### 数学代写|微分方程代写differential equation代考|MATH 2003

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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|Modeling Chemical Reactions

1.2.4. Modeling Chemical Reactions. One of the important uses of differential equations, at least in this book, is to model the dynamics of chemical reactions. The two elementary reactions that are of most importance here are conversion between species, denoted
$$A \stackrel{\alpha}{\rightleftarrows} B,$$

called a first order reaction, and formation and degradation of a product from two component species, denoted
$$A+B \underset{\approx}{\rightleftarrows} C$$
called a second order reaction.
The differential equations describing the first of these are
$$\frac{d a}{d t}=\beta b-\alpha a, \quad \frac{d b}{d t}=-\beta b+\alpha a$$
where $a=[A]$ and $b=[B]$, is the statement in math symbols that $B$ is created from $A$ at rate $\alpha[A]$ and $A$ is created from $B$ at rate $\beta[B]$. Of course, the total of $A$ and $B$ is a conserved quantity, since $\frac{d}{d t}(a+b)=0$.
The second of these reactions is described by the three differential equations
$$\frac{d a}{d t}=-\gamma a b+\delta c, \quad \frac{d b}{d t}=-\gamma a b+\delta c, \quad \frac{d c}{d t}=\gamma a b-\delta c$$
where $c=[C]$, which puts into math symbols the fact that $C$ is created from the combination of $A$ and $B$ at a rate that is proportional to the product $[A][B]$, called the law of mass action. Notice that the units of $\gamma$ are different ((time) ${ }^{-1}$ (concentration $)^{-1}$ ) than those for first order reactions ((time) $\left.{ }^{-1}\right)$. The degradation of $C$ into $A$ and $B$ is a first order reaction. For this reaction there are two conserved quantities, namely $[A]+[C]$ and $[B]+[C]$

An important example of reaction kinetics occurs in the study of epidemics, with the so-called SIR epidemic. Here $S$ represents susceptible individuals, I represents infected individuals, and $R$ represents recovered or removed individuals. We represent the disease process by the reaction scheme
$$S+I \stackrel{\alpha}{\longrightarrow} 2 I, \quad I \stackrel{\beta}{\longrightarrow} R$$
This implies that a susceptible individual can become infected following contact with an infected individual, and that infected individuals recover at an exponential rate.

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

1.3.1. Decay Processes. Now that we have the review of differential equations behind us, we must face the fact that differential equation descriptions of biological processes are at best, highly idealized. This is because biological processes, and in fact many physical processes, are not deterministic, but noisy, or stochastic. This noise, or randomness, could be because, while the process actually is deterministic, we do not have the ability or the patience to accurately calculate the outcome of the process. For example, the flipping of a coin or the spin of a roulette wheel has a deterministic result, in that, if initial conditions were known with sufficient accuracy, an accurate calculation of the end result could be made. However, this is so impractical that it is not worth pursuing. Similarly, the motion of water vapor molecules in the air is by completely deterministic process (following Newton’s Second Law, no quantum physics required) but determining the behavior of a gas by solving the governing differential equations for the position of each particle is completely out of the question.

There are other processes for which deterministic laws are not even known. This is because they are governed by quantum dynamics, having possible changes of state that cannot be described by a deterministic equation. For example, the decay of a radioactive particle and the change of conformation of a protein molecule, such as an ion channel, cannot, as far as we know, be described by a deterministic process. Similarly, the mistakes made by the reproductive machinery of a cell when duplicating its DNA (i.e., the mutations) cannot, as far as we currently know, be described by a deterministic process.

Given this reality, we are forced to come up with another way to describe interesting processes. And this is by keeping track of various statistics as time proceeds. For example, it may not be possible to exactly track the numbers of people who get the flu every year, but an understanding of how the average number changes over several years may be sufficient for health care policy makers. Similarly, with carbon dating techniques, it is not necessary to know exactly how many carbon- 14 molecules there are in a particular painting at a particular time, but an estimate of an average or expected number of molecules can be sufficient to decide if the painting is genuine or a forgery.
1.3.1.1. Probability Theory. To make some progress in this way of describing things, we must define some terms. First, there must be some object that we wish to measure or quantify, also called a random variable, and the collection of all possible outcomes of this measurement is called its state space, or sample space. For example, the flip of a coin can result in it landing with head or tail up, and these two outcomes constitute the state space. Similarly, an ion channel may at any given time be either open or closed, and this also constitutes its state space. The random variable could be a discrete or continuous variable taking on only integer values if it is discrete or a real valued number or vector if it is continuous.

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

1.3.2. Several Reactions. In the example of particle decay there was only one reaction possible. However, this is not typical as most chemical reactions involve a range of possible reactions. For example, suppose a particle (like a bacterium) may reproduce at some rate or it may die at a different rate. The question addressed here is how to do a stochastic simulation of this process.

