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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|数学分析代写Mathematical Analysis代考|Numerical Simulations

To assess the effects of stigma, we considered three regimes: (a) moderate stigma, (b) high stigma, and (c) no stigma. We used sample values chosen within the ranges from the table and varied the effect of stigma using the disclosure rate $\sigma$ to assess moderate versus high stigma regimes and turned off the proportion of individuals entering the stigma compartment for the no-stigma case.

Using recent leprosy-prevalence statistics [4], we conduct approximate order-ofmagnitude estimates of our infection coefficients. Roughly 214,000 cases of leprosy were reported in 2014 , with approximately $81 \%$ of those cases coming from Brazil, India and Indonesia. Approximating the populations of these nations by 200 million, $1.25$ billion and 250 million, respectively, we estimate the coefficient of infection to be
$$\beta_{1}=\frac{0.81 \times 214000}{1700000000}=0.000101965 \approx 0.0001 .$$
We considered a small village of 1000 individuals. Initial conditions were chosen so that there were 1000 susceptibles and a single infected non-stigmatised individual. Figure 2 illustrates the case of moderate stigma, showing an infection wave and a substantial number of uninfected individuals $(S+R=636$ at the end of this simulation).

Next, we used the same parameters and initial conditions as in Fig. 2 except that we changed the rate of disclosure from $\sigma=1$ to $\sigma=0.1$. This reflects the case where individuals remain stigmatised for ten years (since the length of time remaining in a compartment is inversely proportional to the rate of leaving it). In this case, the number of stigmatised individuals exceeds the number of non-stigmatised or asymptomatic individuals, sustaining a high level of infected individuals. The number of uninfected individuals was significantly lower than in Fig. $2(S+R=376$ in this simulation).

## 数学代写|数学分析代写Mathematical Analysis代考|Analysis of the Models of Interacting Species

First notice, that in both cases solutions are positive for positive initial data, while in the cases listed above:

1. For each of the species we have $\dot{x}{i} \leq r{i} x_{i}\left(1-b_{i} x_{i}\right)$, meaning that the set $\left{\left(x_{1}, x_{2}\right)\right.$ $\left.\in \mathbb{R}^{2}: 0 \leq x_{i} \leq K_{i}\right}\left(K_{i}=1 / b_{i}\right.$ is the carrying capacity for the $i$ th species) is positively invariant, implying existence of solutions for all $t \geq 0$;
2. solutions can tend to $\infty$, depending on the parameters.
Notice also that we are able to scale both variables by $b_{i}$ obtaining exactly the same model but with $b_{i}=K_{i}=1$ and $c_{i}$ scaled accordingly. Hence, we assume $b_{i}=1$ for $i=1,2$.

Next, we focus on the existence and stability of steady states. There can be up to 4 steady states. Clearly, $(0,0),(0,1)$ and $(1,0)$ always exist, while existence of a positive steady state depends on the values of the model parameters. This state satisfies the following system of equations:
$$r_{i}\left(1-x_{i}\right)+c_{i} x_{j}=0 \text { for } i, j=1,2, i \neq j .$$
Clearly, linear function $x_{2}=\frac{r_{1}}{c_{1}}\left(x_{1}-1\right)$ is the null-cline for the first variable and $x_{2}=1+\frac{c_{2}}{r_{2}} x_{1}$ is the null-cline for the second one and they have a cross section in $\mathbb{R}_{+}^{2}$ :

