Thresholds for macroparasite infections

Abstract

Abstract. We analyse here the equilibria of an infinite system of partial differential equations modelling the dynamics of a population infected by macroparasites. We find that it is possible to define a reproduction number R 0 that satisfies the intuitive definition, and yields a sharp threshold in the behaviour of the system: when R 0 < 1, the parasite-free equilibrium (PFE) is asymptotically stable and there are no endemic equilibria; when R 0 > 1, the PFE is unstable and there exists a unique endemic equilibrium. The results mainly confirm what had been obtained in simplified models, except for the fact that no backwards bifurcation occur in this model. The stability of equilibria is established by extending an abstract linearization principle and by analysing the spectra of appropriate operators.

The threshold for persistence of parasites with multiple infections

“…We showed in [23] that including hosts' age in the model (as in [17]) does not really introduce big complications, and indeed makes many expressions more transparent. Thus, we rewrite the model for an age-structured host population, allowing for age-dependent host fertility and mortality, and, in order to have a parasite-free equilibrium, for density-dependent host birth rate.…”

“…In Section 2 we present the model, set it in abstract form, and, summarizing some results of [22,23], show (Proposition 2) that the parasite-free-equilibrium is asymptotically stable or unstable, according to whether s(B+A) is negative or positive, where B and A are suitable operators, and s represents the spectral bound [28]. In Section 3, we prove, in an infinite-dimensional setting, a technical lemma on the spectrum of positive operators, well known in finite dimensions [7]: i.…”

“…The variables of the model are p i (t) for i ≥ 0, the density of hosts carrying i parasites at time t. One can write differential equations for these variables, by taking into account the new infections, the deaths of adult parasites, as well as hosts' births and deaths. The method is explained, for instance, in [23]; the main difference lies in the infection mechanism.…”

“…Such systems have proved very difficult to analyse (see, however, [10,17,18,22,23,20]) and much of the analysis, including the formulation of thresholds for parasite persistence in terms of R 0 , has been performed using simplified models consisting of few ordinary differential equations.…”

“…A rigorous treatment of the threshold condition for parasite persistence, written as R 0 > 1, has been carried out, using semigroup methods, in [23] (see also [20] for extensions of the idea). The computations performed in [23], while providing the general setting used here, did not yield an explicit expression for the case considered here.…”