On global and local critical points of extended contact process on homogeneous trees.

We study spatial stochastic epidemic models called households models. The households models have more than two states at each vertex of a graph in contrast to the contact process. We show that, in the households models on trees, two thresholds of infection rates characterize epidemics. The global critical infection rate is defined by epidemic occurrence. However, some households may be eventually disease-free even for infection rates above the global critical infection rate, in as far as they are smaller than the local critical point. Whether the global one is smaller than the local one depends on the graph and the model. We show that, in the households models, the global one is smaller than the local one on homogeneous trees.

[1]  T. Liggett.  Interacting Particle Systems , 1985 .

[2]  F. Ball,et al.  Optimal vaccination policies for stochastic epidemics among a population of households. , 2002, Mathematical biosciences.

[3]  A. Sudbury.  Rigorous lower bounds for the critical infection rate in the diffusive contact process , 2001, Journal of Applied Probability.

[4]  K. Aihara,et al.  Transmission of severe acute respiratory syndrome in dynamical small-world networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  Norio Konno.  Phase Transitions of Interacting Particle Systems , 1995 .

[6]  Frank Ball,et al.  Stochastic multitype epidemics in a community of households: Estimation of threshold parameter R* and secure vaccination coverage , 2004 .

[7]  Stephen M. Krone.  The two-stage contact process , 1999 .

[8]  T. Liggett,et al.  Stochastic Interacting Systems: Contact, Voter and Exclusion Processes , 1999 .

[9]  Roberto H. Schonmann,et al.  Multiplicity of Phase Transitions and Mean-Field Criticality on Highly Non-Amenable Graphs , 2001 .

[10]  R. Schinazi,et al.  On the role of social clusters in the transmission of infectious diseases. , 2002, Theoretical population biology.

[11]  T. E. Harris.  Contact Interactions on a Lattice , 1974 .

[12]  T. Britton,et al.  Stochastic multitype epidemics in a community of households: estimation and form of optimal vaccination schemes. , 2004, Mathematical biosciences.

[13]  Norio Konno,et al.  Coexistence results for a spatial stochastic epidemic model , 2022 .

[14]  Thomas M. Liggett,et al.  Improved Upper Bounds for the Contact Process Critical Value , 1995 .

[15]  R. Durrett.  Lecture notes on particle systems and percolation , 1988 .

[16]  R. Schinazi,et al.  On the importance of risky behavior in the transmission of sexually transmitted diseases. , 2001, Mathematical biosciences.

[17]  F. Ball,et al.  Stochastic and deterministic models for SIS epidemics among a population partitioned into households. , 1999, Mathematical biosciences.

[18]  Frank Ball,et al.  Stochastic multi-type SIR epidemics among a population partitioned into households , 2001, Advances in Applied Probability.

[19]  Ronald Dickman,et al.  Time-dependent perturbation theory for diffusive non-equilibrium lattice models , 1993 .

[20]  Idan Segev,et al.  Methods in Neuronal Modeling , 1988 .

[21]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.