Problem 64 - Calculus for Biology and Medicine

Question

The hazard-rate function of an organism is given by $$ \lambda(x)=0.1+0.5 e^{0.02 x}, \quad x \geq 0 $$ where \(x\) is measured in days. (a) What is the probability that the organism will live less than 10 days? (b) What is the probability that the organism will live for another five days, given that it survived the first five days?

Step-by-Step Solution

Answer

(a) Calculate \( P(X < 10) = 1 - S(10) \). (b) Use \( P(X > 10 \mid X > 5) = \frac{S(10)}{S(5)} \).

Step 1: Understanding the Hazard Rate Function

The function given is the hazard rate function \( \lambda(x) = 0.1 + 0.5 e^{0.02 x} \). This represents the instantaneous failure rate of the organism at time \( x \). It's important because it helps us determine the survival function, which tells us the probability of surviving until a certain time.

Step 2: Determining the Survival Function

The survival function \( S(x) \) is related to the hazard rate by \( S(x) = e^{- ext{H}(x)} \), where \( ext{H}(x) \) is the cumulative hazard function defined as \( ext{H}(x) = \int_0^x \lambda(t) dt \). To find \( ext{H}(x) \), integrate the hazard rate function: \[ \text{H}(x) = \int_0^x (0.1 + 0.5 e^{0.02 t}) dt.\] The integral results in \( ext{H}(x) = 0.1x + 25(e^{0.02x} - 1) \). Thus, the survival function is:\[ S(x) = e^{-\left(0.1x + 25(e^{0.02x} - 1)\right)}.\]

Step 3: Calculating the Probability of Living Less Than 10 Days

The probability that the organism will live less than 10 days can be found using \( P(X < 10) = 1 - S(10) \). Calculate \( S(10) \) using the survival function:\[ S(10) = e^{-\left(0.1 \times 10 + 25(e^{0.2} - 1)\right)}.\]Solve this to find \( S(10) \), and then find:\[ P(X < 10) = 1 - S(10).\]

Step 4: Finding Survival Probability for Another 5 Days Given Survival of First 5 Days

To find the probability that the organism survives another 5 days after already surviving 5 days, use the conditional survival function: \[ P(X > 10 \mid X > 5) = \frac{S(10)}{S(5)}.\]First, calculate \( S(5) \):\[ S(5) = e^{-\left(0.1 \times 5 + 25(e^{0.1} - 1)\right)}.\]Then, calculate \( P(X > 10 \mid X > 5) \) using the values of \( S(10) \) and \( S(5) \).

Key Concepts

Hazard Rate Function

The hazard rate function is a key concept in survival analysis, providing insight into how likely an event (like failure or death) is to occur at a specific moment in time, given survival up until that time. This function is expressed as \( \lambda(x) \), representing the instantaneous rate at which events happen. It's vital because it helps inform the use of other functions like the survival and cumulative hazard functions.

In the exercise provided, the hazard rate function is given as \( \lambda(x) = 0.1 + 0.5 e^{0.02 x} \). The linear component \( 0.1 \) suggests a baseline hazard, while \( 0.5 e^{0.02 x} \) reflects an increasing risk over time as \( x \) increases, influenced exponentially by \( e^{0.02 x} \).

Understanding the hazard rate function helps in understanding the dynamics of the survival process over time, aiding calculations of the probability that the organism survives for various durations.

Survival Function

The survival function, \( S(x) \), is crucial for understanding the likelihood of an entity surviving beyond a certain time \( x \). It is directly derived from the hazard rate function and gives detailed insights into the expected lifespan of organisms under study. The survival function is mathematically linked by the formula \( S(x) = e^{-\text{H}(x)} \), where \( \text{H}(x) \) is the cumulative hazard function.

For our exercise, the cumulative hazard function was integrated from the hazard rate function, yielding \( \text{H}(x) = 0.1x + 25(e^{0.02 x} - 1) \). Thus, the survival function becomes \[ S(x) = e^{-\left(0.1x + 25(e^{0.02x} - 1)\right)}. \]

Calculating \( S(x) \) at specific points allows us to determine the probability of survival beyond these timeframes. For instance, \( S(10) \) can be evaluated to compute how likely the organism is to live for more than 10 days. These probabilities enable predictions and strategies related to organism life expectancy.

Cumulative Hazard Function

The cumulative hazard function, represented as \( \text{H}(x) \), accumulates the hazard rate over time. It quantifies the total amount of risk that has been gathered as time progresses, crucial for deriving the survival function. The formula to define the cumulative hazard function is an integral:

• \( \text{H}(x) = \int_0^x \lambda(t) dt \)

In the exercise, this integration led to \[ \text{H}(x) = 0.1x + 25(e^{0.02x} - 1). \]

Using this cumulative hazard function, we derive \( S(x) \) by exponentiating its negative form \( e^{-\text{H}(x)} \). The importance of the cumulative hazard function lies in its role as a bridge between knowing the hazard at any given time and predicting future survival chances, which requires an aggregate sense of all past hazards. Calculating these functions accurately is critical in survival analysis as it enables precise estimation and comparison of the survival prospects across different time frames.