Dynamical properties and thresholds of an HIV model with super-infection

Super-infection by multiple HIV-1 subtypes, previously thought restricted to high risk groups, has now been reported in the general heterosexual populations at relatively the same incidence rate as in high risk groups. We present a simple deterministic HIV model with super-infection by two HIV-1 subtypes. Mathematical characteristics including the basic reproductive number $(\mathcal{R}_0)$, invasion threshold $(\mathcal{R}_{21},\mathcal{R}_{12})$ and conditions for asymptotic stability are derived. In the absence of super-infection the model exhibits competitive exclusion, and all equilibria are globally attracting if they exist except for the disease free which is a saddle for $\mathcal{R}_0>1.$ The results show that the subtype with the dominant reproductive number exceeding unity dominates the weaker subtype forcing it to extinction regardless of the size of the reproductive number. On the other end, super-infection may promote subtype co-existence whenever the minimum of the subtype specific reproductive numbers $(\mathcal{R}_1,\mathcal{R}_2)$ and the invasion reproductive numbers $(\mathcal{R}_{12},\mathcal{R}_{21})$ exceed unity. Our results demonstrate that if the partial reproductive numbers $(\mathcal{R}_1~\mbox{and}~\mathcal{R}_2 )$ and the invasion reproductive number for the weaker subtype $(\mathcal{R}_{21})$ satisfy $\mathcal{R}_2<1,~\mathcal{R}_1>1~\mbox{and}~\mathcal{R}_{21}>1,$ then primary infection by subtype $1$ may stay the extinction of subtype $2$ despite its relatively low reproductive fitness. For certain parameter ranges, hysteresis (including backward bifurcation) occurs with possible differences in the asymptotic level of disease prevalence. Super-infection may thus facilitate the continued re-generation of reproductively noncompetent subtypes whose subtype specific reproductive numbers will be less than unity while at the same time allowing for the mutual coexistence and persistence of multiple strains. Persistence and co-existence of multiple strains has detrimental effect on vaccine design and development and administration of ART where one or more of the strains are drug resistant.