Suppose the state $S_{j}$ can transition to the state $S_{k}$ at rate $\lambda_{k j}$. To do a stochastic simulation of this process, we must decide when the next reaction takes place and which reaction it is that takes place.

To decide when the next reaction takes place, we use the fact that the probability that the next reaction has taken place by time $t$ is 1 minus the probability that the next reaction has not taken place by time $t$. Furthermore, the probability that the reaction from state $j$ to state $k$ has not taken place by time $t$ is $\exp \left(-\lambda_{k j} t\right)$. So, the probability that no reaction has taken place by time $t$ (since these reactions are assumed to be independent) is
$$\prod_{k} \exp \left(-\lambda_{k j} t\right)=\exp \left(-\sum_{k} \lambda_{k j} t\right) .$$
It follows that the cdf for the next reaction is
$$1-\exp \left(-\sum_{k} \lambda_{k j} t\right)=1-\exp (-r t),$$
where $r=\sum_{k} \lambda_{k j}$. In other words, the next reaction is an exponential process with rate $r$
Next, the probability that the next reaction is the $i$ th reaction $S_{j} \rightarrow S_{i}$ is
$$p_{i j}=\frac{\lambda_{i j}}{\sum_{k} \lambda_{k j}}=\frac{\lambda_{i j}}{r} .$$
To be convinced of this, apply the results of Exercise $1.26$ to the case where either the $S_{j} \rightarrow S_{i}$ reaction occurs first or another reaction occurs first.

With these facts in hand, as we did above, we pick the next reaction time increment to be
$$\dot{\delta} t=\frac{-1}{r} \ln R_{1} \text {, }$$
where $0<R_{1}<1$ is a uniformly distributed random number. Next, to decide which of the reactions to implement, construct the vector $x_{k}=\frac{1}{r} \sum_{i=1}^{k} \lambda_{i j}$, the scaled vector of cumulative sums of $\lambda_{i j}$. Notice that the vector $x_{k}$ is ordered with $0 \leq x_{1} \leq x_{2} \leq \cdots \leq$ $x_{N}=1$, where $N$ is the total number of states. Now, pick a second random number $R_{2}$, uniformly distributed between zero and one, and pick the next reaction to be $S_{j} \rightarrow S_{k}$ where
$$k=\min {j}\left{R{2} \leq x_{j}\right}$$

## 数学代写|微分方程代写differential equation代考|Modeling Chemical Reactions

1.2.4。模拟化学反应。至少在本书中，微分方程的重要用途之一是模拟化学反应的动力学。这里最重要的两个基本反应是物种之间的转化，表示为

d一个d吨=bb−一个一个,dbd吨=−bb+一个一个

d一个d吨=−C一个b+dC,dbd吨=−C一个b+dC,dCd吨=C一个b−dC

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

1.3.1。衰减过程。既然我们已经回顾了微分方程，我们必须面对这样一个事实，即生物过程的微分方程描述充其量是高度理想化的。这是因为生物过程，实际上是许多物理过程，不是确定性的，而是嘈杂的或随机的。这种噪音或随机性可能是因为虽然过程实际上是确定性的，但我们没有能力或耐心来准确计算过程的结果。例如，掷硬币或转动轮盘赌具有确定性结果，因为如果初始条件足够准确，则可以对最终结果进行准确计算。然而，这太不切实际了，不值得追求。相似地，

1.3.1.1。概率论。为了在这种描述事物的方式上取得一些进展，我们必须定义一些术语。首先，必须有一些我们希望测量或量化的对象，也称为随机变量，并且该测量的所有可能结果的集合称为其状态空间或样本空间。例如，抛硬币会导致它头朝上或尾部朝上落地，这两种结果构成了状态空间。类似地，离子通道可以在任何给定时间打开或关闭，这也构成了它的状态空间。随机变量可以是离散变量或连续变量，如果它是离散的，则它可以是仅取整数值的变量，或者如果它是连续的，则可以是实数值或向量。

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

1.3.2. 几个反应。在粒子衰变的例子中，只有一种反应可能。然而，这并不典型，因为大多数化学反应都涉及一系列可能的反应。例如，假设一个粒子（如细菌）可能以某种速度繁殖，或者它可能以不同的速度死亡。这里解决的问题是如何对这个过程进行随机模拟。

∏ķ经验⁡(−λķj吨)=经验⁡(−∑ķλķj吨).

1−经验⁡(−∑ķλķj吨)=1−经验⁡(−r吨),

p一世j=λ一世j∑ķλķj=λ一世jr.

d˙吨=−1rln⁡R1,

k=\min {j}\left{R{2} \leq x_{j}\right}

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