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2006年C题特等奖论文美国大学生数学建模特等奖论文集.rar （7个子文件）

2006年C题特等奖论文美国大学生数学建模特等奖论文集

2006年C题特等奖论文说明.pdf 794KB

2006 C O Managing the HIVAIDS Pandemic.pdf 2.19MB

2006 C O AIDS Modeling a Global Crisis and Australia.pdf 880KB

2006 C O Managing the HIV AIDS Pandemic 2006-2055.pdf 825KB

2006 C O The United Nations and the Quest for the Holy Grail (of AIDS).pdf 705KB

2006 C O The Spreading HIV AIDS Problem.pdf 663KB

2006 C O The United Nations and the Quest for the Holy Grail.pdf 765KB

Team 788 Page 1 of 61

Managing the HIV/AIDS Pandemic:

2006-2055

Team 788

February 6, 2006

Abstract

The HIV/AIDS pandemic is undoubtedly one of the greatest challenges facing the

modern world. Not only is it a fatal virus, it is a master of elusion. After twenty-five

years, no human effort has come near stopping the destruction caused by HIV/AIDS.

Further, the problem is so large that determining how to attack it may be one of the more

insurmountable difficulties faced thus far.

We outline a plan of attack for understanding and controlling the HIV/AIDS crisis. It

begins with a thorough consideration of determining what nations are facing the most

critical situations with respect to HIV/AIDS. We accomplish this by modeling an

adjusted life expectancy, using a short-term logistic differential equation model, and

mathematically defining criticality.

We more deeply analyze the future situations of the most critical nations with a powerful

computer simulation model. The model is very versatile and can provide a great deal of

information about populations, and has the further advantage of dealing directly with

people rather than homogenous populations, as a differential equations model would.

Treatment analysis includes a brief estimation on the amount of foreign aid available

through 2055, and then predicts the effects of Antiretroviral Therapy and the possibilities

of a preventative HIV/AIDS vaccine. We then consider the ramifications of drug resistant

strains and how they may arise.

We conclude with a series of recommendations to the United Nations, for how best to

allocate resources to the HIV/AIDS Pandemic.

By continent, we conclude the most critical nations are: Botswana, Thailand, Tonga,

Ukraine, Bahamas, and Guyana. We recommend intensive spending on research and

development for and HIV/AIDS vaccine in the short term, followed by a global coverage

of ARV treatment with heavy emphasis on maintaining adherence.

- 1 -

数学中国

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本内容由645617861l同学与工作人员共同整理，恭喜同学成功完成数学中国第一期威客项目

数学中国：www.madio.net，最专业的数学建模平台

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Tyler Huffman, Barry Wright III, Charles Staats III

Duke University

2006 Outstanding Winner

数学中国提供

http://www.madio.net

Team 788 Page 2 of 61

Introduction

Since first being identified in 1981, the HIV virus has spread rampantly throughout many

parts of the world. HIV is a sexually transmitted retrovirus that weakens the immune

system, almost inevitably leading to death. It is estimated that approximately 25 million

people have already died from complications related to the HIV virus and AIDS. Despite

a wide variety of global efforts to control this pandemic, the number of infections and

deaths from HIV/AIDS continues to rise. One of the main complications when dealing

with a virus such as HIV is that it affects its host at a rate much slower than the disease is

spread. This allows HIV positive individuals to spread the disease for some time before

they are even aware that they are infected.

HIV and AIDS has become a particularly challenging problem to deal with in the

developing world. In countries where poverty and famine are widespread, methods of

prevention such as condoms and HIV/AIDS education aren’t available. Furthermore,

these countries often lack the necessary funding to test citizens for HIV and to treat

current patients with antiretroviral drugs (ARV treatment). International funding has

become an integral part of managing the HIV/AIDS pandemic in the world today.

Funding from private sectors, non-profit organizations, as well as individual

governments, provides the majority of the resources with which nations are able to

Restatement of the Problem

While almost everyone will agree that the rapid spread of the HIV virus is an enormous

crisis throughout many parts of the world, a highly disputed topic is how funding should

be distributed to deal with the current situation. While some will argue that treatment of

patients is of utmost importance, many feel that research into preventative measures such

as development of a vaccination would provide the most benefit in the long run.

Our goal throughout this model is to access the trade-offs between each of the above

approaches, specifically determining the most beneficial way to distribute funding over

the next fifty years. We will pay specific attention to factors such as the possible

development of a vaccine, changes in ARV treatment, and the possible mutation of the

virus into multiple strains.

1.1 Defining Criticality

1.2 Approach

We were charged with the task of determining a country from each continent “most

critical in terms of HIV/AIDS.” But what exactly makes a country “critical”? The

obvious answer is to choose countries with the greatest number of HIV/AIDS cases, or

- 2 -

数学中国

www.madio.net

本内容由645617861l同学与工作人员共同整理，恭喜同学成功完成数学中国第一期威客项目

数学中国：www.madio.net，最专业的数学建模平台

备战美赛，美赛资料下载：http://www.madio.net/forum-108-1.html

美赛讨论加入群：2017美赛官方群287336869

Team 788 Page 3 of 61

the greatest proportion of cases, but this is not a complete analysis. A critical situation

implies that progress can be made towards a solution. At this point in time, nothing

beyond antiretroviral therapy can be done for an HIV/AIDS patient. Therefore, countries

with high rates of treatment can do little more for their infected population (although it is

important to keep in mind that prevention and education measures are still very

applicable). Thus, such countries should not be deemed most critical. The term critical

also implies that action is urgent, that HIV/AIDS will be very detrimental in the short

term. We believe that the best way to measure the effect of HIV/AIDS on a population is

to determine the cumulative number of years of life lost due to infection.

