2015年美赛O奖论文A题Problem_A_35532.pdf
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35532
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A
2015 Mathematical Contest in Modeling (MCM) Summary Sheet
Summary
The complex epidemic of Zaire ebolavirus has been affecting West Africa. A series of realistic, sensible, and useful mathematical model about Ebola of spreading and medication delivery are developed to eradicating Ebola.
这个复杂的传染病，埃博拉，已经影响西非好久。一系列真实可信的关于抑制埃博拉传播和医药相关的数学模型正在建立。
First, we divide the spreading of disease into three periods: naturally spreading period, spreading period with isolation but without effective medications and spreading period with effective medications. We develop a SEIR (susceptible-exposed-infectious-recovered) model to simulate the spread of the disease in the primary period. Then the model are improved to a SEIQR (susceptible-exposed-infectious-quarantined-recovered) model to adapt to the second and third period and to predict the future trends in 第一，我们把这个疾病划分为三个部分：自然传播过程，没有有效药物控制的独立传播过程和有有效控制的传播过程。我们建立了一个SEIR （易受暴露的感染性恢复）模型来模拟早期疾病的传播。接着我们又把模型升级为一个SEIQR （易受暴露性感染后的隔离恢复）来适应第二和第三阶段的传播同时也用来预测在Guinea, Sierra Leone and Liberia.的传播情况。
According to our plan, drugs are delivered to countries in need separately by air, then to medical centers by highway and be used for therapy of patients there. To solve the problem of location decision of medical centers , which belongs to a set covering problem , we developed a multi-objective optimization model . The model’s goal is minimizing the numbers of medical centers and total patients’ time cost on the road on the condition that all of patients can be sent a medical center in time. We solved the model with genetic algorithm, and get an approximate optimal solution with 7 medical centers.
根据我们的计划，运送药物的国家个别的需要空中运输，然后疾病中心通过高速公路将药物运送到患者手中。为了决定当地医疗中心位置的一个覆盖问题，我们建立了一个多元线性规划的选择模型。这个模型的目的是最小化医疗中心与主要患者之间在路上的距离使得所有患者能够被及时得送去医疗中心。我们用一个演变的算法解决了这个模型并得出一个最优化的结果----7个医疗中心。
Then we built a logistic block growth model to describe the changing speed of drugs manufacturing. Comparing it with the SEIIR model, we considered the two situations: one is in severe shortage of drugs, the other is relatively sufficient in drugs. We built two optimization models for the two situations. The optimization goal is minimizing the number of the infectious and minimizing of death cases and the number of infectious individuals, respectively. The decision variables is the drug allocation for every country, and the constraint conditions is drug production.
接着我们便建立了一个逻辑上的增长模型来描述这个药物生产的改变速度。跟SEIIR 模型比较起来，我们考虑了两个处境：一个就是药物上的严重不足，另一个就是药物充足。也因此建立了两个优化模型针对这两种情况。最优的方案就是各自缩小死亡和感染的人数。这个结论是根据每个国家的制药厂和制药的约束条件决定的。
Finally, by a comprehensive analysis, we made a six-month drug delivery plan for Guinea, Liberia and Sierra Leone, and predict the spreading trends of Ebola in the next six month with the efficient medication’s blockage.
最后通过一个复杂的分析，我们为Guinea, Liberia and Sierra Leone制定了一个六个月的送药计划，并且预测了埃博拉疾病在有效药物控制情况下的发展趋势。
The sensitivity analysis of our models has pointed out that the transmission rate and the initial value setting will affect the result greatly. We find that the speed of drug manufacturing’s growth rate may con trol of the epidemic. The minimum daily output of drugs G m must greater than 4000. Otherwise ,the epidemic will be out of control.
我们这个模型分析的准确性极大地取决于运输速度和重要地点的设立。我们分析药物的生产速度r会控制这个疾病。每天药物的生产量必须得有4000.否则，这个疾病就会失去控制。
Team # 35532 1 / 34 Introduction（介绍）
Problem background
A complex epidemic of Zaire Ebola virus (EBOV) has been affecting West Africa since approximately December 2013, with first cases likely occurring in southern Guinea [1] and facilitating several transmission chains to progress essentially unchecked in the region and to cross porous borders with neighboring Sierra Leone and Liberia and seed a limited outbreak in Nigeria via commercial airplane on 20 July 2014 [2]. Then the number of new cases appear an exponential growth. While public health interventions have been introduced in all affected countries, the numbers of infected cases and deaths from EBOV continue to increase due to the loss of effective medication. A total of 22,495 cases, with 8,981 deaths, have been reported to the World Health Organization as of 4 February 2015[3].
