# American Institute of Mathematical Sciences

2010, 7(3): 579-602. doi: 10.3934/mbe.2010.7.579

## On the interaction between the immune system and an exponentially replicating pathogen

 1 European Institute of Oncology, Division of Epidemiology and Biostatistics, Via Ripamonti 435, 20141 Milano

Received  July 2009 Revised  October 2009 Published  June 2010

In this work, we generalize the Pugliese-Gandolfi Model [A. Pugliese and A. Gandolfi, Math Biosc, 214,73 (2008)] of interaction between an exponentially replicating pathogen and the immune system. After the generalization, we study the properties of boundedness and unboundedness of the solutions, and we also give a condition for the global eradication as well as for the onset of sustained oscillations. Then, we study the condition for the uniqueness of the arising limit cycle, with numerical applications to the Pugliese-Gandolfi model. By means of simulations, we also show some alternative ways to reaching the elimination of the pathogen and interesting effects linked to variations in aspecific immune response. After shortly studying some pathological cases of interest, we include in our model distributed and constant delays and we show that also delays may unstabilize the equilibria.
Citation: Alberto d'Onofrio. On the interaction between the immune system and an exponentially replicating pathogen. Mathematical Biosciences & Engineering, 2010, 7 (3) : 579-602. doi: 10.3934/mbe.2010.7.579
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