Bifurcation analysis of a family of multi-strain epidemiology models

Abstract

In this paper we use bifurcation theory to analyze three multi-strain compartmental models, motivated by dengue fever epidemiology. The models are extensions of the classical SIR model where the notion of two different strains is needed to describe primary and secondary infections. Ferguson et al. formulate and analyzed in Ferguson et al. (1999) [1] a two-strain model with Antibody Dependent Enhancement (ade), where the pre-existing antibodies to previous dengue infection cannot neutralize but rather enhance the new infection. No cross-immunity is assumed and therefore co-infection is possible and moreover individuals can belong to multiple compartments. In the modeling approach proposed by Billings et al. in (2007) [2] cross-infection is not possible as long as an individual is primarily infected. As a result all compartments are distinct. In the third model Aguiar et al. (2011) [9] starting from the Billings et al. model, temporary cross-immunity is introduced by additional compartments for the recovered from the primary infection. The study of the different models gives insight into how different long-term dynamic behavior originate. We use the same parameter set for all models, except the duration of the cross-immunity period in the last model which will be a bifurcation parameter. Another bifurcation parameter is the ade factor. Besides endemic equilibria and periodic solutions there is also chaotic behavior originating via different routes. Numerical bifurcation analysis, Lyapunov exponent calculation and simulation techniques (quantitative results) as well as symbolic analysis (qualitative results) are used to unravel the different types of long-term dynamics. When the cross-immunity period is moderately long or short the predictions show, besides similar chaotic dynamics as in the two models without temporal cross-immunity, a new chaotic attractor with a different origin.