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# 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 $R_0 < 1$, where $R_0$ 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 $\rho(B(A)^{-1} )< 1$ [$>1$], where $B$ is a positive operator, and $A$ generates a positive semigroup of negative type. Finally, we discuss how $R_0$ depends on the parameters of the system, especially on the mean size of infecting clumps.
Mathematics Subject Classification: Primary: 92D30; Secondary: 35L60, 47B65.

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