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A contact process with mutations on a tree by Thomas M Liggett Rinaldo B Schinazi Jason Schweinsberg University of California at Los Angeles University of Colorado at Colorado Springs and University of California at San Diego December 18 2006 Abstract Consider the following stochastic model for immune response Each pathogen gives birth to a new pathogen at rate When a new pathogen is born it has the same type as its parent with probability 1 r With probability r a mutation occurs and the new pathogen has a different type from all previously observed pathogens When a new type appears in the population it survives for an exponential amount of time with mean 1 independently of all the other types All pathogens of that type are killed simultaneously Schinazi and d Schweinsberg 2006 have shown that this model on Z behaves rather differently from its non spatial version In this paper we show that this model on a homogeneous tree captures d features from both the non spatial version and the Z version We also obtain comparison results between this model and the basic contact process 1 Introduction Schinazi and Schweinsberg 2006 have recently introduced a non spatial and a spatial version of the following stochastic model for immune response For the non spatial version Model 2 in their paper each pathogen gives birth to a new pathogen at rate When a new pathogen is born it has the same type as its parent with probability 1 r With probability r a mutation occurs and the new pathogen has a different type from all previously observed pathogens When a new type appears in the population it survives for an exponential amount of time with mean 1 independently of all the other types All pathogens of that type are killed simultaneously For the spatial version Model S2 in Schinazi and Schweinsberg 2006 let S be the square lattice Zd or the homogeneous tree Td in which every vertex has d 1 neighbors Let x be a site in S occupied by a pathogen and y be one of its nearest neighbors There are 2d such neighbors in Zd and d 1 in Td After a random exponential time with rate the pathogen on x gives Partially supported by NSF grant DMS 0301795 Partially supported by NSF grant DMS 0504882 Key words and phrases mutation immune system branching process spatial stochastic model contact process 2000 Mathematics Subject Classification 60K35 1 birth on y provided y is empty if y is occupied nothing happens With probability 1 r the new pathogen on y is of the same type as the parent pathogen on x With probability r the new pathogen is of a different type We assume that every new type that appears is different from all types that have ever appeared As in the non spatial model we assume that each type survives for an exponential amount of time with mean 1 independently of all other types and that all pathogens of a type are killed simultaneously Biological motivation has been provided in Schinazi and Schweinsberg 2006 There is also some obvious motivation for such a model for the spread of a virus in computer networks In particular when a virus is discovered in a computer network it is usually destroyed at once It is also interesting to test the influence of particular network topologies on the spread of a virus see for instance Pastor Sattoras and Vespignani 2001 Berger et al 2005 Ganesh et al 2005 and Draief et al 2006 Schinazi and Schweinsberg 2006 have shown that the pathogens survive in the non spatial model with positive probability if and only if 1 and r 0 see their Theorem 1 2 In contrast for the spatial model on Zd they have shown that if is large enough then the pathogens survive with positive probability for large r and die out with probability 1 for small r see their Theorem 4 Given that a ball of radius R has of the order of dR sites in the homogeneous tree Td one might conjecture that the behavior of the spatial model on Td is similar to that of the non spatial model there are so many sites on the tree that space may not be a limitation As the reader will see in the next section this is not so In fact the model on the tree captures both features from the model on Zd and from the non spatial model These results will be stated in Section 2 and proved in Sections 4 and 5 The basic contact process may be thought of as being a particular case of this model with r 1 See Liggett 1999 for background and results on the contact process For r 1 all pathogens are of different types and therefore only one pathogen dies at a time Hence one might conjecture that the model with r 1 has a better chance of surviving than any model with r 1 This turns out to be true but requires nontrivial arguments This will be done in Section 3 This comparison result applies to any graph 2 Phase transitions We say that the pathogens survive on a graph S if there is a positive probability that at all times there is at least one pathogen somewhere in S If the pathogens do not survive they are said to die out Theorem 1 Consider the contact process with mutations on the homogeneous tree Td with d 2 started with a single pathogen of type 1 1 If 1 d 1 then the pathogens survive for all r 0 1 1 2 If d 1 2r then the pathogens die out In particular if d 1 the pathogens die out for all r 0 1 d d 1 7 9d 1 3 If the pathogens die out for r close to 0 and survive for r close d 1 2 d2 1 to 1 2 These results will be proved in Sections 4 and 5 Note that for all d 2 the left hand side of the inequality in Theorem 1 3 is strictly smaller than the right hand side Therefore Theorem 1 3 shows that for intermediate values of there is a phase transition in r Moreover the difference between the right and left sides of Theorem 1 3 is asymptotic to 3d22 as d Combining Theorem 1 1 and 1 2 we see that the set of s for which there is a phase transition in r is of size asymptotically at most d22 So up to a factor of three we have found the correct size of this set We now turn to another type of phase transition Let At be the set of sites that are occupied by any pathogen The process is started with a single pathogen at the site x Recall that pathogens are said to survive if P At 6 t 0 0 1 The pathogens are said to survive weakly if 1 holds and P x At for sufficiently large t 1 2 That is with positive probability there are pathogens somewhere in S for all times but almost surely a given …