xue ◽  
Tiantian Li
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Threshold Value ◽  
Endemic Equilibrium ◽  
Logistic Growth ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic

We study a delayed SIR epidemic model and get the threshold value which determines the global dynamics and outcome of the disease. First of all, for anyτ, we show that the disease-free equilibrium is globally asymptotically stable; whenR0<1, the disease will die out. Directly afterwards, we prove that the endemic equilibrium is locally asymptotically stable for anyτ=0; whenR0>1, the disease will persist. However, for anyτ≠0, the existence conditions for Hopf bifurcations at the endemic equilibrium are obtained. Besides, we compare the delayed SIR epidemic model with nonlinear incidence rate to the one with bilinear incidence rate. At last, numerical simulations are performed to illustrate and verify the conclusions.

scholarly journals Local and Global Dynamics in a Discrete Time Growth Model with Nonconcave Production Function

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Serena Brianzoni ◽  
Cristiana Mammana ◽  
Elisabetta Michetti
Keyword(s):  
Production Function ◽  
Growth Model ◽  
Discrete Time ◽  
Production Functions ◽  
Elasticity Of Substitution ◽  
Global Dynamics ◽  
Production Factors ◽  
Local And Global Dynamics ◽  
Complex Features

We study the dynamics shown by the discrete time neoclassical one-sector growth model with differential savings while assuming a nonconcave production function. We prove that complex features exhibited are related both to the structure of the coexixting attractors and to their basins. We also show that complexity emerges if the elasticity of substitution between production factors is low enough and shareholders save more than workers, confirming the results obtained while considering concave production functions.

scholarly journals The thalamic low-threshold Ca 2+ potential: a key determinant of the local and global dynamics of the slow (<1 Hz) sleep oscillation in thalamocortical networks

2011 ◽  
Vol 369 (1952) ◽  
pp. 3820-3839 ◽  
Author(s):  
Vincenzo Crunelli ◽  
Adam C. Errington ◽  
Stuart W. Hughes ◽  
Tibor I. Tóth
Keyword(s):  
Action Potential ◽  
Slow Oscillation ◽  
Global Dynamics ◽  
Action Potential Firing ◽  
Local And Global Dynamics ◽  
Up States ◽  
Potential Firing ◽  
Low Threshold

During non-rapid eye movement sleep and certain types of anaesthesia, neurons in the neocortex and thalamus exhibit a distinctive slow (<1 Hz) oscillation that consists of alternating UP and DOWN membrane potential states and which correlates with a pronounced slow (<1 Hz) rhythm in the electroencephalogram. While several studies have claimed that the slow oscillation is generated exclusively in neocortical networks and then transmitted to other brain areas, substantial evidence exists to suggest that the full expression of the slow oscillation in an intact thalamocortical (TC) network requires the balanced interaction of oscillator systems in both the neocortex and thalamus. Within such a scenario, we have previously argued that the powerful low-threshold Ca 2+ potential (LTCP)-mediated burst of action potentials that initiates the UP states in individual TC neurons may be a vital signal for instigating UP states in related cortical areas. To investigate these issues we constructed a computational model of the TC network which encompasses the important known aspects of the slow oscillation that have been garnered from earlier in vivo and in vitro experiments. Using this model we confirm that the overall expression of the slow oscillation is intricately reliant on intact connections between the thalamus and the cortex. In particular, we demonstrate that UP state-related LTCP-mediated bursts in TC neurons are proficient in triggering synchronous UP states in cortical networks, thereby bringing about a synchronous slow oscillation in the whole network. The importance of LTCP-mediated action potential bursts in the slow oscillation is also underlined by the observation that their associated dendritic Ca 2+ signals are the only ones that inform corticothalamic synapses of the TC neuron output, since they, but not those elicited by tonic action potential firing, reach the distal dendritic sites where these synapses are located.

COMPETITION AND COOPERATION ON PREDATION: BIFURCATION THEORY OF MUTUALISM

2021 ◽  
pp. 1-25
Author(s):  
SRIJANA GHIMIRE ◽  
XIANG-SHENG WANG
Keyword(s):  
Quantitative Information ◽  
Model Systems ◽  
Exclusion Principle ◽  
Positive Equilibrium ◽  
Global Dynamics ◽  
Severe Condition ◽  
Predator Species ◽  
Competition And Cooperation ◽  
Local And Global Dynamics ◽  
Predator Prey Models

In this paper, we investigate two predator–prey models which take into consideration hunting cooperation (i.e., mutualism) between two different predators and within one predator species, respectively. Local and global dynamics are obtained for the model systems. By a detailed bifurcation analysis, we investigate the dependence of predation dynamics on mutualism (cooperative predation). From our study, we prove that mutualism may enhance the survival of mutualist predators in a severe condition and break the competitive exclusion principle. We further provide quantitative information about how the cooperative predation (mutualism) may (i) establish multiple stability switches on the positive equilibrium; (ii) generate backward bifurcation on equilibria; (iii) induce supercritical or subcritical Hopf bifurcations; and (iv) establish bi-stability phenomenon between the predator-free equilibrium and a positive equilibrium (or a limit cycle).

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