一维格上SIR-Bass传染病的精确描述
Exact description of SIR-Bass epidemics on 1D lattices
摘要：
本文研究了一个随机流行病模型，它是SIR模型的一个变种，我们在从易感到感染的转移率中增加了一个额外的因子，该因子考虑了由于移民或环境传染源引起的感染流入。这个因素产生了新的传染集群的形成，而不必事先明确地指定它们出现的日期和地点。我们在一维格上建立了这种随机过程的{精确确定性描述（有限线，半无限线，无限线，圆）通过证明在空间和时间中给定点的感染概率可以作为确定性ODE系统的解来获得，晶格我们的结果允许随机初始条件和参数的任意时空不均匀性，然后将我们的结果应用到一些具体的情形，得到了有用的定性结果和关于宏观动力学和每个个体的局部时间行为的显式公式.特别是，我们提供了一个精细的分析，通过研究{病人零问题}和{随时间变化的点源}的影响的集群形成的某些方面。最后，我们表明，空间-离散模型产生新的空间连续模型，这是常微分方程或偏微分方程，这取决于重新缩放制度上的参数假设。
英文摘要：
This paper is devoted to the study of a stochastic epidemiological model which is a variant of the SIR model to which we add an extra factor in the transition rate from susceptible to infected accounting for the inflow of infection due to immigration or environmental sources of infection. This factor yields the formation of new clusters of infections, without having to specify a priori and explicitly their date and place of appearance.We establish an {exact deterministic description} for such stochastic processes on 1D lattices (finite lines, semi-infinite lines, infinite lines, circles) by showing that the probability of infection at a given point in space and time can be obtained as the solution of a deterministic ODE system on the lattice. Our results allow stochastic initial conditions and arbitrary spatio-temporal heterogeneities on the parameters.We then apply our results to some concrete situations and obtain useful qualitative results and explicit formulae on the macroscopic dynamics and also the local temporal behavior of each individual. In particular, we provide a fine analysis of some aspects of cluster formation through the study of {patient-zero problems} and the effects of {time-varying point sources}.Finally, we show that the space-discrete model gives rise to new space-continuous models, which are either ODEs or PDEs, depending on the rescaling regime assumed on the parameters.
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