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# 数据科学代写|复杂网络代写Complex Network代考|CS7280 Epidemic outbreaks

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## 数据科学代写|复杂网络代写Complex Network代考|Epidemic outbreaks

Figure 7.8, sketching the temporal evolution of prevalence for three basic epidemic models above their epidemic threshold $\left(\lambda_c=0\right.$ for the SI model), provides a qualitative view of epidemic outbreaks. The evolution starts from a tiny fraction of infective vertices, uniformly randomly selected in a network, and its initial stage with a yet-small prevalence is noisy (Moreno, PastorSatorras, and Vespignani, 2002; Barthélemy, Barrat, Pastor-Satorras, and Vespignani, 2005b; Pastor-Satorras, Castellano, Van Mieghem, and Vespignani, 2015). During this period, if the initial number of infective vertices is finite, the SIS model has a chance to fall into the absorbing state. The next stage of the evolution of an epidemic is an exponential growth of prevalence. This stage determines the time scale of an epidemic outbreak. During the third period of the evolution, the models exponentially (with negative exponential) converge to their final endemic states. ${ }^{16}$

Within the exponential expansion period of an epidemic, a prevalence grows as $\rho(t) \sim e^{t / \tau}$, where the characteristic time $\tau$ is related with a key number in epidemiology, a reproduction number $\mathcal{R}_0$,
$$\frac{d \ln I(t)}{d t}=\frac{1}{\tau}=\mathcal{R}_0-1$$

## 数据科学代写|复杂网络代写Complex Network代考|Epidemics in a metapopulation

A metapopulation is a complex network of coupled populations. Of numerous metapopulation models of disease spreading and various processes (Colizza, Pastor-Satorras, and Vespignani, 2007), here we touch upon the one explored by Brockmann and Helbing (2013). Let a metapopulation consist of $M$ well-mixed populations of large sizes $N_n$, where $n=1,2, \ldots, M$. Let $S_n=N_n^{\text {(susceptible) }} / N_n, I_n=N_n^{\text {(infective) }} / N_n$, and $R_n=N_n^{(\text {removed })} / N_n$ be, respectively, the fractions of susceptible, infective, and removed (recovered) individuals in population $n$, and $S_n+I_n+R_n=1$. The model treats the epidemic process within each of the populations in the spirit of the SIR model. In addition, it assumes that there is a given stationary flux of individuals (irrespective of their states) between different populations. Let $F_{m n}$ be the flux of individuals from population $n$ to population $m$, and the full set of the flows $\left{F_{m n}\right}$ be known. This set of flows between populations forms a directed weighted network. It is convenient to introduce a matrix $P$ with the entries $P_{m n}=F_{m n} / \sum_m F_{m n}$, so that $0 \leq P_{m n} \leq 1, P_{m m}=0$. That is, the entry $P_{m n}$ is the fraction of travellers that leave population $n$ and arrive at population $m$. In fact, the model is defined by the following phenomenological evolution equations for the full set of $I_n(t)$ and $S_n(t)$, $n=1,2, \ldots M$ :
\begin{aligned} & \frac{d}{d t} I_n=\beta_n S_n I_n \sigma\left(I_n / \epsilon\right)-\mu_n I_n+\gamma \sum_m P_{m n}\left(I_m-I_n\right), \ & \frac{d}{d t} S_n=-\beta_n S_n I_n \sigma\left(I_n / \epsilon\right)+\gamma \sum_m P_{m n}\left(S_m-S_n\right) \end{aligned}

## 数据科学代写|复杂网络代写Complex Network代考|Epidemic outbreaks

$$\frac{d \ln I(t)}{d t}=\frac{1}{\tau}=\mathcal{R}_0-1$$

## 数据科学代写|复杂网络代写Complex Network代考|Epidemics in a metapopulation

《left 缺少或无法识别的分隔符 被知道。人口之间的这组流动形成了一个有向加权网络。引入一个矩阵很方便 $P$ 与条目 $P_{m n}=F_{m n} / \sum_m F_{m n}$ ，以便 $0 \leq P_{m n} \leq 1, P_{m m}=0$. 也就是说， $入 \square P_{m n}$ 是离开人口的旅行者的比例 $n$ 并到达人 口 $m$. 事实上，该模型由以下现象学演化方程定义为全镸 $I_n(t)$ 和 $S_n(t), n=1,2, \ldots M$ :
$$\frac{d}{d t} I_n=\beta_n S_n I_n \sigma\left(I_n / \epsilon\right)-\mu_n I_n+\gamma \sum_m P_{m n}\left(I_m-I_n\right), \quad \frac{d}{d t} S_n=-\beta_n S_n I_n \sigma\left(I_n / \epsilon\right)+\gamma \sum_m P_{m n}\left(S_m-S_n\right)$$

## MATLAB代写

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