# 反应扩散模型和亚种群模型

## 反应扩散过程

(a) Schematic illustration of the simplified modeling framework based on the particle-network scheme. At the macroscopic level the system is composed of a heterogeneous network of subpopulations. The contagion process in one subpopulation can spread to other subpopulations because of particles diffusing across subpopulations. (b) At the microscopic level, each subpopulation contains a population of individuals. The dynamical process, for instance, a contagion phenomenon, is described by a simple compartmentalization (compartments are indicated by different colored dots in the picture). Within each subpopulation, individuals can mix homogeneously or according to a subnetwork and can diffuse with probability p from one subpopulation to another following the edges of the network. (c) A critical value $p_c$ of the individuals or particles diffusion identifies a phase transition between a regime in which the contagion affects a large fraction of the system and one in which only a small fraction is affected.

More in general models of social behavior and human mobility are often framed as reaction-diffusion processes where each node $i$ is allowed to host any nonnegative integer number of particles $\mathcal{N}(i)$, so that the total particle population of the system is $\mathcal{N}=\sum_i\mathcal{N}(i)$. This particle-network framework considers that each particle diffuses along the edges connecting nodes with a diffusion coefficient that depends on the node degree and/or other node attributes. Within each node particles may react according to different schemes characterizing the interaction dynamics of the system. A simple sketch of the particle-network framework is represented in the Figure.

In order to have an analytic description of reaction-diffusion systems in networks one has to allow the possibility of heterogeneous connectivity patterns among nodes. A first analytical approach to these systems considers the extension of the degree-based mean-field approach to reaction-diffusion systems in networks with arbitrary degree distribution. For simplicity, we first consider the DBMF approach to the case of a simple system in which noninteracting particles (individuals) diffuse on a network with arbitrary topology. A convenient representation of the system is therefore provided by quantities defined in terms of the degree k:

## 亚种群模型

The above approach can be generalized to reacting particles with different states by adding a reaction term to the above equations [3]. We now describe a generalization to this setting of the standard SIS model in discrete time, with probability per unit time $\beta$ of infection and probability $\mu$ of recovery. We consider $\mathcal{N}$ individuals diffusing in a heterogeneous network with $N$ nodes and degree distribution $P(k)$. Each node i of the network has a number $I(i)$ of infectious and $S(i)$ of susceptible individuals. The occupation numbers $I(i)$ and $S(i)$ can have any integer value, including $I(i)=S(i)=0$, that is, void nodes with no individuals. This modeling scheme describes spatially structured interacting subpopulations, such as city locations, urban areas, or defined geographical regions [4][5] and is usually referred to as the metapopulation approach. Each node of the network represents a subpopulation and the compartment dynamics accounts for the possibility that individuals in the same location may get into contact and change their state according to the infection dynamics. The interaction among subpopulations is the result of the movement of individuals from one subpopulation to the other. We have thus to associate with each individual’s class a diffusion probability $p_I$ and $p_S$ that indicates the probability for any individual to leave its node and move to a neighboring node of the network. In general the diffusion probabilities are heterogeneous and can be node dependent; however, for simplicity we assume that individuals diffuse with probability $p_I=p_S=1$ along any of the links departing from the node in which they are. This implies that at each time step an individual sitting on a node with degree k will diffuse into one of its nearest neighbors with probability $1/k$. In order to write the dynamical equations of the system we define the following quantities:

where the sums $\sum_{i\in\mathcal{V}(k)}$ are performed over nodes of degree $k$. These two quantities express the average number of susceptible and infectious individuals in nodes with degree $k$. Clearly, $\mathcal{N}_k=I_k+S_k$ is the average number of individuals in nodes with degree $k$. These quantities allow one to write the discrete-time equation describing the time evolution of $I_k(t)$ for each class of degree k as

## 参考文献 References

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3. Colizza V, Pastor-Satorras R, Vespignani A. Reaction–diffusion processes and metapopulation models in heterogeneous networks[J]. Nature Physics, 2007, 3(4): 276-282.
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5. Ecology, genetics and evolution of metapopulations[M]. Academic Press, 2004.