Analysis of symmetries in models of multi-strain infections

Blyuss, Konstantin B.

doi:10.1007/s00285-013-0740-0

Cited by 5 publications

(17 citation statements)

References 64 publications

(86 reference statements)

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“…It should be noted that, using the symmetry property of the 2-locus-2-allele system and isotopic decomposition of the phase space to diagonalize the Jacobian, Blyuss [3] also obtained similar expressions for the fully symmetric steady state. We observe that the characteristic polynomials P k (ζ ) have positive coefficient A k,l , k = 1, 2, 3, l = 0, 1, .…”

Section: Strong Endemicity: No Strain Structurementioning

confidence: 80%

“…The study presented generalizes the results of [10] and [3], for our approach introduces additional constants that capture the topology of the strain spaces and are applicable to multiple strain systems beyond four strains. In the proceeding sections, we first describe the model including boundedness and positivity of solutions in Sections 2 and 3, respectively.…”

Section: Introductionmentioning

confidence: 76%

“…Using the multi-locus-allele framework, Gupta et al [26,27] and Recker et al [41] characterized individuals based on their immunological history and divided the host population into three partially overlapping epidemiological compartments for each strain ij: the proportion of the population that has been exposed to pathogenic type ij and/or is now immune to strain ij, z ij ; the proportion of the individuals who have been exposed and are immune to any antigenic type sharing alleles with strain ij, w ij and the proportion of the population infectious with strain ij, y ij . The model is given by the following set of differential equations: (3) where the force of infection is given by λ i j = βy i j , and β is the transmission rate. In addition, the model assumes that individuals leave the population at a constant rate μ due to natural death (i.e., 1 μ is the average life-expectancy of the host), and recover from the infectious class at the rate of σ (where 1 σ is the average period or séjour of infectiousness).…”

Section: Model Description: Multi-locus-allele Systemsmentioning

confidence: 99%

“…Herein, we present a different approach to perform a detailed study of the multi-locus-allele system of [26,27] with particular emphases on stability analysis of the equilibria and the existence of a Hopf bifurcation, which have not been out until recently by Chan and Yu [10] and Blyuss [3]. The study presented generalizes the results of [10] and [3], for our approach introduces additional constants that capture the topology of the strain spaces and are applicable to multiple strain systems beyond four strains.…”

Section: Introductionmentioning

confidence: 99%

“…Recker and Gupta [40] have also considered a variant model of [27] and studied the stability of the partially synchronous and fully synchronous steady states via an ε-fluctuation method and numerically showed the existence of cyclic dynamics. More recently, Chan and Yu [10] and Blyuss [3] used the groupoid formalism of [23] and center-manifold reduction, and an equivariant dynamical system approach to study synchronous clustering and symmetric dynamics, respectively.…”

Section: Introductionmentioning

confidence: 99%

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Mathematical analysis of a multiple strain, multi-locus-allele system for antigenically variable infectious diseases revisited

Cherif

2015

Mathematical Biosciences

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Many important pathogens such as HIV/AIDS, influenza, malaria, dengue and meningitis generally exist in phenotypically distinct serotypes that compete for hosts. Models used to study these diseases appear as meta-population systems. Herein, we revisit one of the multiple strain models that have been used to investigate the dynamics of infectious diseases with co-circulating serotypes or strains, and provide analytical results underlying the numerical investigations. In particular, we establish the necessary conditions for the local asymptotic stability of the steady states and for the existence of oscillatory behaviors via Hopf bifurcation. In addition, we show that the existence of discrete antigenic forms among pathogens can either fully or partially self-organize, where (i) strains exhibit no strain structures and coexist or (ii) antigenic variants sort into non-overlapping or minimally overlapping clusters that either undergo the principle of competitive exclusion exhibiting discrete strain structures, or co-exist cyclically.

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Section: Strong Endemicity: No Strain Structurementioning

confidence: 80%

Section: Introductionmentioning

confidence: 76%

Section: Model Description: Multi-locus-allele Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Mathematical analysis of a multiple strain, multi-locus-allele system for antigenically variable infectious diseases revisited

Cherif

2015

Mathematical Biosciences

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An age-structured multi-strain epidemic model for antigenically diverse infectious diseases: A multi-locus framework

Cherif

Dyson

Maini

et al. 2017

Nonlinear Analysis: Real World Applications

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The establishment of cross-protective responses and development of immunity within a host exert pressure on pathogens through cross-immunity mediated competition between antigenic forms. In this paper, we incorporate age-specificity in the multi-locus epidemic model used to study the pathogen-specific dynamic behaviors for infectious diseases with diverse co-circulating antigenic types. We establish results on the existence of a unique mild solution, and on the necessary conditions for local stability of the steady-states. In particular, we find that, when the reproductive number R 0 < 1, all strains go to extinction. When R 0 > 1, we show that there exist additional conditions which determine the stability of different types of endemic equilibria, namely weak and strong endemicity, where the weak endemic equilibria correspond to the existence of principle of competitive exclusions of pathogen-specific clusters, while strong endemicity represents the coexistence of all strains. Using numerical simulations, we also show that weak endemic equilibria yield dynamic features in which only one of the clusters containing discrete strain structures (e.g., of minimally, or non-overlapping antigenic types) persists while others go to extinction. For unique strong endemicity, we observe no strain structure, where antigenic types co-exist or exhibit cyclical strain structure with diverse dynamical behaviors (e.g., (quasi-)periodicity, intermittency, chaos). This demonstrates that pathogenic-specific dynamic features are ubiquitous and shows how cross-immunity between antigenic variants shape the maintenance and evolution of strain structures.

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Symmetry, Hopf bifurcation, and the emergence of cluster solutions in time delayed neural networks

Wang

Campbell

2017

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We consider the networks of N identical oscillators with time delayed, global circulant coupling, modeled by a system of delay differential equations with Z symmetry. We first study the existence of Hopf bifurcations induced by the coupling time delay and then use symmetric Hopf bifurcation theory to determine how these bifurcations lead to different patterns of symmetric cluster oscillations. We apply our results to a case study: a network of FitzHugh-Nagumo neurons with diffusive coupling. For this model, we derive the asymptotic stability, global asymptotic stability, absolute instability, and stability switches of the equilibrium point in the plane of coupling time delay (τ) and excitability parameter (a). We investigate the patterns of cluster oscillations induced by the time delay and determine the direction and stability of the bifurcating periodic orbits by employing the multiple timescales method and normal form theory. We find that in the region where stability switching occurs, the dynamics of the system can be switched from the equilibrium point to any symmetric cluster oscillation, and back to equilibrium point as the time delay is increased.

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Analysis of symmetries in models of multi-strain infections

Cited by 5 publications

References 64 publications

Mathematical analysis of a multiple strain, multi-locus-allele system for antigenically variable infectious diseases revisited

Mathematical analysis of a multiple strain, multi-locus-allele system for antigenically variable infectious diseases revisited

An age-structured multi-strain epidemic model for antigenically diverse infectious diseases: A multi-locus framework

Symmetry, Hopf bifurcation, and the emergence of cluster solutions in time delayed neural networks

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