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(36448)2015美赛A题论文

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Team Control Number 36448

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2015Mathematical Contest in Modeling(MCM)Summary Sheet

Abstract

This paper aims to establish a model to distribute drug to the various affected areas rationally and efficiently.Our model includes the following factors.

●Spread of the virus:We select some data about the number of infections in the area of

West Africa.We calculate the sum of the numbers of infections,forecast total number of infections in the coming six weeks via the analysis of Time Series Models.

●The quantity of the medicine needed and speed of manufacturing of the drug:We access

to relevant information and compare the demand to the supply for the drug at at this stage, We can conclude that drug is in short supply at this stage.

●Selection of the distribution centers:We take ETC as the delivery destinations.We take

the airport as the distribution center.According to the current number of infections and the predicted number of infections in the coming six weeks,we classify ETCs into three categories:A level,B level and C level.Then we establish Goal Programming Model to select the most appropriate8airports as distribution centers(result is shown in Table

4)based on the level of ETC and distances between the airports and ETCs.

●Drug distribution mechanism:According to levels and number of ETC which is within

the distribution circles and the speed of manufacturing of the drug,we can select the appropriate distribution centers and the total quantity of drug distributed to various airports.The quantity of drug which should be delivered to various ETCs is determined by the level of the ETC.Finally,ETC distribute the drug to nearby epidemic areas.The drug distribution mechanism is economical and efficient with such optimization.The result is shown in Table4.

Innovations of our model:

1.We predict the number of infections over the next six weeks by the ARIMA model.Thus we can calculate the amount of drug of various epidemic areas in advance and effectively.In this way we can make preparation for Follow-up treatment.

2.We distribute the drug by the level of ETC to make the best use of the shortage drug.

3.We reduce the cost of delivery under the premise of taking the least time to distribute drugs.

Background

The Ebola virus causes an acute,serious illness which is often fatal if untreated.The current outbreak in west Africa,(first cases notified in March2014),is the largest and most complex Ebola outbreak since the Ebola virus was first discovered in1976.There have been more cases and deaths in this outbreak than all others combined.According to the report from the World Health Organization,in West Africa,cases number of infection in countries such as Liberia,Sierra Leone and Guinea,the most three severe countries there with the plague of Ebola hemorrhagic fever,reach more than twenty thousand(including suspected cases),above one million deaths included,meaning that the mortality is higher than50%.And after comparison between countries,the infected cases,diagnosed or suspected,and deaths in Guinea,Liberia,Sierra Leone,Mali,United States as well as the other three countries which have finished this epidemic,including Nigeria,Senegal and Spain,are even more.The numbers of infected cases and deaths in West Africa have reached a record high,and the situation is still in a state of deterioration.Guinea,Sierra Leone and Liberia have very weak health systems,lacking human and infrastructural resources,having only recently emerged from long periods of conflict and instability.On August8,The director-general of the World Health Organization announced that this epidemic is a public health emergency with international concern.Several aid agencies and international organizations,including the US Centers for Disease and Prevention(CDC),the European Commission and the Economic Community of West African States and other units are trying to extenuate investment in human disease.[www.who.int]Now American experts have developed an effective drug treatment which can prevent disease and cure disease not too serious.So now it is necessary to deliver the drug to all infected areas reasonably and efficiently after consideration of the infection and traffic conditions in all regions,to mitigate the current epidemic pressure and eradicate Ebola virus eventually,thus reduce the losses to a minimum.

Problem Description

According to the World Medical Association's report,there is new drug that can prevent the spread of Ebola and cure the disease is not serious patients now.We combine spread of the virus,quantity of required medicine,feasible delivery systems,locations of delivery and speed of manufacturing of the vaccine or drug to establish a viable model to optimize the eradication of Ebola,or at least its current strain.Also,a1-2page non-technical letter should be prepared for the world medical association to use in their announcement.

Terminology and Definition

Table1.

Model parameters.

The natural logarithm of the number of patients per week

B Backward shift operator

Random interference

The number of patients with Week（i=1,2,3……34）

αA level ETC demand coefficient（Highest level）

βB level ETC demand coefficient

γC level ETC demand coefficient（Lowest level）

μVaccine or drug demand intensity

m Number of A level ETC in the distribution circle

n Number of B level ETC in the distribution circle

k Number of C level ETC in the distribution circle

x Drug production in per week

Quantity of drug which is distributed to airport

L The summations of distances from various ETCs which are in the intersection of various distribution circles to the airports.

d Th

e distance between the ETC and airport Assumptions

1.During the use of drug therapy,there is no inflow or outflow of population in disaster area.

