Abstract: AIDS is the destruction of human immune system by AIDS virus, which makes the human body lose all kinds of life-threatening pathogens.
A serious infectious disease that causes a variety of infections or tumors and eventually leads to death. So far, the international medical community has
There are no effective drugs and treatments for AIDS. Therefore, effective prevention and control of AIDS should be the focus of our fight against AIDS.
The most effective treatment. In this paper, the mathematical model of parametric programming is established, and the Matlab software package is used as the tool platform to study the auxiliary tools.
The spreading process and epidemic trend of AIDS are expected to provide theoretical reference for the government to do a good job in AIDS prevention and control in the new period.
Keywords: AIDS; Communication mode; Parametric programming; matlab
China library classification number: O22 1.8 document identification number: a.
Step 0 introduce
AIDS is spreading more and more rapidly around the world, which seriously threatens human health and social development.
As the fourth biggest killer threatening people's health. UNAIDS was announced on may 30th, 2006, and since June 198 1.
Since AIDS was first confirmed, 65 million people around the world have been infected with HIV in 25 years, of which 2.5 million have died. special
What is worrying is that about 95% of AIDS patients in the world come from developing countries with weak prevention and control capabilities, such as Africa and South Asia.
Southeast Asia, Central America and other places. In China, the Joint Assessment Report on AIDS Prevention and Control in China (2007) shows that as of 2007,
By the end of the year, there will be about 700,000 HIV-infected people and patients. The overall epidemic situation is low, and the specific population and local areas are high.
Under the epidemic situation, sexual transmission has gradually become the main route of HIV transmission, which means that the situation of AIDS prevention and control is more severe. The future me
Whether the AIDS epidemic in China is growing steadily or rapidly depends on whether AIDS prevention activities can be actively carried out on a large scale.
And provide effective treatment.
It has been more than a century since the establishment of mathematical models to study the development mechanism and transmission process of epidemics, and AIDS has emerged.
Later, it attracted the attention of biological mathematicians and did more research in this field. This paper is completed on the basis of previous studies.
On this basis, through the establishment of parametric programming mathematical model, with the help of Matlab software package to explore the AIDS transmission model.
I hope it can provide theoretical reference for the government to do a good job in AIDS prevention and control in China in the new period.
1. Introduction to AIDS
The full medical name of AIDS is "acquired immunodeficiency syndrome" in Chinese and "acquired immunity" in English.
Deficieney Syndrome ",referred to as AIDS for short, is caused by AIDS (the full name in Chinese and English is" human immunodeficiency virus ",
"Human immunodeficiency virus" (HIV for short). This virus is a lifelong infection and destroys people's immune system.
System, make the human body lose the ability to resist various diseases, thus seriously endangering people's lives [1].
After HIV enters the human body, it targets the most important CD4+T lymphocytes in the human immune system.
The amount of phagocytosis destroys CD4+T lymphocytes, thus destroying the human immune system, which eventually collapses the human immune system and makes people
The body became ill and died because it lost its resistance to various diseases. The process of AIDS from infection to attack can be roughly divided into three stages.
Segment: initial stage, incubation period and attack period of infection. In the initial stage of infection, HIV enters infected human macrophages and brings the virus to.
Local lymph nodes cause various acute symptoms, and then CD8+T lymphocytes and antibodies react to control the disease.
With the development of HIV in the blood, the disease has entered the incubation period. When HIV infects macrophages,
With the infection of CD4+T cells, the immune system is gradually destroyed, and the infected CD4+T cells are lysed, resulting in a large number of HIV.
1 This project is supported by Guangxi education and scientific research project "Research on Theory and Method of Fuzzy Multi-attribute Decision-making in Discrete Space" (No.:200707LX037).
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When normal CD4+T cells decrease sharply and HIV increases rapidly, diseases will break out and various pathogens will appear at any time.
It will cause infection, and the patient will eventually die of various functional failures.
Generally speaking, without treatment, the average time from infection to attack is about 10 years, but different patients
There is a big difference. Clinically, the incubation period of 2 to 18 years can be observed, which mainly depends on the decrease of CD4+T cell concentration and HIV.
The speed at which the concentration rises. Usually, a healthy person has 1000 CD4+T cells per 1mm3 blood on average, and is a carrier of HIV.
When CD4+T cells decreased to 200 /mm3, the disease broke out.
2. AIDS transmission model
There are four main ways of transmission of AIDS: sexual transmission, blood transmission, needle transmission and mother-to-child transmission, among which sexual transmission
Transmission has become the main way of AIDS transmission today. Therefore, the following model mainly studies the spread of AISD through sexual activities.
