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Predicting the Spread of AIDS

We use differential equations to predict the spread of diseases through a population. The growth of AIDS is an example that follows the curve of the logistic equation, derived from solving a differential equation. We will see how to solve differential equations later in this chapter.

The HIV Virus invades a white blood cell...
Image source.

Populations usually grow in an exponential fashion at first:

However, populations do not continue to grow forever, because food, water and other resources get used up over time. Differential equations are used to predict populations of people, animals, bacteria and viruses that are being affected by external events.

The logistic equation (developed in the mid-19th century) allows for a growth term AND an inhibition term. It is predicted that the AIDS epidemic will follow the pattern of the logistic equation.

If

A = number of people affected by the virus at time t,

P = the total population (a constant), and

c is a constant,

then the rate of growth of the virus at time t is given by the differential equation:

The term cPA is the growth term and -cA² is the inhibition term.

AIDS Example

AIDS is spreading through a city of 50,000 people who take no precautions. The virus was brought to the town by 100 people and it was found that 1000 people were infected after 10 weeks. How long will it take for half of the population to be infected?

Answer

So the number affected at time t is given by:

The graph of the number of people affected by AIDS after t years is:

We see in the graph the S-shaped curve which the logistic equation can take.

Now to find how long it takes for half the population (25000) to be infected:

Solution: t = 26.769

So we conclude that half of the population will be infected after about 27 weeks (marked on the graph above).

Newly Reported AIDS Cases in Australia to 2005

The following table shows actual reported AIDS cases (cumulative) in Australia.

Year	1981	1984	1987	1990	1993	1996	1999	2002	2005
AIDS cases	2	55	802	2623	5060	7493	8415	9118	9609

There is a strong bias towards men acquiring AIDS, with around 95% of the cases being male.

Men: 9119

Women: 490

Total: 9609

Data Sources: Dept of Health and Ageing, Australia, and Avert.

We see that the data roughly follows the expected S-shaped logistic equation curve. The decrease in growth from the late 1990s is largely due to suppressant drugs and education.

Reported AIDS Cases - Australia

Let's now move on to see what differential equations are and then learn how to solve them.

 

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Chapter Contents

• Differential Equations
• Predicting AIDS - a DEs example
• 1. Solving Differential Equations
• 2. Separation of Variables
• 3. Integrable Combinations
• 4. Linear DEs of Order 1
• 5. Application: RL Circuits
• 6. Application: RC Circuits
• 7. Second Order DEs - Homogeneous
• 8. Second Order DEs - Damping - RLC
• 9. Second Order DEs - Forced Response
• 10. Second Order DEs - Solve Using SNB

Following are the original SNB files (.tex or .rap) used in making this chapter. For more information, go to SNB info.

SNB files

Predicting AIDS - a DEs example

7. Second Order DEs - Homogeneous

8. Second Order DEs - Damping - RLC

9. Second Order DEs - Forced Response

10. Second Order DEs - Solve Using SNB

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