Problem 14 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Based on data compiled by WHO, the number of people living with HIV (human immunodeficiency virus) worldwide from 1985 through 2006 is estimated to be $$ N(t)=\frac{39.88}{1+18.94 e^{-0.2957}} \quad(0 \leq t \leq 21) $$ where \(N(t)\) is measured in millions and \(t\) in years, with \(t=0\) corresponding to the beginning of 1985 . a. How many people were living with HIV worldwide at the beginning of 1985 ? At the beginning of 2005 ? b. Assuming that the trend continued, how many people were living with HIV worldwide at the beginning of \(2008 ?\)

Step-by-Step Solution

Answer

At the beginning of 1985, there were approximately 2.08 million people living with HIV worldwide. At the beginning of 2005, there were approximately 34.37 million people living with HIV worldwide. Assuming the trend continued, about 36.37 million people were living with HIV worldwide at the beginning of 2008.

Step 1: (a) Number of people living with HIV worldwide at the beginning of 1985 and 2005:

To find the number of people living with HIV worldwide at the beginning of 1985, we need to find N(0): $$ N(0) = \frac{39.88}{1 + 18.94 e^{-0.2957 \cdot 0}} $$ Since \(e^0 = 1\), we get: $$ N(0) = \frac{39.88}{1 + 18.94} \approx 2.08 $$ So, there were approximately 2.08 million people living with HIV worldwide at the beginning of 1985. Now, let's find the number of people living with HIV worldwide at the beginning of 2005. Since 2005 corresponds to t = 20, we need to find N(20): $$ N(20) = \frac{39.88}{1 + 18.94 e^{-0.2957 \cdot 20}} $$ Now compute the value of N(20): $$ N(20) \approx 34.37 $$ So, there were approximately 34.37 million people living with HIV worldwide at the beginning of 2005.

Step 2: (b) Number of people living with HIV in 2008:

We are asked to find the number of people living with HIV worldwide at the beginning of 2008, assuming the trend continued. This corresponds to t = 23. So we need to find N(23): $$ N(23) = \frac{39.88}{1 + 18.94 e^{-0.2957 \cdot 23}} $$ Now compute the value of N(23): $$ N(23) \approx 36.37 $$ Hence, assuming the trend continued, about 36.37 million people were living with HIV worldwide at the beginning of 2008.

Key Concepts

Exponential Decay

In the context of HIV infection rates and other areas of epidemiology, exponential decay is a crucial concept that helps to describe how the number of new infections might decrease over time under certain conditions. This mathematical function characterizes a process in which a quantity decreases at a rate proportional to its current value.

For the exponential decay function, we use the formula: $$ N(t) = N_0 e^{-kt} $$ where:

• N(t) is the quantity at time t,
• N_0 is the initial quantity,
• k is the decay constant.

In the case of the HIV infection rates from the exercise, a modified version of the exponential decay model is used in the given equation to represent the change in infection rates over time. The model includes a logistic function that captures how the infection rate evolves in a population that does not have an infinite capacity for growth due to factors like limited resources or behavior changes that reduce transmission. Understanding exponential decay is essential for interpreting trends in epidemiology and effectively strategizing public health interventions.

Mathematical Models in Epidemiology

The power of mathematics extends into the field of health sciences through mathematical models in epidemiology. These models are tools that scientists use to understand the spread and control of diseases in populations. They can capture complex realities with a set of equations, helping us predict future outbreaks, evaluate risk factors, and design strategies to mitigate the spread of diseases like HIV.

The equation provided in the HIV infection exercise is a prime example of such modeling. Epidemiological models typically incorporate variables like infection rates, recovery rates, and contact patterns to simulate how diseases spread. These models often fall into different classes such as deterministic or stochastic, with the former being predictable and having a fixed set of inputs leading to a fixed output, while the latter includes probabilities and variable outcomes. Models are calibrated using real-world data to give the most accurate predictions. Informed by these predictions, public health officials can make better decisions to protect communities.

Logistic Growth Function

The logistic growth function is widely used in biological sciences, especially in situations where the growth of a population is capped due to limited resources or other constraining factors. Unlike exponential growth, which assumes infinite resources and space, logistic growth models more accurately represent the slowed growth as a population approaches its carrying capacity, which is the maximum population size that the environment can sustain indefinitely.

The logistic growth function is given by the formula: $$ N(t) = \frac{K}{1 + \frac{K-N_0}{N_0} e^{-rt}} $$ where:

• N(t) represents the population at time t,
• K is the carrying capacity,
• N_0 is the initial population size,
• r is the intrinsic growth rate, and
• e is the base of the natural logarithm.

This function exhibits a sigmoid curve, where the growth rate initially increases when the population size is small, then declines as the population reaches its carrying capacity. In the context of the HIV infection exercise, the logistic function helps to forecast the number of people living with HIV, taking into account the natural constraints on the spread of the virus within the population, thus providing a more realistic depiction than exponential models.