Bifurcation, stability, and cluster formation of multi-strain infection models

Abstract

Clustering behaviours have been found in numerous multi-strain transmission models. Numerical solutions of these models have shown that steady-states, periodic, or even chaotic motions can be self-organized into clusters. Such clustering behaviours are not a priori expected. It has been proposed that the cross-protection from multiple strains of pathogens is responsible for the clustering phenomenon. In this paper, we show that the steady-state clusterings in existing models can be analytically predicted. The clusterings occur via semi-simple double zero bifurcation from the quotient networks of the models and the patterns which follow can be predicted through the stability analysis of the bifurcation. We calculate the stability criteria for the clustering patterns and show that some patterns are inherently unstable. Finally, the biological implications of these results are discussed.

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

“…ρ p = 0 and ρ ∅ = 0 because they violate the necessary stability condition in Theorem 4.3. These observations are consistent with numerical simulations presented herein and in [1,7,10,14,19,26,27,39,40] and [41].…”

“…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.…”

“…That is, in the particular 2-locus-2-allele case where the number of discordant strains is ρ = 2, the threshold condition for the local stability of DSS is given by γ > 1 − 1/2R 0 which is equivalent to the condition given by Gupta et al [26,27]. When R 0 = 1, Chan and Yu [10] showed, using quotient network and central manifold reduction, that discrete strain structures consisting of discordant sets are stable provided γ ρ > 1, where ρ was given in terms of the quotient network parameters (i.e., difference between the number of each strain in D p sharing alleles with strains in D ∅ and number of strains sharing alleles in the same set D p ). This is not surprising since there is a natural pairing of strains.…”

“…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.…”

“…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.…”