### 数学代写|微分方程代写differential equation代考|MAST20030

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|微分方程代写differential equation代考|Multiple Species

This method of simulation and analysis generalizes readily to the situation where there are multiple species and multiple reactions. At any given time, the state vector is the vector of integers $\mathcal{S}=\left(n_{1}, n_{2}, \ldots, n_{K}\right)$, and there are $J$ reactions with rates $r_{j}$ that depend on the state of the system $\mathcal{S}$. For each reaction there is a change in the state vector $c(j, k)$, meaning that if reaction $j$ occurs, the $k$ th integer $n_{k}$ changes by the amount $c(j, k)$.
As an example, consider the SIR reactions
$$S+I \stackrel{\alpha}{\longrightarrow} 2 I, \quad I \stackrel{\beta}{\longrightarrow} R .$$
Here $S$ represents susceptible individuals in a population, $I$ represents the infected and contagious individuals, and $R$ represents those individuals who are removed and no longer contagious. The deterministic differential equations for these reactions are given by (1.63), however, as we all know from experience with COVID-19, the evolution of an epidemic is highly stochastic.

An interesting question to ask is how many individuals have been infected and how many susceptibles remain (or survive) after an infection has run its course, and we can address this question using a stochastic simulation. The setup for this stochastic simulation is straightforward. The state space is identified by the three integers $n_{s}, n_{i}$, and $n_{r}$, and the two reactions are at rates
$$r_{1}=\alpha n_{s} n_{i}, \quad r_{2}=\beta n_{i},$$
and the change matrix $C=c(j, k)$ is
$$C=\left(\begin{array}{ccc} -1 & 1 & 0 \ 0 & -1 & 1 \end{array}\right)$$
This is easily implemented in Matlab code and in fact, the code that does this is titled stochastic_SIR.m.

Scatter plots of recovery times vs. number of survivors for the SIR stochastic process shown in Figure $1.10$ are surprising, and are certainly different than what is predicted by the deterministic model. (Recovery time refers to the first time at which there are no more infected individuals.) The deterministic model predicts a unique outcome (recall (1.67)), with an epidemic spreading if $R_{0}=\frac{\alpha s(0)}{\beta}>1$ and not spreading if $R_{0}=\frac{\alpha s(0)}{\beta}<1$. However, in Figure $1.10(\mathrm{a})$, where $R_{0}=2.5$, the results of the stochastic simulation show a biphasic outcome, with many of the trials, as expected, having a large epidemic with few survivors and long recovery times, but also with a significant number of trials with little spread of the infection, a large percentage of survivors, and a short recovery time. Similarly, in Figure $1.10(\mathrm{~b})$, where $R_{0}=0.9$, most of the trials result in a short-lived epidemic with a high percentage of survivors. However, there are nonetheless quite a few trials showing a substantial epidemic with few survivors and long recovery times, noticeably different than the prediction of the deterministic model.

## 数学代写|微分方程代写differential equation代考|The Conservation Law

The purpose of this text can be summarized as learning how to count biological objects as they change over time. Demographers do this by taking a census of their population of interest from time to time and then making a plot of the pointwise values and connecting the points with lines. This approach is of limited value because it gives no explanation, or mechanism, for the observed changes, and it has no predictive value. Even if data points are fit to a regression curve, there is no confidence that the fit curve can be extrapolated to values outside the range of times for which data was collected.
The approach taken here is to recognize that for any quantity of some material with density $u$ (i.e., number per unit volume) which is changing in time, it must be that the total amount of the material in some region of space can change only because of flux (i.e., movement) across the boundary or production/destruction in the interior of the region. In mathematical language, this can be stated as
$$\frac{d}{d t} \int_{\Omega} u d V=-\int_{\partial \Omega} \mathbf{J} \cdot \mathbf{n} d S+\int_{\Omega} f d V$$
where $\Omega$ is a closed region in space, $\partial \Omega$ is its boundary surface, and $\mathbf{n}$ is the outward unit normal to the boundary of $\Omega$. Here, $f$ is the rate of production (or destruction, if $f$ is negative) of $u$. Since the units of the terms on the left and right hand side of this equation must match, $f$ must have units of $u$ /time. Since $d V$ has units of volume, and $d S$ has units of area, the flux $\mathbf{J}$ is the vector-valued quantity with units of $u$ times length/time, i.e., units of $u$ times velocity, which is the same as number per unit area per time. ${ }^{1}$ The minus sign here is to recognize that if $\mathbf{J} \cdot \mathbf{n}$ is positive, then the flux of material is outward across the boundary, hence decreasing the amount of material in the domain.

## 数学代写|微分方程代写differential equation代考|Examples of Flux—How Things Move

There are several examples of flux that are important in biology.
Advection. Suppose particles with concentration $u$ are dissolved in water and the water is moving with velocity $\mathbf{v}$ and that the dissolved particles are moving with the same velocity. The flux of concentration at any point is the velocity of the water times the concentration
$$\mathbf{J}=\mathbf{v} u$$
This flux is a pointwise object having units of concentration times velocity. If this is constant in a tube like a vein or artery. with crossectional area $A$, then the flow in the tube is given by
$$Q=A \mathbf{J}=A \mathbf{v} u,$$
which has units of volume times concentration per unit time $=$ number of particles per unit time. This formula will be useful for Exercise 2.3.

Fick’s law. If individual particles have a velocity that is different than that of the water in which they are dissolved, for example, a random motion, then we might reasonably expect that they would tend to spread out, by moving, on average, down their concentration gradient. This is certainly what happens in our ordinary experience. For example, if you put a drop of ink into water, it will very quickly disperse, or diffuse, away, and eventually the ink will be uniformly distributed throughout the water, with no regions with higher or lower concentration. In math language, this is stated as
$$\mathbf{J}=-D \nabla u,$$
and is called Fick’s law, and $D$ is called the diffusion coefficient. Notice that $D$ must have units of (length) ${ }^{2}$ /time, since the flux must have units of velocity times units of $u$.
Fick’s law is not truly a law, but a model, hence appropriate in certain contexts. For example, it applies if the particles are diffuse with no self-interactions, but not so few that $u$ cannot be viewed as a continuous variable.

## 数学代写|微分方程代写differential equation代考|Multiple Species

r1=一个nsn一世,r2=bn一世,

C=(−110 0−11)

SIR 随机过程的恢复时间与幸存者数量的散点图如图所示1.10令人惊讶，并且肯定与确定性模型所预测的不同。（恢复时间是指第一次没有更多受感染的个体。）确定性模型预测一个独特的结果（回忆（1.67）），如果流行病传播，R0=一个s(0)b>1如果不传播R0=一个s(0)b<1. 然而，在图1.10(一个)， 在哪里R0=2.5，随机模拟的结果显示出双相结果，正如预期的那样，许多试验具有大规模流行病，幸存者很少，恢复时间长，但也有大量试验，感染传播很少，大存活率高，恢复时间短。同样，在图1.10( b)， 在哪里R0=0.9，大多数试验导致一种短暂的流行病，幸存者比例很高。然而，仍有相当多的试验表明，大规模流行病几乎没有幸存者，恢复时间很长，这与确定性模型的预测明显不同。

## 数学代写|微分方程代写differential equation代考|The Conservation Law

dd吨∫Ω在d在=−∫∂ΩĴ⋅nd小号+∫ΩFd在

Ĵ=在在

Ĵ=−D∇在,

## 有限元方法代写

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## MATLAB代写

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