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We construct graphs (trees of bounded degree) on which the contact process has critical rate (which will be the same for both global and local survival) equal to any prescribed value between zero and $lambda_c(mathbb{Z})$, the critical rate of the one-dimensional contact process. We exhibit both graphs in which the process at this target critical value survives (locally) and graphs where it dies out (globally).
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We are interested in the spread of an epidemic between two communities that have higher connectivity within than between them. We model the two communities as independent Erdos-Renyi random graphs, each with n vertices and edge probability p = n^{a-1} (0<a<1), then add a small set of bridge edges, B, between the communities. We model the epidemic on this network as a contact process (Susceptible-Infected-Susceptible infection) with infection rate lambda and recovery rate 1. If nplambda = b > 1 then the contact process on the Erdos-Renyi random graph is supercritical, and we show that it survives for exponentially long. Further, let tau be the time to infect a positive fraction of vertices in the second community when the infection starts from a single vertex in the first community. We show that on the event that the contact process survives exponentially long, tau |B|/(np) converges in distribution to an exponential random variable with a specified rate. These results generalize to a graph with N communities.
We consider the extinction time of the contact process on increasing sequences of finite graphs obtained from a variety of random graph models. Under the assumption that the infection rate is above the critical value for the process on the integer line, in each case we prove that the logarithm of the extinction time divided by the size of the graph converges in probability to a (model-dependent) positive constant. The graphs we treat include various percolation models on increasing boxes of Z d or R d in their supercritical or percolative regimes (Bernoulli bond and site percolation, the occupied and vacant sets of random interlacements, excursion sets of the Gaussian free field, random geometric graphs) as well as supercritical Galton-Watson trees grown up to finite generations.
We consider the contact process on the model of hyperbolic random graph, in the regime when the degree distribution obeys a power law with exponent $chi in(1,2)$ (so that the degree distribution has finite mean and infinite second moment). We show that the probability of non-extinction as the rate of infection goes to zero decays as a power law with an exponent that only depends on $chi$ and which is the same as in the configuration model, suggesting some universality of this critical exponent. We also consider fini
In this paper we give an improved upper bound for critical value $lambda_c$ of the basic contact process on the lattice $mathbb{Z}^d$ with $dgeq 3$. As a direct corollary of out result, [ lambda_cleq 0.384. ] when $d=3$.
We consider a contact process on $Z^d$ with two species that interact in a symbiotic manner. Each site can either be vacant or occupied by individuals of species $A$ and/or $B$. Multiple occupancy by the same species at a single site is prohibited. The name symbiotic comes from the fact that if only one species is present at a site then that particle dies with rate 1 but if both species are present then the death rate is reduced to $mu le 1$ for each particle at that site. We show the critical birth rate $lambda_c(mu)$ for weak survival is of order $sqrt{mu}$ as $mu to 0$. Mean-field calculations predict that when $mu < 1/2$ there is a discontinuous transition as $lambda$ is varied. In contrast, we show that, in any dimension, the phase transition is continuous. To be fair to physicists the paper that introduced the model, the authors say that the symbiotic contact process is in the directed percolation universality class and hence has a continuous transition. However, a 2018 paper asserts that the transition is discontinuous above the upper critical dimension, which is 4 for oriented percolation.
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