Rinaldo B. Schinazi

Branching random walks on trees

Let p(x, y) be the transition probability of an isotropic random walk on a tree, where each site has d [greater-or-equal, slanted]3 neighbors. We define a branching random walk by letting a particle at site x give birth to a new particle at site y at rate [lambda]dp(x, y), jump to y at rate vdp(x, y), and die at rate [delta]. Let [lambda]2 (respectively, [mu]2) be the infimum of [lambda] such that the process starting with one particle has positive probability of surviving forever (respectively, of having a fixed site occupied at arbitrarily large times). We compute [lambda]2 and [mu]2 exactly, proving that [lambda]2

THE CRITICAL CONTACT PROCESS ON A HOMOGENEOUS TREE

We prove that the expected number of particles of the critical contact process on a homogeneous tree is bounded above. This is the first graph for which the behavior of the expected number of particles of the critical contact process is known. As an easy corollary of our result we get that the critical contact process dies out on any

A Cellular Automaton

Cellular automata are widely used in mathematical physics and in theoretical biology. These systems start from a random state and evolve using deterministic rules. We concentrate on a specific model in this chapter. The techniques we use are similar to the ones used in percolation.

A COMPLETE CONVERGENCE THEOREM FOR AN EPIDEMIC MODEL

We use an interacting particle system on Z to model an epidemic. Each site of Z can be in either one of three states: empty, healthy or infected. An empty site x gets occupied by a healthy individual at a rate βn 1 (x) where n 1 (x) is the number of healthy nearest neighbors of x. A healthy individual at x gets infected at rate αn 2 (x) where n 2 (x) is the number of infected nearest neighbors of x. An infected individual dies at rate δ independently of everything else. We show that for all α, β and δ > 0 and all initial configurations, all the sites of fixed finite set remain either all empty or all healthy after, an almost surely finite time. Moreover, if the initial configuration has infinitely many healthy individuals then the process converges almost surely (in the sense described above) to the all healthy state. We also consider a model introduced by Durrett and Neuhauser where healthy individuals appear spontaneously at rate β > 0 and for which coexistence of 1s and 2s was proved in dimension 2 for some values of α and β. We prove that coexistence may occur in any dimension.

A stochastic model for cancer risk

We propose a simple stochastic model based on the two successive mutations hypothesis to compute cancer risks. Assume that only stem cells are susceptible to the first mutation and that there are a total of D stem cell divisions over the lifetime of the tissue with a first mutation probability μ1 per division. Our model predicts that cancer risk will be low if m = μ1D is low even in the case of very advantageous mutations. Moreover, if μ1D is low the mutation probability of the second mutation is practically irrelevant to the cancer risk. These results are in contrast with existing models but in agreement with a conjecture of Cairns. In the case where m is large our model predicts that the cancer risk depends crucially on whether the first mutation is advantageous or not. A disadvantageous or neutral mutation makes the risk of cancer drop dramatically.

On the role of reinfection in the transmission of infectious diseases

We introduce a spatial stochastic model for the spread of tuberculosis. After a primary infection, an individual may become sick (and infectious) through an endogenous reinfection or through an exogenous reinfection. We show that even in the absence of endogenous reinfection an epidemic is possible if the exogenous reinfection parameter is high enough. This is in sharp contrast with what happens for a mean field model corresponding to our spatial stochastic model.

A Spatial Stochastic Model for Rumor Transmission

We consider an interacting particle system representing the spread of a rumor by agents on the d-dimensional integer lattice. Each agent may be in any of the three states belonging to the set {0,1,2}. Here 0 stands for ignorants, 1 for spreaders and 2 for stiflers. A spreader tells the rumor to any of its (nearest) ignorant neighbors at rate λ. At rate α a spreader becomes a stifler due to the action of other (nearest neighbor) spreaders. Finally, spreaders and stiflers forget the rumor at rate one. We study sufficient conditions under which the rumor either becomes extinct or survives with positive probability.

On the Spread of Drug-Resistant Diseases

We introduce an interacting particle system to model the emergence of drug-resistant diseases, one of the most serious health problems in modern society. We are interested in diseases for which a natural strain may mutate into a drug-resistant strain. This happens, for instance, when antibiotics are misused. The main result of our analysis is that with an efficient drug against the natural strain, if there is even a small chance that the natural strain mutates into the drug-resistant one, then there will eventually be an outbreak of the drug-resistant strain throughout the population. In that case the natural strain disappears and is replaced by the drug-resistant strain. The disturbing part of this is that an efficient treatment of the natural strain gives an edge to the drug-resistant strain.

On an interacting particle system modeling an epidemic

We consider an interacting particle system onZdto model an epidemic. Each site ofZdcan be in either one of three states: empty, healthy or infected. Healthy and infected individuals give birth at different rates to healthy individuals on empty sites. Healthy individuals get infected by infected individuals. Infected and healthy individuals die at different rates. We prove that in dimension 1 and with nearest-neighbor interactions the epidemic may persist forever if and only if the rate at which infected individuals give birth to healthy individuals is high enough. This is in sharp contrast with models analysed by Andjel and Schinazi (1994) and Sato et al. (1994) where infected individuals do not give birth. We also show that some results in the latter reference can be obtained easily and rigorously using probabilistic coupling to the contact process.

Survival and coexistence for a multitype contact process

We study the ergodic theory of a multitype contact process with equal death rates and unequal birth rates on the d-dimensional integer lattice and regular trees. We prove that for birth rates in a certain interval there is coexistence on the tree, which by a result of Neuhauser is not possible on the lattice. We also prove a complete convergence result when the larger birth rate falls outside of this interval.