# 统计代写|生存模型代写survival model代考|PROGRESSION OF A DISEASE

#### Doug I. Jones

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## 统计代写|生存模型代写survival model代考|PROGRESSION OF A DISEASE

In Panjer’s model, a total of six states are defined. A healthy person not infected by the AIDS virus is said to be in State la, and a person infected by the virus (i.e., testing HIV+) but having no symptoms of the disease is in State $1 \mathrm{~b}$. The next three states, denoted $2 \mathrm{a}, 2 \mathrm{~b}$, and 3 , reflect progressive stages of the disease, and the final state is that of death. Without concerning ourselves with medical meanings, we identify the three stages of the disease as follows:
State 2b: AIDS-Related Compex (ARC)
State 3: Acquired Immune Deficiency Syndrome (AIDS)
This classification system is consistent with the Walter Reed Staging Method (see Redfield, et al. [62]).

It is assumed that persons progress through the successive stages of the disease without returning to an earlier stage. For simplicity it is also assumed that death prior to State 3, although certainly possible, is a risk small enough to be ignored. This means that a person in any state can leave that state only by progressing to the next state in the model (i.e., a single-decrement environment), as illustrated in Figure 10.3.

Since the only decrement in a given state is progressing to the next state, then the model is completely defined by the hazard rates, which we shall call forces of progression, operating in each state. A simplifying assumption is that these forces of progression, denoted $\mu_j$ for $j=1 a, 1 b, 2 a$, $2 b, 3$, are constant within each state, not depending on the age or sex of the person nor the length of time the person has been in that particular state.

## 统计代写|生存模型代写survival model代考|Properties of the Model

Since the force of progression is constant, then the random variable for length of time spent in State $j$, denoted $T_j$, is exponentially distributed, with
$$\begin{gathered} S_{T_j}(t)=\operatorname{Pr}\left(T_j>t\right)=e^{-t \cdot \mu_j}, \ F_{T_j}(t)=\operatorname{Pr}\left(T_j \leq t\right)=1-e^{-t \cdot \mu_j}, \ f_{T_j}(t)=\frac{d}{d t} F_{T_j}(t)=\mu_j \cdot e^{-t \cdot \mu_j}, \end{gathered}$$

$$E\left[T_j\right]=\frac{1}{\mu_j},$$
and
$$\operatorname{Var}\left(T_j\right)=\frac{1}{\mu_j^2} .$$
Recall the “memoryless” property of the exponential distribution, which says that $T_j$ is the random variable for future time spent in State $j$, whether measured from the time of entry into State $j$ or any later time while in State $j$. Furthermore, the random variable $T_j$ is independent of the random variables associated with the other states, so that progression to the next state does not depend on how long the person was in a previous state.

The model assumes that persons leave State 3 (AIDS) only through death, so $\mu_3$ is actually the force of mortality for persons with AIDS, and $T_3$ is the future lifetime random variable for such persons. Then the random variable
$$R=T_{2 b}+T_3$$
denotes the future lifetime of a person in State $2 b$ (ARC), with expected value
$$E[R]=E\left[T_{2 b}\right]+E\left[T_3\right]=\frac{1}{\mu_{2 b}}+\frac{1}{\mu_3}$$
and
$$\operatorname{Var}(R)=\operatorname{Var}\left(T_{2 b}\right)+\operatorname{Var}\left(T_3\right)=\frac{1}{\mu_{2 b}^2}+\frac{1}{\mu_3^2},$$

# 生存模型代考

## 统计代写|生存模型代写survival model代考|Properties of the Model

$$\begin{gathered} S_{T_j}(t)=\operatorname{Pr}\left(T_j>t\right)=e^{-t \cdot \mu_j}, \ F_{T_j}(t)=\operatorname{Pr}\left(T_j \leq t\right)=1-e^{-t \cdot \mu_j}, \ f_{T_j}(t)=\frac{d}{d t} F_{T_j}(t)=\mu_j \cdot e^{-t \cdot \mu_j}, \end{gathered}$$

$$E\left[T_j\right]=\frac{1}{\mu_j},$$

$$\operatorname{Var}\left(T_j\right)=\frac{1}{\mu_j^2} .$$

$$R=T_{2 b}+T_3$$

$$E[R]=E\left[T_{2 b}\right]+E\left[T_3\right]=\frac{1}{\mu_{2 b}}+\frac{1}{\mu_3}$$

$$\operatorname{Var}(R)=\operatorname{Var}\left(T_{2 b}\right)+\operatorname{Var}\left(T_3\right)=\frac{1}{\mu_{2 b}^2}+\frac{1}{\mu_3^2},$$

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