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
Team # 38915 Page 10 of 21
fixed sites. To meet the optimization objective (minimizing the sum of I in each district after a short period of days), a corresponding vaccine offering plan can be reached by intelligent algorithm. The solution is a plan describing ideal quantities of vaccine gave out in each site per day (vaccinating speed).
3.4.1 Ascertainment of terminology
? Immune Population
To address the problem, we introduced 1/γv as the vaccine period, like 3 days of vaccine and once per day. σ is the effective probability of the vaccine, for we consider the probability of people vaccinated acquiring the resistant ability may not be 100％ but somewhat near it. Considering the people taking vaccine are either susceptibles or exposed, the population range of vaccine is S + E, while the vaccine has no effect on exposed individuals, so the effective rate is S/(S + E). When offering vaccine, there is a limit on the capability of one site per day, which is indicated as Ai/p, so the number of new immune people per day is
ASIm=?v??i (5)
S?Epwhere Ai is the medication quantity delivered in one site; p is the period of medication delivery. The properties of Im group are the same as the recovered group, thus R becomes bigger, which is one impact brought by vaccine. Simultaneously, S becomes smaller.
? Medication Demand
The medication needed in each district is depended on the epidemic stage. To be specific, a district with bigger E(t) and I(t) is on a more serious stage than those with smaller. Therefore, to achieve our objective, more medication should be distributed to districts on more serious stage. The needs for one person is set as one unit. The whole demand for medication is named as D. It is assumed that the site can equally give out all vaccine during one delivery period, while the constraint is manufacture speed, that is Ai ? Vaccine Manufacture Speed Another feature in this model is the medication supplement. Only with steady and sufficient manufacture process of vaccine, can the timely and sufficient delivery be assured. Then the constraint mainly comes from the total quantity of medication made in one period (p), which we set as A. As is mentioned before, there is a limited number of delivering vaccine in one site per period, which is indicated as Ai and we have Ai Team # 38915 Page 11 of 21 Our objective of this model is the medication distribution. That is, the values of Ai. Our objective function is min (?I(jp)) j?1n(6) where n is the total number of districts concerned; the values of I is used that for one period later. And we have ?n??Ai?A ,i=1,2,?,ns.t. ?i?1 ?D?A?0i?(7) The connection between I and Ai can be referred to formula (1) and (4). Here we introduce Particle Swarm Optimization (PSO) to solve the problem, which is an intelligent algorithm used to find optimal solution. According to the formula (5), fitness value (V) used to evaluate possible distributions is V??I(jp) j?1nSo the objective is indicated by the fitness value, which is the sum of I for several days (a period) later. Initialize random position and velocityCalculate fitness valueFind peak values of individual and swarmCalculate fitness valueUpdate positionUpdate velocity （modified）Update peak values of individual and swarmReach iteration timesYesOutput resultNo Fig 4: Flow chart of MPSO algorithm MPSO algorithm is illustrated as flow chart above. The modified algorithm enables faster running speed.[8] We may set proper iteration times to balance between the running time of the program and the quality of the solution. More times it updates, the better is the solution. 3.4.2 Solution and Result We take Sierra Leone, a country with 14 districts, as a study case, so we have n = 14. Other settings are: A = 1,000,000; p = 4; 1/γv = 3 and σ = 1. Based on epidemic model, we Team # 38915 Page 12 of 21 calculate I in all 14 districts of Sierra Leone as the initial value: Table 4: Initial value of I in initial value District Kailahun Kenema Kono I for 4th Feb 565 502 246 District I for 4th Feb Tonkolili 448 Bo 314 Bombali 990 Kambia 150 Pujehun 31 Port Loko 846 Freetown 2029 Koinadugu 104 Western Rural Area 1118 Bonthe Moyamba 5 205 The initial values are extracted from the last day’s real data. So we get results: Fig 5: Pie chart of vaccine distribution in Sierra Leone The pie chart illustrates how vaccine should be delivered to each district. Theoretically, according to this plan, four days later, the sum of I should be a relevant small number. The results may not reach the overall best solution, for the iteration times are limited and the new plans generated are basically random. And the unit of quantities of vaccine is one share for one person at a time. For example, the optimal quantities delivered to West Rural Area is 209508 shares, which tops among all other districts of Sierra Leone. It is mainly resulted from the largest density of infected population there, that is the infected stage of the district. Team # 38915 Page 13 of 21 Fig 6: Influence of vaccine on I in Sierra Leone The bar chart shows there is a clear connection between I and vaccine quantities. And when offered more vaccine, I tends to become smaller, thus the vaccine has certain effect on reducing infected population. 3.4.3 Analysis of the Result In the part of ‘Medication Demand’, we mentioned that there is a relation between the districts’ epidemic stage and the quantities of vaccine for each districts. Comparing figure 5 and table 4, it is obvious that our results is reasonable. Bombali, Port Loko, Freetown and Western Rural Area, all has the large number of I, and large quantities of vaccine as well. Bontheand Pujehun, which with the lowest number of I, also have the lowest number of vaccine. The initial value of this model are extracted from the statistical data on WHO website. From the statistical data of Sierra Leone, we could draw a conclusion that the epidemic stage of Sierra Leone is serious now. So that the influence of vaccine maybe more effective on the early epidemic stage. 3.5 Medication Delivery Model In distribution model we ascertained the medication distribution for every district. If the departure location is confirmed, a medication delivery system is therefore developed. We build a delivery model to get an optimal plan of transporting corresponding amount of medication to each district. ? Departure Locations