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# 数学代写|数学建模代写Mathematical Modeling代考|SIS Model with Constant Number of Carriers

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## 数学代写|数学建模代写Mathematical Modeling代考|SIS Model with Constant Number of Carriers

In this model, only carriers spread the disease and their number decreases exponentially with time as these are identified and eliminated, so that we get
\begin{aligned} & \frac{d S}{d t}=-\beta S(t) C(t)+\gamma I(t), \quad \frac{d I}{d t}=-\beta C(t) S(t)-\gamma I(t), \ & \frac{d C}{d t}=-\alpha C \end{aligned}
so that
$$\begin{gathered} S(t)+I(t)=S_0+I_0=N(\text { say }), C(t)=C_0 \exp (-\alpha t) \ \frac{d I}{d t}=\beta C_0 N \exp (-\alpha t)-\left[\beta C_0 \exp (-\alpha t)+\gamma\right] I \end{gathered}$$

Here infected persons are removed by death or hospitalization at a rate proportional to the number of infectives, so that the model is
\begin{aligned} \frac{d S}{d t} & =-\beta S I, \quad \frac{d I}{d t}=\beta S I-\gamma I=\beta I\left[S-\frac{\gamma}{\beta}\right] \ & =\beta I(S-\rho) ; \rho=\frac{\gamma}{B} \end{aligned}
with initial conditions
\begin{aligned} S(0) & =S_0>0, \quad I(0)=I_0>0, \quad R(0)=R_0=0, \ S_0+I_0 & =N \end{aligned}

## 数学代写|数学建模代写Mathematical Modeling代考|COMPARTMENT MODELS THROUGH SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Pharmokinetics (also called drug kinetics or tracer kinetics or multi-compartment analysis) deals with the distribution of drugs, chemicals, tracers, or radioactive substances among various compartments of the body where compartments are real or fictitious spaces for drugs.

Let $x_i(t)$ be the amount of the drug in the $i$ th compartment at time $t$. We shall assume that the amount that can be transferred from the $i$ th to the $j$ th compartment $(j \neq i)$ in the time interval $(t, t+\Delta t)$ is $k_{i j} x_i(t) \Delta t+0(\Delta t)$, where $k_{i j}$ is called the transfer coefficient from the $i$ th to the $j$ th compartment. The total change $\Delta x_i$. in time $\Delta t$ is given by the amount entering the $i$ th compartment from the other compartment which is reduced by the amount leaving the $i$ th compartment for other compartments including the zeroeth compartment that denotes the outside system.
Thus we get
$$\Delta x_i=-\sum_{\substack{j=0 \ j \neq i}}^n k_{i j} x_i \Delta t+\sum_{\substack{j=1 \ j \neq i}}^n k_{i j} x_i \Delta t+0(\Delta t)$$
Dividing by $\Delta t$ and proceeding to the limit as $\Delta t \rightarrow 0$, we get
$$\begin{gathered} \frac{d x_i}{d t}=-x_i \sum_{\substack{j=1 \ j \neq i}}^n k_{i j}+\sum_{\substack{j=1 \ j \neq i}}^n k_{i j} x_j \ =\sum_{j=1}^n k_{i j} x_j,(i=1,2, \ldots, n) \end{gathered}$$
where we define
$$k_{i i}=-\sum_{\substack{j=1 \ j \neq i}}^n k_{i j},(i=1,2, \ldots, n)$$

## 数学代写|数学建模代写Mathematical Modeling代考|SIS Model with Constant Number of Carriers

\begin{aligned} & \frac{d S}{d t}=-\beta S(t) C(t)+\gamma I(t), \quad \frac{d I}{d t}=-\beta C(t) S(t)-\gamma I(t), \ & \frac{d C}{d t}=-\alpha C \end{aligned}

$$\begin{gathered} S(t)+I(t)=S_0+I_0=N(\text { say }), C(t)=C_0 \exp (-\alpha t) \ \frac{d I}{d t}=\beta C_0 N \exp (-\alpha t)-\left[\beta C_0 \exp (-\alpha t)+\gamma\right] I \end{gathered}$$

\begin{aligned} \frac{d S}{d t} & =-\beta S I, \quad \frac{d I}{d t}=\beta S I-\gamma I=\beta I\left[S-\frac{\gamma}{\beta}\right] \ & =\beta I(S-\rho) ; \rho=\frac{\gamma}{B} \end{aligned}

\begin{aligned} S(0) & =S_0>0, \quad I(0)=I_0>0, \quad R(0)=R_0=0, \ S_0+I_0 & =N \end{aligned}

## 数学代写|数学建模代写Mathematical Modeling代考|COMPARTMENT MODELS THROUGH SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

$$\Delta x_i=-\sum_{\substack{j=0 \ j \neq i}}^n k_{i j} x_i \Delta t+\sum_{\substack{j=1 \ j \neq i}}^n k_{i j} x_i \Delta t+0(\Delta t)$$

$$\begin{gathered} \frac{d x_i}{d t}=-x_i \sum_{\substack{j=1 \ j \neq i}}^n k_{i j}+\sum_{\substack{j=1 \ j \neq i}}^n k_{i j} x_j \ =\sum_{j=1}^n k_{i j} x_j,(i=1,2, \ldots, n) \end{gathered}$$

$$k_{i i}=-\sum_{\substack{j=1 \ j \neq i}}^n k_{i j},(i=1,2, \ldots, n)$$

## MATLAB代写

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