Abstract Branching processes are widely used to model the viral epidemic evolution. For more adequate investigation of viral epidemic modeling, we suggest to apply branching processes with transport of particles usually called branching random walks (BRWs). This allows to investigate not only the number of particles (infected individuals), but also their spatial spread. We consider two models of continuous-time BRWs on a multidimensional lattice in which the transport of infected individuals is described by a symmetric random walk on a multidimensional lattice whereas the processes of birth and death of infected individuals are represented by a continuous-time Bienayme–Galton–Watson processes at the lattice points (branching sources). A special attention is paid to the properties of branching random walks with one branching source on the lattice and finitely or infinitely many initial particles. We show that there exists a kind of duality between the branching random walk with a finite number of initial particles and the branching random walk with an infinite number of initial particles, which is associated with the possibility of their twofold description. The fact of duality is useful from the biological point of view. Each of the models can be considered taking into account the vaccination process. We suppose the vaccination to be a proportion of immune individuals in the population, who are resistant to disease. For simplicity, in all our BRW models, we assume that the vaccination process does not depend on time, what allows to investigate spatial properties of viral evolution.

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分支随机游走及其在流行病建模中的应用

摘要分支过程被广泛用于模拟病毒流行的演变。为了对病毒流行模型进行更充分的研究，我们建议应用通常称为分支随机游走 (BRW) 的粒子传输的分支过程。这不仅可以调查粒子（受感染的个体）的数量，还可以调查它们的空间分布。我们考虑多维格子上的两个连续时间 BRW 模型，其中受感染个体的运输由多维格子上的对称随机游走描述，而受感染个体的出生和死亡过程由连续时间 Bienayme 表示 - Galton-Watson 在晶格点（分支源）处进行处理。特别注意了在晶格上有一个分支源和有限或无限多个初始粒子的分支随机游走的特性。我们证明了初始粒子数有限的分支随机游走和初始粒子数无限的分支随机游走之间存在一种二元性，这与它们的双重描述的可能性有关。从生物学的角度来看，二元性的事实是有用的。考虑到疫苗接种过程，可以考虑每个模型。我们假设疫苗接种是人口中对疾病有抵抗力的免疫个体的一部分。为简单起见，在我们所有的 BRW 模型中，我们假设疫苗接种过程不依赖于时间，