# American Institute of Mathematical Sciences

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November  2008, 7(6): 1483-1496. doi: 10.3934/cpaa.2008.7.1483

## The threshold for persistence of parasites with multiple infections

 1 Dipartimento di Matematica, Università di Trento, Via Sommarive 14, Povo (TN), 38100, Italy, Italy

Received  September 2007 Revised  February 2008 Published  August 2008

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.
Citation: M. P. Moschen, A. Pugliese. The threshold for persistence of parasites with multiple infections. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1483-1496. doi: 10.3934/cpaa.2008.7.1483
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