发布时间 : 星期六 文章数学建模 美赛 2015 A题更新完毕开始阅读
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To address the meditation delivery system, we have to determine the locations for material receiving. A proper pick of locations will definitely benefits the delivery system.
Our model is based on real world situation, so we choose international airports for medication transportation. The airports receive the supply from medication manufacture sites, and send them to the capitals of each district to meet the optimal distribution plan.
Here we ignored the delivery process among manufacture sites and airports. Because the manufacture sites are located in other continents, the delivery system is affected by many unknown factors, which makes the problem rather complicated.
? Transportation System
Now the sending sites and the receiving sites are determined, so are the quantities needed in each receiving sites. The optimal delivery system should has the shortest total travelled distance. It is described as our model objective:
min (??cijdij)
i?1j?1nm(8)
and
?n??cij?yj?i?1?ms.t. ??cij?xi (9)
?j?1?c?0?ij?where i identifies different airports; j identifies different districts; dij is the distance between airport i and district j; xi is the quantities of medication sent from airport i; yi is the quantities of medication received by district j; cij is the quantities of medication delivered form airport i to district j.
The solution should be a matrix of cij, which describes the whole delivery system. 3.5.1 Solution and Result
There are two international airports in Sierra Leone: SherBro International Airport and Lungi International Airport. We acquire the geographic coordinates of them, alone with the information of capitals of each district. So the distances among them can be calculated.
Table 5: Distances among sites KailahuKeneKambiPort KoinaduDistance(km) Kono Bombali
n ma a Loko gu
SherBro International
229.6 151.2 210.5 158.9 181.6 140.5 245.8
Airport
Lungi International
290.8 235.5 244.6 129.9 63.51 47.86 210.5
Airport
Team # 38915 Page 15 of 21 Distance(km)
Tonkolili
Bo
Bonthe Moyamba Pujehun Freetown
Western Rural Area
SherBro International
146.2 97.89 1.59 70.52 90.51 132 107.6
Airport
Lungi International
138.1 176.2 143.1 97.98 215.1 15.28 34.52
Airport The airports are only for sending out vaccine, and the capitals are only for receiving materials. So the distances needed exclude those among capitals.
We implement Lingo to solve the problem, and we get the delivery system as follows:
Table 6: Delivery system
KailahuKeneKambiPort Koinadu
Vaccine delivery amount Kono Bombali
n ma a Loko gu
SherBro International 4971
60240 44925 161781 0 0 22859
Airport 5 Lungi International
0 0 0 0 2842 137868 2815
Airport
Western
TonkoliBonthMoyamPujehuFreeto
Vaccine delivery amount Bo Rural
li e ba n wn
Area
SherBro International 1449
54310 35045 50093 1518 0 0
Airport 5 Lungi International
0 0 0 0 0 141948 209508
Airport
The table above indicates our final plan for Sierra Leone. Theoretically, the plan produced by the algorithm is the best solution of the problem.
IV. Sensitivity Analysis
4.1Influence of β
βI refers to the possibilities of susceptibles getting Ebola from infectious individuals, with βQ and βF meaning from quarantined individuals and funeral respectively. Although these three parameters all refer to the effective contact rate, the influence power on epidemic model of each other is different, because of the different infection source and other factors. Now we give a sensitivity analysis by changing the value of these parameters, then we get the evaluation of each contact rate.
Continuing our work on Sierra Leone and comparing with the results in first model, we reduce 0.2 in βI, βQ and βF respectively and observe the change under three circumstances. 4.1.1 Influence of βI
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The value of βI in SEIQFR model decreases by 0.2, while other parameters are unchanged. We use MATLAB to draw the prediction curves with changed and unchanged parameters, the figure is as follows:
Fig 7: The spread situation with βI changed
It is obvious that the value of peak decreased by 100 thousands approximately, and the arrival time of peak is delayed about 3 months after changing the value of βI. The decrease of peak means that the epidemic situation is controlled effectively with the decrease of infection rate of I. And the delay of peak time gives us more chance to take other measures to control the spread of Ebola. 4.1.2 Influence of βQ
With only the value of βQ decreasing by 0.2, we compare the result and that in model one.
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Fig 8: The spread situation with βQ changed
We find that the curve tendency is similar with former one. The difference between the values of two peaks is about 5 thousands, and the time delayed is 50 days. 4.1.3 Influence of βF
With only the value of βF decreasing by 0.2, we compare the result and that in model one.
Fig 9: The spread situation with βF changed
The difference between curve peak is about 2.5 thousands, and the time delayed is about 20 days.
4.1.4 Analysis of results
The dip of peak value between changed and former ones is due to a change of I. And the difference between peak time of changed and former curves means the deadline for we to take controlling measures. Both the peak value dip and the peak time delay are the bigger the better. So we can draw the conclusion that the influence of βF is the least, while βI is the greatest, which means that the contact rate between S and I is the most impactful parameter for controlling the epidemic. 4.2 Time begin intervention
When the outbreak happens, measurements will be took by the government. We focus on the intervention time of quarantine. To compare the difference caused by various intervention time, we simulate three situations with the same level of quarantine but only at different moment. And one case without intervention for reference.