The threshold for persistence of parasites with multiple infections

We analyse a model for macro-parasites in an age-structured host population, with infections of hosts occurring in clumps of parasites. The resulting model is an infinite system of partial differential equations of the first order, with non-local boundary conditions. We establish a condition for the parasite-free equilibrium to be asymptotically stable, in terms of R < 1, where R is a quantity interpreted as the reproduction number of parasites. To show this, we prove that s(B − A) < 0 [> 0] if and only if ρ(B(A)−1) < 1 [> 1], where B is a positive operator, and A generates a positive semigroup of negative type. Finally, we discuss how R depends on the parameters of the system, especially on the mean size of infecting clumps.