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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


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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