And there is no change of population density.

2.Some individual cases,which is not in the three worst-hit countries:Guinea,Liberia and

Sierra Leone won’t impact of Ebola Treatment.

3.During the treatment,the virus won’t mutate.Drug can suppress the virus.

4.The infection mechanism of Ebola virus won’t change.

5.The drug won’t be damaged during the transport.

6.The airplanes,cars and staff for drug delivery are adequate.

7.When the ETC is very closed to the airport,we regard the distance between them as0km.

The Model

Figure1:Factors included in the model Preparations for model

Spread of the virus

We review statistics(Table1)of the infections and deaths from March1,2014to February4, 2015in Guinea,Liberia and Sierra Leone---the three worst-hit countries.

1.Handling missing data

Because of missing data segment data has little impact on forecast,we omit the missing data directly.

2.Sum

Calculate the sum of infections of the three countries to get the total number of infections.

3.Handling redundant data

We take‘week’as a unit because the date of published data is irregular,and keep weekly data when we omit the redundant data directly.

4.Establishment of ARIMA model

We establish ARIMA model with SPSS21.0.After the identification of model,parameter estimation and model diagnostics establish comprehensive model---ARIMA(0,2,1)

Model:

Predict the number of patients over the next six week:

Table2.

The actual and predicted values of the total number of patients since June31,2014 Serial

Date Actual Cases Predicted Cases number

131-Jul-141322.001243.69 28-Aug-141766.001555.38 315-Aug-142115.002186.61 422-Aug-142599.002591.12 528-Aug-143052.003189.52 66-Sep-143944.003697.86 712-Sep-144366.004875.72 818-Sep-145325.005226.20 926-Sep-146553.006414.24 103-Oct-147470.007949.28 1110-Oct-148376.008902.65 1217-Oct-149191.009810.84 1325-Oct-1410114.0010567.08 1431-Oct-1413540.0011486.63 157-Nov-1413241.0016155.43 1614-Nov-1414383.0014906.53 1721-Nov-1415319.0016033.19 1828-Nov-1416899.0016859.90 193-Dec-1417111.0018624.72 2010-Dec-1417908.0018404.49 2117-Dec-1418569.0019120.08 2224-Dec-1419463.0019668.97 2331-Dec-1420171.0020566.36 247-Jan-1520712.0021207.46 2514-Jan-1521261.0021639.82 2621-Jan-1521689.0022113.30 2728-Jan-1522057.0022445.42 284-Feb-1522460.0022724.75 2911-Feb-152**** ****-Feb-152**** ****-Feb-1524488 324-Mar-152**** ****-Mar-152**** ****-Mar-1527296

The quantity of the drug needed and the speed of manufacturing of the drug

We take‘week’as a unit to calculate the amount of drug

the coming week needed according to predicted increasing the quantity of coming weeks and the total number of infections.

Then the number of the increased patients weekly since February4,2015:

(i=29,30,31,32,33,34)

By calculating，They are351,671,740,826,929,1053.According to past experience of treatment,we need2doses of this drug to cure one patient.[Cai,August29,2014].So,the quantity of the drug needed weekly is2*.Obviously,the demand for the drug is much larger than the production of the drug in the early production of drug(the next few weeks).It’s in short supply.So it is important to distribute the drug consciously.

Classification of the ETC

Figure2:The distribution of epidemic(Source:https://www.wendangku.net/doc/6b1217011.html,/)

Figure3：The distribution of ETCs and International Airports (Source:https://www.wendangku.net/doc/6b1217011.html,)‘A’represents International Airport in the figure According to the severity of the epidemic which is shown in Figure2and Figure3where the ETC is located,we classify ETCs into A level,B level,C level,

A level:501-4000

B level:101-500

C level:1-100

Demand coefficient of ETC of each level:

A level：α=4000/(100+500+4000)=0.87

B level：β=500/(100+500+4000)=0.11

C level：γ=100/(100+500+4000)=0.02

Basic model

The selection of airports

Searching for the distances from each ETC to the airports,then build table3(sources: https://www.wendangku.net/doc/6b1217011.html,/）,when they are within200kilometers from airports in usual,as a result,we call it an airport as the center point that the radius of circle is200

Kilometers’distribution circle.