2014年4月3日- 世卫组织支持国家当局对埃博拉病毒病疫情作出应对（埃博拉病毒病；以往称作埃博拉出血热）。现在已经确定，该疫情由一种与扎伊尔型埃博拉病毒十分接近（98％）的埃博拉病毒引起。这是首次在西非发现该病。在几内亚东南部的森林地带首次报告出现病例。疫情发展很快，若干地区以及科纳克里已报告发生由埃博拉病毒病引起的病例和死亡情况。在邻近国家也报告发生了少量疑似病例和死亡，这些病例都曾经由几内亚。几内亚和利比里亚已报告发生确诊病例。（世界卫生组织组织的话，暂不翻译）
Our work
Since new anti-Ebola drug has been developed, a realistic, sensible, and useful mathematical model about drug allocation and delivery is necessary. Depending on the goal, we divide the spread of disease into three periods: naturally spreading period, spreading period with intervention and spreading period with effective medications. And then we develop a SEIR (susceptible-exposed-infectious-recovered) model to simulate the spread of the disease in the primary period, based on the changing trend of the numbers of infected cases and deaths. Finally, the model are improved to a SEIIR (susceptible-exposed-infectious-isolated-recovered) model to adapt to the second and third period and to predict the future trends in Guinea, Sierra Leone and Liberia.
自从新的埃博拉疫苗被研发出来，一个有效实用的药物运输和存储数学模型就是必要的。基于这个目的，我们将疾病传播分为3个时期：自然传播，不受药物控制的传播和收药物控制的传播。同时我们基于这些不断变化的感染和死亡的数据，建立了一个SEIR模型来模拟疾病早期的蔓延。然后，最终的一个升级模型就是SEIIR，这是适用于二三阶段并预期在这几个地区的发展趋势。
A delivery system include two main parts: numbers and locations of medical central delivering drugs decision and drug allocation of every medical center. For this reason, an evaluation system of every country’s long-term epidemic situation, two multi-objective optimization are built separately. We developed a detailed plan via our models.
一个运输系统应当包括两个方面：一个就是医疗运输中心位置的数量，另一个就是每个医疗中心的药物存储位置。由于这个原因每个国家对于流行性的长期疾病的评估系统就要结合这两个多元的优化结果而独立建立。
Symbols and definitions
(1) Symbol for SEIIR model with effective medication
Symbol definitions
S susceptible individuals（易感
人群）
E exposed individuals（被暴露
的人群）
I infectious individuals（被感
染人群）
I isolated isolated infectious individuals（被感染后的隔离人群）
I free non- isolated infectious individuals（感染后没
有被隔离的人群）
R recover and survive individuals（恢复率和存
活率）
D death case（死亡率））
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Team # 35532
N
β
1 σ
1 γ
f
μ
f 1
f 2
G (t ) (2)Symbol for model II
Symbol p
q
V =
{
}
v , v , , v ,
, v
p
1 2
i
X = {x , x ,
x
j
, x
}
1
2
q
d ( x , y)
h
i
R
n
M n
r n :
m
:
n
B n
I
n
2 / 34
total population of the area （该地区总人口）
The transmission rate of EBOV （疾病传播速率）
the average durations of incubation.（潜伏时长）
the average durations of infectiousness （感染时长）
the case fatality rate （致死率）
the ratio of new non-isolated infective
patients （新的未被隔离的患病率）
the non- isolated individuals fatality rate （未
被隔离的死亡率）
the isolated individuals fatality rate （隔离后的死亡率）
number of drug supply （药物供给
量）
Definitions （定义）
the number of capitals of first-level
administrative divisions in a country （这个国家一级管理部门的数量）
the number of medical centers for EVD
set in a country;（全国为EVD 的医疗救护中心）
the set of capitals （中心的建立）
the set of medical centers （医疗救护
中心的建立）
he shortest distance from x to y(X 与Y 的最短距离)
the metric of quantity of drug needed of the i th administrative in the coming period （在
未来一段时间内药物需求量的管理距离）
the population of the n th administrative
division
the area of the n th administrative division
the population of the main city
of the n th
administrative
division.（n个管理部门的分
布）
the area of the main city of the
n th administrative division.