1. if $\left|c_{i}\right|>r_{i}$ or $\left|c_{i}\right|<r_{i}$ for both $i=1,2$;
2. if $r_{1} r_{2}>c_{1} c_{2}$.
Using the Dulac-Bendixson Criterion it is easy to check that the models have no periodic orbits in positive quadrant. Clearly, let define $B\left(x_{1}, x_{2}\right)=\frac{1}{x_{1} x_{2}}$ and calculate the divergence of the vector field $\left(B F_{1}, B F_{2}\right)$. We obtain
$$\frac{\partial}{\partial x_{1}}\left(\frac{r_{1}\left(1-x_{1}\right)}{x_{2}}+c_{1}\right)+\frac{\partial}{\partial x_{2}}\left(\frac{r_{2}\left(1-x_{2}\right)}{x_{1}}+c_{2}\right)=-\frac{r_{1}}{x_{2}}-\frac{r_{2}}{x_{1}}<0 \text { for } x_{1}, x_{2}>0 .$$
Thus, we can conclude that if solutions remain in the bounded region, then any solution tend to one of the steady states. This implies the following dynamics:
3. either there is exactly one stable steady state and it is globally stable, or there are two stable steady states and we observe bi-stability;
4. either there exists a positive steady state and it is globally stable, or there is no such state and solutions are unbounded.

## 数学代写|数学分析代写Mathematical Analysis代考|Numerical Simulations

$$\beta_{1}=\frac{0.81 \times 214000}{1700000000}=0.000101965 \approx 0.0001 .$$

## 数学代写|数学分析代写Mathematical Analysis代考|Analysis of the Models of Interacting Species

1. 对于我们拥有的每个物种 $\dot{x} i \leq r i x_{i}\left(1-b_{i} x_{i}\right)$, 意味着集合 $\$ left 的分隔符缺失或无法识别
是承载能力 $i$ th物种）是正不变的，这意味着存
在所有解决方案 $t \geq 0$;
2. 解决方案可以倾向于 $\infty$ ，取决于参数。
还请注意，我们能够通过以下方式缩放两个变量 $b_{i}$ 获得完全相同的模型，但具有 $b_{i}=K_{i}=1$ 和 $c_{i}$ 相应地缩放。因此，我们假设 $b_{i}=1$ 为了 $i=1,2$.
接下来，我们关注稳态的存在和稳定性。最多可以有 4 个稳态。清楚地， $(0,0),(0,1)$ 和 $(1,0)$ 总是存在的，而正稳态的存在取决于模型参数的值。该状态满足以 下方程组:
$$r_{i}\left(1-x_{i}\right)+c_{i} x_{j}=0 \text { for } i, j=1,2, i \neq j .$$
显然，线性函数 $x_{2}=\frac{r_{1}}{c_{1}}\left(x_{1}-1\right)$ 是第一个变量的空斜线，并且 $x_{2}=1+\frac{c_{2}}{r_{2}} x_{1}$ 是第二个的零雓线，它们的横截面在 $\mathbb{R}_{+}^{2}$ :
3. 如果 $\left|c_{i}\right|>r_{i}$ 或者 $\left|c_{i}\right|<r_{i}$ 对彼此而言 $i=1,2$;
4. 如果 $r_{1} r_{2}>c_{1} c_{2}$.
使用 Dulac-Bendixson 准则很容易检查模型在正象限中没有周期性轨道。显然，让我们定义 $B\left(x_{1}, x_{2}\right)=\frac{1}{x_{1} x_{2}}$ 并计算矢量场的散庹 $\left(B F_{1}, B F_{2}\right)$. 我们获得
$$\frac{\partial}{\partial x_{1}}\left(\frac{r_{1}\left(1-x_{1}\right)}{x_{2}}+c_{1}\right)+\frac{\partial}{\partial x_{2}}\left(\frac{r_{2}\left(1-x_{2}\right)}{x_{1}}+c_{2}\right)=-\frac{r_{1}}{x_{2}}-\frac{r_{2}}{x_{1}}<0 \text { for } x_{1}, x_{2}>0 .$$
因此，我们可以得出结论，如果解保持在有界区域内，那么任何解都趋向于其中一种稳态。这意味着以下动态:
5. 要么只有一个稳定的稳态，它是全局稳定的，要么有两个稳定的稳态，我们观察到双稳态；
6. 要么存在正稳态并且全同稳定，要么不存在这样的状态并且解是无界的。

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

assignmentutor™您的专属作业导师
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