Backward bifurcation, equilibrium and stability phenomena in a three-stage extended BRSV epidemic model

Greenhalgh, David; Griffiths, Martin

doi:10.1007/s00285-008-0206-y

Cited by 29 publications

(22 citation statements)

References 12 publications

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“…This could be interpreted as saying that a particular endemic equilibrium was LAS if, and only if, the gradient of the bifurcation at that point was positive. Interestingly enough, Greenhalgh and Griffiths (2009) found that these straightforward stability patterns did not always occur when the BRSV model was extended to three stages. The vertical turning points of the more complicated bifurcation curves were not necessarily the points at which a change in the stability of the endemic equilibria occurred.…”

Section: Locally Asymptotically Stable Endemic Equilibria and The Bifmentioning

confidence: 92%

“…We now investigate the possibility that more complicated bifurcation diagrams, of the type observed for the three-stage BRSV model by Greenhalgh and Griffiths (2009) The type of bifurcation curve as given in Figure 5. β .…”

Section: Bifurcation Diagrams For the Extended Cg Modelmentioning

confidence: 99%

“…When investigating backward bifurcation and associated phenomena in the two and three-stage BRSV models, Greenhalgh and Griffiths (2009) obtained, both from the deterministic and the stochastic point of view, a number of potentially interesting results. It cannot be assumed however, that such results, obtained by studying one model in isolation, will automatically carry over to other epidemic models.…”

Section: Introductionmentioning

confidence: 99%

“…In such cases it might be possible to extend the model in a natural way, thereby giving a better approximation to the true disease structure. Indeed, Greenhalgh and Griffiths (2009) argued that a three-stage model for the spread of bovine respiratory syncytial virus (BRSV) in cattle may be more realistic than the two-stage model studied by Greenhalgh et al (2000). It might then be asked whether this increase in the complexity of the model provides scope for yet more complicated bifurcation diagrams and hence more complicated system dynamics in the region of the parameter space near 1 0 = R .…”

Section: Introductionmentioning

confidence: 99%

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Dynamic phenomena arising from an extended Core Group model

Greenhalgh

Griffiths

2009

Mathematical Biosciences

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In order to obtain a reasonably accurate model for the spread of a particular infectious disease through a population, it may be necessary for this model to possess some degree of structural complexity. Many such models have, in recent years, been found to exhibit a phenomenon known as backward bifurcation, which generally implies the existence of two subcritical endemic equilibria. It is often possible to refine these models yet further, and we investigate here the influence such a refinement may have on the dynamic behaviour of a system in the region of the parameter space near 1 0 = R .We consider a natural extension to a so-called core group model for the spread of a sexually transmitted disease, arguing that this may in fact give rise to a more realistic model.From the deterministic viewpoint we study the possible shapes of the resulting bifurcation diagrams and the associated stability patterns. Stochastic versions of both the original and the extended models are also developed so that the probability of extinction and time to extinction may be examined, allowing us to gain further insights into the complex system dynamics near 1 0 = R . A number of interesting phenomena are observed, for which heuristic explanations are provided.

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Section: Locally Asymptotically Stable Endemic Equilibria and The Bifmentioning

confidence: 92%

Section: Bifurcation Diagrams For the Extended Cg Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Dynamic phenomena arising from an extended Core Group model

Greenhalgh

Griffiths

2009

Mathematical Biosciences

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“…This is the reason for the Hopf-bifurcation occurring at δ = δ H is mentioned here as backward bifurcation. We like to mention here that the notion of 'backward bifurcation' used here is completely different compared to the same phrase used in literature for epidemic models [19]. Fig.…”

Section: Backward Hopf-bifurcationmentioning

confidence: 99%

Deterministic Chaos vs. Stochastic Fluctuation in an Eco-epidemic Model

Mandal

Banerjee

2012

Math. Model. Nat. Phenom.

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Abstract. An eco-epidemiological model of susceptible Tilapia fish, infected Tilapia fish and Pelicans is investigated by several author based upon the work initiated by Chattopadhyay and Bairagi (Ecol. Model., 136, 103 -112, 2001). In this paper, we investigate the dynamics of the same model by considering different parameters involved with the model as bifurcation parameters in details. Considering the intrinsic growth rate of susceptible Tilapia fish as bifurcation parameter, we demonstrate the period doubling route to chaos. Next we consider the force of infection as bifurcation parameter and demonstrate the occurrence of two successive Hopfbifurcations. We identify the existence of backward Hopf-bifurcation when the death rate of predators is considered as bifurcation parameter. Finally we construct a stochastic differential equation model corresponding to the deterministic model to understand the role of demographic stochasticity. Exhaustive numerical simulation of the stochastic model reveals the large amplitude fluctuation in the population of fish and Pelicans for certain parameter values. Extinction scenario for Pelicans is also captured from the stochastic model.

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Stability and bifurcation in epidemic models describing the transmission of toxoplasmosis in human and cat populations

Ferreira

Echeverry

Peña

2017

Math Methods in App Sciences

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A five-dimensional ordinary differential equation model describing the transmission of Toxoplamosis gondii disease between human and cat populations is studied in this paper. Self-diffusion modeling the spatial dynamics of the T. gondii disease is incorporated in the ordinary differential equation model. The normalized version of both models where the unknown functions are the proportions of the susceptible, infected, and controlled individuals in the total population are analyzed. The main results presented herein are that the ODE model undergoes a trans-critical bifurcation, the system has no periodic orbits inside the positive octant, and the endemic equilibrium is globally asymptotically stable when we restrict the model to inside of the first octant. Furthermore, a local linear stability analysis for the spatially homogeneous equilibrium points of the reaction diffusion model is carried out, and the global stability of both the disease-free and endemic equilibria are established for the reaction-diffusion system when restricted to inside of the first octant. Finally, numerical simulations are provided to support our theoretical results and to predict some scenarios about the spread of the disease.

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Backward bifurcation, equilibrium and stability phenomena in a three-stage extended BRSV epidemic model

Cited by 29 publications

References 12 publications

Dynamic phenomena arising from an extended Core Group model

Dynamic phenomena arising from an extended Core Group model

Deterministic Chaos vs. Stochastic Fluctuation in an Eco-epidemic Model

Stability and bifurcation in epidemic models describing the transmission of toxoplasmosis in human and cat populations

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