（主要城市的n个管理部
门的分布）
the temperature of the n th administrative
division
the Infected people number of the n th administrative division is represented （分隔地区下的受感染人群数量）
鉃??*??錡鍦醟w釱?/鈅???錓嶞?`oo醷?鎌
Team # 35532 3 / 34
F
open degree of Ebola treatment centers （埃博拉医疗中心的公开程度）
n
C n the temperature of the n th administrative
Division （每个分离地区的温度）
F
open degree of Ebola treatment centers
（埃博拉医疗中心的公开程度）
n
c
the number of open ETCs in the country （在该国家ETCs 的公开程度）
open
c
the number of cities in the country （）
all
E n
The economy of the n th country （）
J
The education level of the n th country i
n
K
age coefficient （年龄因素）
(3)Symbol for model
III
Symbol
definitions
G The average daily production of drugs （每日用药量）.
G
the maximum production rate of the drugs
（每日药物的最大生产量）.
m
rt
growth trends in drug production speed （药物的生产趋势）
P
the aggregate of drug （药物总数）
t n
t n
the first day of the time period of drug
Delivery.（第一天的药物投递速度）
T
p
the time period of drug delivery （运输周
期）
I
t he number of non-isolated infectious
i , free
Individuals （未被隔离的受感染人群数量）
G (t )
i th country’s number of the drug （该国家
的药物量）
i
N
i
i th country’s the total population （该国总
人口
D
i th country’s death cases （该国死亡数）
i
Assumptions （假设）
1. In early transmission period, EBOV spread in absence of control interventions.在周期传播过程中疾病是失去控制的）
2. The average duration of the incubation and infectious period were fixed to previous estimates
from an outbreak of the same EBOV subtype in Congo in 1995 (1σ= 5 .3 days and1γ = 5 .61 days) [3].这个潜伏和感染的平均周期是和1995年相同疾病爆发时是一样的
3. Infection only occurs between the patients who are not isolated and susceptible.
感染只发生在患者和未隔离的易感人群之间
Team # 35532 4 / 34 4.Even if without effective medication, the mortality rate of isolated patients is lower than
that of free patients.即使没有有效的药物，隔离人群的死亡率比不隔离的死亡率低
5.People who have recovered from Ebola will obtain the long-term immunity against Ebola.已经恢复的人会获得易感长期的免疫能力
6.Only being sent to medical center and isolated, affected people have the chance to receive therapy.只有被送到医疗中心救治和隔离，感染人群才有机会能被救治
7.Ignoring patient’s different condition, the quantity of the medicine needed by all affected people whose disease is not advanced during the whole therapy is equal to one portion of drugs.忽略患者之间的差异性，用药量在整个救护期间是平均分配的，只要其病情为加重
8.The population of the region we study is a constant N该地人口的数量为常数N
9.The transmission only occurs btween only the susceptible and the infective.
传播只在患者和易感人群之间
10.Patients isolated will not affect the suspective. In other word,their effective contact rate equals to zero.被隔离的患者不会产生影响，换句话说，他们的的影响规律为0
11.The medication only refers to drugs used therapy of Ebola, not include vaccine.提到的药
物仅仅只包括埃博拉疫苗，不包括其他的
12.If affected people want to accept therapy, they must get to a medical center set as delivering medication.如果感染人群想要接受治疗，他们必须去救治中心登记
13.If a patient in whom symptoms of Ebola has appeared don’t get a medical central (usually the nearest medical center) in 24 hours, the new medication won’t affect his disease.如果患者的症状表现出来后再4小时内没有被最近的救护站救治，新药将失去药效
14.We set all the medical centers at the capitals of first-level administrative divisions.我们设
立全部的救护中心在每个隔离区的重要位置
15.We use the capital of an administrative division as a representation of the administrative
我们用隔离区的中央位置来代表隔离区的疫情
Model I
SEIR model
After transmission of the virus, susceptible individuals S enter the exposed class E
they become infectious individuals I that either recover and survive R or die D.（在病
毒的传播过后，易感人群S在他们没有变成感染人群I并没有恢复R或死亡D的情
况下进入暴露区E，）
Fig.1 shows the process.
Fig.1 the process of the model
According to the it，people can be divided into four class:
● the susceptible ( S ),who are susceptible to infection.（易被感染
● the exposed(E), who are affected but in the incubation.（被感染但潜伏
● the infectious ( I ), who are infected and have the symptom.
（被感染并有征兆
●
the recovered(R), who recover or survive.
The total population of the area is N, and
N = S + E + I + R ,
before
according to the assumption 8. Numbers of all kinds of people in the initial time are shown as follows:（所有人群的数量在最初的时间里如下所示
S (0)= S0>0, E (0) = E0> 0 , I (0) = I0> 0 , R (0) = R0 =0 Letβ be the contact rate in absence of control interventions. S N is the proportion of the 让 β为未受控制时的系数 S N为易感人群在总人群中的比率
susceptible in total population. Hence, the effective contact rate is βS N , and in a unit of time,