Analysis of an asymmetric two-strain dengue model

Abstract

In this paper we analyse a two-strain compartmental dengue fever model that allows us to study the behaviour of a Dengue fever epidemic. Dengue fever is the most common mosquito-borne viral disease of humans that in recent years has become a major international public health concern. The model is an extension of the classical compartmental susceptible-infected-recovered (SIR) model where the exchange between the compartments is described by ordinary differential equations (ode). Two-strains of the virus exist so that a primary infection with one strain and secondary infection by the other strain can occur. There is life-long immunity to the primary infection strain, temporary cross-immunity and after the secondary infection followed by life-long immunity, to the secondary infection strains. Newborns are assumed susceptible. Antibody Dependent Enhancement (ade) is a mechanism where the pre-existing antibodies to the previous dengue infection do not neutralize but rather enhance replication of the secondary strain. In the previously studied models the two strains are identical with respect to their epidemiological functioning: that is the epidemiological process parameters of the two strains were assumed equal. As a result the mathematical model possesses a mathematical symmetry property. In this manuscript we study a variant with epidemiological asymmetry between the strains: the force of infection rates differ while all other epidemiological parameters are equal. Comparison with the results for the epidemiologically symmetric model gives insight into its robustness. Numerical bifurcation analysis and simulation techniques including Lyapunov exponent calculation will be used to study the long-term dynamical behaviour of the model. For the single strain system stable endemic equilibria exist and for the two-strain system endemic equilibria, periodic solutions and also chaotic behaviour.

Keywords: Antibody Dependent Enhancement; Asymmetric two-strain model; Bifurcation analysis; Dengue fever; Lyapunov exponents; Temporary cross-immunity.