Global dynamics of a two-strain avian influenza model

Gumel, Abba B.

doi:10.1080/00207160701769625

Cited by 64 publications

(36 citation statements)

References 23 publications

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“…Further, let R 02 < 1 < R 01 , so that the boundary equilibrium E 1 exists (Theorem 4) and strain-2 dies out (Lemma 3). Consider the following non-linear Lyapunov function, of Goh-Volterra type (functions of this type have been used in the mathematical ecology/epidemiology literature, such as those in [7,22,26,28]):…”

Section: Theorem 4 the Model (1) Has A Unique And Las Strain 1-only Bmentioning

confidence: 99%

“…Consequently, the mathematical modeling of diseases with multiple pathogen strains, such as dengue fever, HIV/AIDS, influenza, malaria and West Nile virus, has received considerable attention (see, for instance, [1,2,12,15,19,27,29,23,26,34,39] and some of the references therein). These studies have, in general, focussed on the determination of threshold condition(s) for the co-existence of the strains, as well as the evaluation of the role of cross-immunity (defined as a scenario where infection with one strain confers partial or complete protection against infection with another strain) in the transmission dynamics of the disease strains.…”

Section: Introductionmentioning

confidence: 99%

“…In a multi-strain dynamics situation, infection by one or more of the strains may modify the sensitivity to infection by the other strains [15,23,26,29,38]. Some of the main questions of epidemiological interest, in studies of modeling multi-strain dynamics, are:…”

Section: Introductionmentioning

confidence: 99%

“…Past research has estimated that the reproduction number of the 1918-1919 influenza pandemic and other seasonal strains of influenza ranged between 1.5 and 5.4 [15,16,17,18,21,26,35,36].…”

Section: Introductionmentioning

confidence: 99%

“…Bremmerman and Thieme [8] shows the phenomenon of competitive exclusion (where the strain with the largest reproduction number persists and eliminates the remaining strains) in a simple model with multiple strains (similar results were obtained in [11,20,23]). Gumel [26] shows co-existence of two strains (avian and mutant) of influenza when their reproduction numbers exceed unity using avian-human model in the presence of isolation (system does not undergo competitive exclusion).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Cross-immunity-induced backward bifurcation for a model of transmission dynamics of two strains of influenza

Garba

Safi

Gumel

2013

Nonlinear Analysis: Real World Applications

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A new deterministic model for the transmission dynamics of two strains of influenza is designed and used to qualitatively assess the role of cross-immunity on the transmission process. It is shown that incomplete cross-immunity could induce the phenomenon of backward bifurcation when the associated reproduction number is less than unity. The model undergoes competitive exclusion (where Strain i drives out Strain j to extinction whenever R 0i > 1 > R 0j ; i, j = 1, 2, i ̸ = j). For the case where infection with one strain confers complete immunity against infection with the other strain, it is shown that the disease-free equilibrium of the model is globally-asymptotically stable whenever the reproduction number is less than unity. In the absence of cross-immunity, the model can have a continuum of co-existence endemic equilibria (which is shown to be globally-asymptotically stable for a special case). When infection with one strain confers incomplete immunity against the other. Numerical simulations of the model show that the two strains co-exist, with Strain i dominating (but not driving out Strain j), whenever R 0i > R 0j > 1.

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Section: Theorem 4 the Model (1) Has A Unique And Las Strain 1-only Bmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Past research has estimated that the reproduction number of the 1918-1919 influenza pandemic and other seasonal strains of influenza ranged between 1.5 and 5.4 [15,16,17,18,21,26,35,36].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Cross-immunity-induced backward bifurcation for a model of transmission dynamics of two strains of influenza

Garba

Safi

Gumel

2013

Nonlinear Analysis: Real World Applications

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Model Based Economic Assessment of Avian Influenza Vaccination in an All-in/All-out Housing System

Coffey

Eberl

Greer

2018

Springer Proceedings in Mathematics &Amp; Statistics

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An avian-only Filippov model incorporating culling of both susceptible and infected birds in combating avian influenza

2016

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Depopulation of birds has always been an effective method not only to control the transmission of avian influenza in bird populations but also to eliminate influenza viruses. We introduce a Filippov avian-only model with culling of susceptible and/or infected birds. For each susceptible threshold level [Formula: see text], we derive the phase portrait for the dynamical system as we vary the infected threshold level [Formula: see text], focusing on the existence of endemic states; the endemic states are represented by real equilibria, pseudoequilibria and pseudo-attractors. We show generically that all solutions of this model will approach one of the endemic states. Our results suggest that the spread of avian influenza in bird populations is tolerable if the trajectories converge to the equilibrium point that lies in the region below the threshold level [Formula: see text] or if they converge to one of the pseudoequilibria or a pseudo-attractor on the surface of discontinuity. However, we have to cull birds whenever the solution of this model converges to an equilibrium point that lies in the region above the threshold level [Formula: see text] in order to control the outbreak. Hence a good threshold policy is required to combat bird flu successfully and to prevent overkilling birds.

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Global dynamics of a two-strain avian influenza model

Cited by 64 publications

References 23 publications

Cross-immunity-induced backward bifurcation for a model of transmission dynamics of two strains of influenza

Cross-immunity-induced backward bifurcation for a model of transmission dynamics of two strains of influenza

Model Based Economic Assessment of Avian Influenza Vaccination in an All-in/All-out Housing System

An avian-only Filippov model incorporating culling of both susceptible and infected birds in combating avian influenza

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