Broadcasting, through the spread caused by other factors, can establish a similar model.
2. 1 model building
The target population (sexually active population) is divided into three categories, x(t) is the number of people who are susceptible to infection within t years, and y(t) is the number of people infected with HIV.
Number of people, z(t) is the number of people suffering from AIDS. In the absence of specific objectives, assume that the initial values of x, y and z are respectively
15× 106,3× 106,0.05× 106。 Referring to other infectious disease transmission models and parametric programming model cases, the models are [2] and [3].
( )， 1 s c x
Trembling insanity (abbreviation for Delirium Tremens)
dx =？ λ + μ ( 1)
(), 2 degrees Celsius x years
Trembling insanity (abbreviation for Delirium Tremens)
dy = λ？ γ + μ (2)
( )、3 y z
Trembling insanity (abbreviation for Delirium Tremens)
dz = γ？ μ +δ (3)
The definition and fixed value of each parameter are as follows: S ~ the ratio of susceptible population to target population (106/ year); c ~
Average speed of acquiring new sexual partners (2/ year); The probability of λ ~ sexual partners infected with HIV (0.2); 1
μ ~ is susceptible to infection
The proportion of people who quit the target population (0.025 people/year); The proportion of γ ~ HIV infected people entering AIDS (0. 1/ year); 2 μ ~HIV
The proportion of infected people leaving the target population (0.025/ year); The proportion of 3μ~ AIDS withdrawal from the target population (0.025/ year); δ ~
AIDS mortality ratio (0.95/ year).
Although the linear constant coefficient differential equations (1) ~ (3) have analytical solutions, we just want to know the numerical results and observe the straight.
The change of viewpoint follows the trend, so under the above parameters and initial values, the graph 1 is obtained by using Matlab7.5 software package [4].
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It can be seen that people are most concerned about the number of people infected with HIV, y, which will reach the maximum value of about120,000,20 in about five years.
The stable value after the year is about 7.5 million.
2.2 Model analysis
According to Routh-Hurwitz criterion, it is easy to get that the only equilibrium point of equation (1) ~ (3) is
,
1
*
λ + μ
=
c
x s *，
2
y* c x
γ μ
λ
+
= *
three
z* y
μ δ
γ
+
= 。 (4)
Because the three characteristic roots of equations (1) ~ (3) are all negative, the equilibrium point is globally stable, regardless of the initial value.
The proportion of HIV-infected people in the target population in a stable state is
λ
γ μ
μ δ
γ
β
c
x y z
y
2
three
* * *
*
1
1
+
+
+
+
=
+ +
= (5)
When a parameter value increases, the three coordinates of the equilibrium point and the change of β value are shown in the following table 1 (for example, when γ increases, x* remains unchanged and y* remains unchanged.
Decrease, z* increase, β decrease).
Table 1 Changes of three coordinates and β value caused by adding a parameter value
Parameter x* y* z* β
s↑↑↑~
λ ↓ ↑ ↑ ↑
c ↓ ↑ ↑ ↑
γ ￣ ↓ ↑ ↓
Figure 1 Numerical Solutions of Equation (1) ~ (3)
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1 μ
↓ ↓ ↓ ￣
2 μ ￣ ↓ ↓ ↓
3 μ ￣ ￣ ↓ ↑
δ ￣ ￣ ↓ ↑
Because of the global stability of the equilibrium point, as long as appropriate measures are taken to make each parameter move in the right direction (increase or decrease)
In the long run, changes can reduce the number of HIV-infected people and AISD, regardless of the current situation.
2.3 Vaccination mode
It is an artificial intervention to use attenuated HIV as a vaccine to help the human body establish immunity to the virus, so as to establish
under the circumstances
In the model, two functions need to be added: the number of people vaccinated in the target population () 1 y t, and the number of people infected by the virus after vaccination.