When the ETC only belong to just the distribution circle not the other distribution circle, the distribution circle that corresponds to the airport is necessary.

When the ETC belong to many distribution circles,we are supposed to lead into necessary degree coefficientμ,and whether the distances to the airports are necessary to analyze.

μ=m*0.87+n*0.11+k*0.02

As to a ETC that belongs to i distribution circles at the same time,we choose airport to the distribution circle by comparing toμ.

Max｛,……｝

The distribution circle which is max that is necessary.

If the number of airports is j that eachμis equal,we will count these airports distances L in total whichμis equal,supposing k ETCs are in the intersection of j distribution circles, Then

Then

Min{,,……}

L which is minimum is necessary.

Combining with the Table3,using the methods to all the ETC,we will reach a conclusion that KATOUROU Airport,BADALA Airport,LUNGI Airport,KENEMA Airport, KISSIDOUGOU Airport,BALLA YELLA Airport,JAMES SPRIGGS PAYNE Airport, NIMBA Airport.These8airports are the most proper distribution center.

Quantity of drug which is distributed to airports

Quantity of drug which is distributed to airport

Table3.

The distances from ETCs to the nearby airport

.Airport

ETC KATOU

ROU

CONA

KRY

BAD

ALA

LUN

HASTI

NGS

GBANGH

ATOK

BO KENE

TELIMELE(

118km262km

CONAKRY(

160km0km

DABOLA(C)161k

KISSIDOUG

OU(C)

FREETOWN

(A)

0km38.6km193km

MAKENI(A)133k

m 167km178km199

197km

BO(B)0km73km KENEMA(B)73k

0km

DARU(B)126

km 51.3k m

KAILAHUN( A)181

107km

KOINDU(B)143km BEUDU(A)110km FOYA(A)123km GUECKEDO

U(B)

MACENTA(

VOINJAMA(

BOMI(B)

JFK

HOSPITAL(

BONG

COUNTY(B)

SANNIQUE

LLIE(B)

NIMBA

COUNTY(B)

Table3(continued)

Airport ETC DAR

YENGE

KISSIDOU

GOU

MACE

NTA

BALL

YELL

JAME

SPRIG

PAYN

ROBERTS

INTERNATI

ONAL

NIM

TELIMELE(C)

CONAKRY(B)

DABOLA(C)

KISSIDOUGO

U(C)

0km134km

FREETOWN(

MAKENI(A)149km

BO(B)125k

165km

KENEMA(B)52.2

107km

DARU(B)0km103km

KAILAHUN( A)57.5

159km

KOINDU(B)92.4

195km

BEUDU(A)67k

173km

FOYA(A)82k

227km

GUECKEDOU

(B)

83.1km91.1km MACENTA(A)134km0km

VOINJAMA(A )120.5 km

BOMI(B)64.6k

117.2km

JFK

HOSPITAL(A)

3km54.7km

BONG COUNTY(B)71.1k m

SANNIQUEL LIE(B)22k m

NIMBA COUNTY(B)113k m

Extended Model

There are many factors that affect the choice of the airport and the quantity of drug needed to be delivered to each airport,such as the spread of the virus.We predict a change in the number of infections caused by transmission of the virus by the time series model.According to forec asts of the spread of disease,we recognize that the number of infections in Liberia will decreas e.After some days,we will reduce part of drug supply to Liberia,this part of the drug can be de livered to the worsening epidemic area such as Guinea and Sierra Leone.Considering the the d istances between ETC and the epidemic areas,drug delivery from ETC to the epidemic of can be further optimized.We classify epidemic areas according to the severity of the epidemic.Th en we build model according to the regional classification and distance to calculate the optima l delivery routes.This will make our distribution system better.In order to effectively curb the spread of the virus,uninfected people should be vaccinated,while infected people are being tre ated,and vaccine distribution should be determined by the total number of people infected and population in epidemic areas.

Figure4：Changes in the number of infections due to the spread of the virus

Solution to the Requirements

Table4.