1.3 Assumptions & Terms

• We are not considering people currently receiving treatment, as it is assumed that

there is nothing more that can be done with them. Criticality is relevant to

determining how urgent it is to act on a situation, so there is no reason to consider

those for whom nothing more can be done.

• ARV patients have 100% adherence. That is, a patient either has ARV treatment

or does not, there is no middle ground.

• No further intervention occurs within the next five years.

• ARV treatment percentages remain constant.

• No other major causes of death affect the population. Since we are predicting over

a relatively short interval of time, it is unlikely that major events such as natural

disaster, wars, or other pandemics will significantly affect the population.

• People-year: A unit equivalent to one person times one year. The number of

people-years of a population is equal to the sum of all the lifetimes of people in

the population.

• To measure the immediate effects of HIV/AIDS on a population receiving no

further intervention, we define criticality over the next five years (2006-2010):

o Absolute Criticality: The total number of people-years lost by a population

over the next five years due to HIV/AIDS.

o Relative Criticality: The average number of people-years lost by a person

over the next five years, in other words, the change in life expectancy over

the next five years.

1.4 Development

Based on our assumptions and definition of criticality, we then derived a mathematical

expression for criticality in terms of various parameters. Mathematically, relative

criticality, ζ, is given by:

(, ,,,) ( ) ()

αβγδε αγ δ βε

- 3 -

数学中国

www.madio.net

本内容由645617861l同学与工作人员共同整理，恭喜同学成功完成数学中国第一期威客项目

数学中国：www.madio.net，最专业的数学建模平台

备战美赛，美赛资料下载：http://www.madio.net/forum-108-1.html

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Team 788 Page 4 of 61

Where:

• α – The average loss of life expectancy due to contracting HIV and without

receiving ARV treatment.

• β – The average loss of life expectancy due to contracting HIV and receiving

ARV treatment.

• γ - Number of current untreated cases divided by population.

• δ - Number of untreated cases contracted over the next five years divided by

population.

• ε - Number of treated cases contracted over the next five years divided by

population.

Absolute criticality is then given by:

Abs

),,,,(

Where P is the total population of a given country.

It seems counterintuitive that a country should be considered “less critical” if its life

expectancy is innately lower, as our model would conclude. What this really means is

that for the given country, spending money on HIV/AIDS may be less relevant than

spending money on other causes of death. If for instance, most people in a country die at

30 of tuberculosis (without AIDS) then tuberculosis is likely to be a bigger concern for

that country even than a raging HIV epidemic. Further, we did not include a parameter

for current HIV cases undergoing treatment, because, as we stated above, nothing more

can be done for these people, so they should have no impact on the criticality.

1.5 Model A: Adjusting the Life Expectancy

1.6 Approach

In order to determine the effects of HIV/AIDS on a population, we initially determined

the life expectancy of a population as if HIV/AIDS did not exist. Furthermore, we then

adjusted for the fact that life expectancy is a function of year of birth.

1.7 Assumptions

• The life expectancy does not significantly change within a period of five years, as

we assume people of each age group are the same age.

• Life expectancy data does not exist for birth years before 1950, so we assume any

person born before 1950 has the life expectancy of someone born in 1950.

• No immigration or emigration occurs.

- 4 -

数学中国

www.madio.net

本内容由645617861l同学与工作人员共同整理，恭喜同学成功完成数学中国第一期威客项目

数学中国：www.madio.net，最专业的数学建模平台

备战美赛，美赛资料下载：http://www.madio.net/forum-108-1.html

美赛讨论加入群：2017美赛官方群287336869

Team 788 Page 5 of 61

1.8 Method

Again, we employed the idea of a person-year as the unit for population life expectancy,

which is simply a person times one year. Using 2005 population data, we multiply the

population for each age group by the life expectancy of the corresponding birth years.

This yields the total number of person-years for a given population. Dividing this by the

population will then give the age-adjusted life expectancy, denoted

We then determined a life expectancy value for people infected with HIV/AIDS.

Research showed that worldwide, the average age for contracting HIV is 23 years old.

We then had to distinguish between life expectancy with and without ARV treatment. We

assumed here that a person on ARV treatment has 100% adherence and never stops

treatment. In developed nations, a person contracting HIV typically lives twelve years,

untreated. Few people treated with ARV have died; however, the program began only ten

years ago. Current projections claim ARV treated patients will live twenty years after

contraction in developed nations. To determine the average life expectancy for a person

contracting HIV/AIDS, we used the following formula, where

is the percentage of

people currently receiving ARV treatment;

))(43)1(35(

−

=Γ

HIV

We take a weighted average of the life expectancies for untreated and treated patients,

and then we account for the difference in life expectancy due to causes other than HIV by

multiplying by the unadjusted life expectancy and dividing by 70, an assumed average for

the life expectancy of a developed nation.

Now we can derive an expression for the HIV-adjusted life expectancy. Rather than using

an algebraic definition, we take a more intuitive approach using set theory.

We have data for the total number of people-years of a population, and given the number

of HIV cases, , and the average life expectancy for HIV patients, we know the total

number of HIV people-years. If these people did not have HIV, the number of people-

years they contribute to the population would increase. Therefore, adding the number of

people-years the HIV infected population loses due to premature death to the unadjusted

number of people-years for a population will yield the adjusted people-years, and

consequently an HIV-adjusted life expectancy for that population.

HIV

- 5 -

数学中国

www.madio.net

本内容由645617861l同学与工作人员共同整理，恭喜同学成功完成数学中国第一期威客项目

数学中国：www.madio.net，最专业的数学建模平台

备战美赛，美赛资料下载：http://www.madio.net/forum-108-1.html

美赛讨论加入群：2017美赛官方群287336869

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