The number of people () is 2 y t, assuming that the initial values of 1 y and 2 y are 1000 and 0 respectively. The model is
( 1)()， 1 1 p s c c x
Trembling insanity (abbreviation for Delirium Tremens)
dx = λ + λ + μ (6)
(), 2 degrees Celsius x years
Trembling insanity (abbreviation for Delirium Tremens)
dy = λ？ γ + μ (7)
( 1 ) ( ) , 1 1 1 4 1
1 ps c x c y y
Trembling insanity (abbreviation for Delirium Tremens)
dy = + λ？ λ ? γ + μ (8)
( 1 ) ( ) , 1 2 5 2
2 degrees Celsius year
Trembling insanity (abbreviation for Delirium Tremens)
dy = λ？ γ + μ (9)
( )， 1 1 2 2 3 y y y z
Trembling insanity (abbreviation for Delirium Tremens)
dz = γ +γ +γ？ μ +δ ( 10)
Among them, the newly added parameters and their set values are as follows: P ~ the vaccination rate of the target population (0.4); 1
λ ~ sexual partner
The probability of vaccination (0.5); ? ~ probability of preventive effect after vaccination (0.93); 1γ ~ vaccination population
Proportion of AIDS patients (0.005/ year); The proportion of 4μ ~ vaccinated people dropping out of the target population (0.025/ year); 2 γ ~
The proportion of vaccinated people infected with the virus entering AIDS (0.95/ year); 5 μ ~ Inoculators were infected by the virus and withdrew from the target population.
The proportion of (0.025/ year).
Although vaccination can't prevent the initial infection with HIV, we can do a simple calculation to predict whether it is prevalent.
Influence of seedling raising progress [4]. For example, when p =0.4, use Matlab7.5 software package to find (6) when =0.93 ~
The numerical solution of (10) leads to Figure 2. As can be seen from Figure 2, the number of people infected with HIV Y reached the maximum in about 3 years.
About 6 million, compared with the results of model (1) ~ (3)
If it is reduced by half, the positive stability value will be about 2 million after 20 years, which is much less than that of the model (1) ~ (3).
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2.4 conclusion analysis
Although the above AIDS transmission model is relatively simple in concept, it involves many parameters. For these models,
The key is how to determine the parameters. The main difficulty in applying it is to determine a set of parameters for a specific country or region.
Although African countries such as Uganda have done a lot of statistical research in this field, these parameters are still uncertain.
In addition, it is difficult to get the number of people infected with HIV and suffering from AIDS more accurately in clinic.
It's hard.
3. Concluding remarks
Although the rising speed of AIDS epidemic in China has slowed down, there is no large-scale AIDS epidemic.
However, we should be soberly aware that there is a potential epidemic, and the transmission route of HIV has evolved into a sexual transmission route.
Mainly it has been the same as the international fashion trend. The regional distribution of AIDS epidemic varies greatly, and AIDS epidemic factors are widespread.
Now, the situation is getting more and more serious, which may have disastrous consequences. So, it's me from now until the end of this century.
In the critical period of AIDS prevention and control in China, if we do not take active prevention and control measures now, we will miss the opportunity.
The urgent task is to fully understand the changes in the route of transmission of AIDS in China, the epidemic trend and the risks related to the affected population.
Behavior, etc. Establish an effective monitoring system to provide accurate data on AIDS transmission for decision makers and predict AIDS.
The epidemic situation and opportunities of AIDS provide a basis for formulating national AIDS prevention and control planning strategies. With the spread of AIDS in different parts of China.
With the continuous spread and expansion, its popular mode will become more and more complicated. Therefore, our monitoring system is not only data acquisition,
Attention should be paid to analyzing data to help formulate countermeasures.
Fig. 2 Numerical solutions of equations (6) ~ (10)
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refer to
A grass. Beware of AIDS [M]. Beijing: Tsinghua University Publishing House, 2005.
[2] Tan Yongji, Cai Zhijie. Mathematical model [M]. Shanghai: Fudan University Press, 2005.4438+00 ~ 320.
[3] Crystal, Christian Prince, Mark Sevox. Operational research case [M]. Beijing: Beijing Linsen Technology Development Co., Ltd., 2007.
Zhao Dongfang. Mathematical model and calculation [M]. Beijing: Science Press, 2007.
Estimation-based AIDS transmission model
Programming; arrange
bright red
Department of Mathematics, Nanning Normal University; Computer science, People's Republic of China (PRC).
Chongzuo City, Guangxi Province (532200)
abstract
AIDS is a serious infectious disease, which is caused by HIV infection, and it destroys human immunity.
System, so that the body loses resistance to various life-threatening pathogens. international
There are no effective drugs and treatments to prevent or treat AIDS in the medical field. So, do it
Doing a good job in AIDS prevention and treatment should be the most effective means to fight AIDS. Our newspaper
To explore the spread process of AIDS by establishing a mathematical model of estimation planning.
Based on Matlab, in addition to the government can take it as a reference for AIDS prevention in the new.
Era.