The quantity of drug should be distributed to each airport and ETC

Airport Supply of

drug

account

for(%)

Quantity

of drug

ETC Demand for

drug

account

for(%)

Quantity

of drug

to ETC

KATOU 1.60.016x TELIMELE(C)26.70.004272x

CONAKRY(B)73.30.011728x BADALA0.30.003x DABOLA(C)100.00.003x LUNGI21.40.214x FREETOWN(A)50.00.107x

MAKENI(A)50.00.107x KENEMA37.50.375x BO(B) 3.60.0135x

KENEMA(B) 3.60.0135x

DARU(B) 3.60.0135x

KAILAHUN(A)28.60.10725x

KOINDU(B) 3.60.0135x

BEUDU(A)28.60.10725x

FOYA(A)28.60.10725x KISSIDOUGOU12.30.123x KISSIDOUGOU(C) 2.10.002583x

GUECKEDOU(B)10.90.013407x

MACENTA(A)87.00.10701x BALLA YELLA12.10.121x VOINJAMA(A)88.90.107569x

BONG COUNTY(B)11.10.013431x

JAMES SPRIGGS PAYNE 12.10.121x BOMI(B)11.10.013431x

JFK HOSPITAL(A)88.90.107569x

NIMBA 2.70.027x SANNIQUELLIE(B)50.00.0135x

NIMBA

COUNTY(B)

50.00.0135x

Result is shown in Table4.Firstly,we should distribute drug to airports,the quantity of drug

is calculated by the level and number of ETC.We assume that the quantity of drug can be pro vided by pharmaceutical factory is x per week.The quantity of drug which should be delivered to each airport is shown in Table4.Secondly,we distribute drug to ETCs according to the leve l of ETC.

Advantages and disadvantages

The model considers various aspects of factors.It clarifies the causal relationship between the factors,provides the basis for the establishment and optimization of the model

1.We use small amount forecast.The predicted values are more accurate.

2.We have the necessary data to prove Conclusions.

3.The model is targeted，it is very suitable for the West African epidemic.

Of course there are also some shortcomings.

1.The model is general poor,,only suitable for the West African epidemic.

2.Due to lack of data,we made certain assumptions with a certain subjectivity,such as drug

production rates.

3.We didn’t take therapeutic effects of drug into account.

References

J.B.Wu,L.X.Ye,Er,K.You.2006.ARIMA model to predict the incidence of infectious diseases.

Journal of Mathematical Medicine2007（1）

W.T.Zhang.SPSS Statistical Analysis Tutorial.Beijing Hopes Electronic Press.2002, 250~289

Wikipedia.Ebola hemorrhagic fever

https://www.wendangku.net/doc/6b1217011.html,/wiki/%E5%9F%83%E5%8D%9A%E6%8B%89%E5%87%BA%E8 %A1%80%E7%83%AD#cite_note-Ruz2014-2

Xu J ingjing，W ang Ha iyan.2009.Dynam ic vacc ine distr ibution modelba sed on epidem ic diffusion rule and cluster ing approach，Journal of Southeast University2010（3）

Letter

Current situation

The current EVD outbreak began in Guinea in December2013.This outbreak now involves transmission in Guinea,Liberia,Nigeria,and Sierra Leone.As of4January2015,countries have reported22560cases(13888confirmed,8672suspect),including9019deaths.This is currently the largest EVD outbreak ever recorded.In response to the outbreak,we have taken many measures to mitigate the epidemic.Now the epidemic has been controlled to a degree. However,due to many infections,our work is still arduous.

Progress

Fortunately,our new medication can stop Ebola and cure patients whose disease is not advanced.With the new medication,we can cure lots of infections and reduce the risk of infection of normal.Our next work is to deliver drug and vaccine to infected areas quickly. We have developed a detailed plan to do this work.We believed Epidemic will be controlled in a short time.

Advice

At the same time,in areas of intense transmission(e.g.the cross border area of Sierra Leone, Guinea,Liberia),the provision of quality clinical care,and material and psychosocial support for the affected populations should be used as the primary basis for reducing the movement of people,but extraordinary supplemental measures such as quarantine should be used as considered necessary.Also,Guinea,Liberia,Sierra Leone should organize some staff some vehicles to assist to receive and deliver drug and vaccines.

With these efforts,Ebola can be beaten.