2015年美赛A题优秀论文

For office use onlyT1________________ T2________________ T3________________ T4________________ Team Control Number35798Problem ChosenAFor office use onlyF1________________F2________________F3________________F4________________2015 Mathematical Contest in Modeling (MCM) Summary Sheet(Attach a copy of this page to your solution paper.)Farewell to EbolaSummaryThe current outbreak of Ebola has caused great harm to the people in West Africa since it began in December 2013. Fortunately, the world medical association has created new medication to stop Ebola. In this paper, we set up mathematic models in two parts in order to control and even eradicate the Ebola virus disease.On the one hand, we establish two transmission dynamical models, the medicine model and the medicine-vaccine model, which are based on the basic model of infectious disease. First, we divide people into different types. Then, we build the conversion relationship between different types of people, on basis of which we get the iterative equations. At last, we iterate the equations to obtain the predicting results. We find that the medicine has larger effect on the control of Ebola virus disease than vaccine, because the contact rate is pretty small. Therefore, it is urgent to deliver medicine to the infected districts. Besides, we obtain the optimal maximal quantity of medicine produced every day is 2000 pieces.On the other hand, we set up two optimal delivery routes for medicine and vaccine separately. The former focuses on the shortest time and it is optimized by the improved Dijkstra Algorithm. The latter concentrates on the least distance and it is optimized by the refined Saving Algorithm. Finally, we conclude that there are 11 trucks required to transport the medicine for 20 days and 7 trucks transporting the vaccine for 30 days when the daily number of vaccine transported is 0.3 million pieces.Additionally, we analyze the strengths and weaknesses of our models and write a letter for the world medical association to propose our suggestions.Content1Introduction (3)1.1Analysis of the Problem (3)1.2Literature Review (3)2Assumptions (4)3The Transmission Model (4)3.1The Basic Model of Epidemic Dynamics (4)3.2The Medicine Model (6)3.2.1Partition of People (6)3.2.2Assumptions (6)3.2.3Establishment of the Model (7)3.2.4Parameter of the Model (9)3.2.5Result of the Model (10)3.3The Medicine-vaccine Model (11)3.3.1Assumptions (12)3.3.2Establishment of the Model (12)3.3.3Result of the Model (13)4The Delivery Systems (17)4.1Collection of Data (17)4.2The Delivery Systems for Medicine (18)4.2.1Assumptions (18)4.2.2Establishment of the Delivery systems (18)4.2.3Result of the Model (19)4.3The Delivery Systems for Vaccine (20)4.3.1Assumptions (20)4.3.2Establishment of the Delivery Systems (20)4.3.3Result of the Model (22)5Conclusion (23)6Strengths and Weaknesse s (24)7 A Letter for the World Medical Association (25)8References (26)1IntroductionSince Ebola virus was first identified in 1976, no previous Ebola outbreak has been as large or persistent as the current epidemic. [1] From December 2013 when the virus began to spread to 1 February 2015, the World Health Organization (WHO) and respective governments have reported a total of 22,522 suspected cases and 8,994 deaths which are believed to be less than the actual numbers. [2]However, the epidemic has been controlled in a way by some measures such as changing funeral customs, isolating patients and so on. More fortunately, the world medical association has created new medication that could stop Ebola and cure patients whose disease is not advanced. In this paper, we establish a model to obtain the optimal delivery system of the new drug for eradicating Ebola under the least expense.1.1Analysis of the ProblemWe are required to set up a model for eradicating Ebola. Taking into consideration the spread of the epidemic, the quantity of the medicine needed, the delivery systems and other factors, we divide the problem into two parts.One part is the transmission of the epidemic. We use the SEIR model as the basic model and then establish the medicine model which considers the effect of medicine and the medicine-vaccine model considering the both effect of the medicine and vaccine.The other part is the delivery systems. Since the effect of medicine is distinct from that of vaccine, we set up two delivery systems for medicine and vaccine. The former focuses on the shortest time and the latter concentrates on the shortest distance.1.2Literature ReviewIn 1927, Kermack and Anderson created an epidemic model in which they considered a fixed population with only three compartments, susceptibles,S, infected, I, and removed, R, that is the SIR model.[3]Then, the SIR model was developed to many extensions, the SIR models with births and deaths, the SEIR model with latent phase considered, the MSIR model considering babies born with a passive immunity from their mother, etc.[4]After the current epidemic began in December 2013, many scientists estimated or predicted the number of cases in the Ebola epidemic. Nishiura et al. established early transmission dynamics of Ebola virus disease (EVD) in West Africa from March to August 2014. [5] Several scientists set up a mathematic model analyzing dynamics and control of Ebola virus transmission in Montserrado, Liberia. [6]The WHO team analyzed the epidemic and predicted the future epidemic. [7]There are also scientists researching the clinical manifestations of patients with Ebola. Mpia A. Bwaka and their companions researched the 1995 outbreak of Ebola in Congo, and obtained the clinical observations in 103 patients. [8] Schieffelin et al. observed patients at Kenema Government Hospital in Sierra Leone in 2014 and obtained the clinical symptoms. [9]2AssumptionsTo simplify the problem, we make the following assumptions.●We just consider Guinea, Liberia and Sierra Leone. Up to 1 February 2015,99.86 percent of the patients with Ebola are in Guinea, Liberia or Sierra Leone, [10]and Ebola will not spread widely in other countries for security check.●We divide the three countries into many districts and the epidemic will notspread between two districts. Actually, the countries have been divided into districts and there is security check on the boundaries of districts.3The Transmission Model3.1The Basic Model of Epidemic DynamicsSince the Ebola transmission has the same characteristics as the universal epidemic, it is suitable for the epidemic dynamic model. Meanwhile, Ebola has the following features. [11](1) Ebola has high fatality rate which of ZEBOV is 89% while SUDV’s is 53%.(2) There is latent period in Ebola which lasts 2~21 days, and the patients don’t have the infectious during latent period.(3) The Ebola patients are immune after recovery.Therefore, we use the SEIR model with latent period. [12]Let S denote susceptibles, E denote exposed individuals in the latent period, I denote infectives, R denote recovered people and N denote the total population whose value is 8612862/person, so we can the following relation figure.Figure 1: Conversions between each category in the SEIR model Where βNis the contact rate, 1/ωis the average latent period and γis the recovery rate.According to the relationship, we can the following equations.{dSdt=−β/NSIdEdt=βNSI−ωEdIdt=ωE−γIdRdt=γI(1)However, the above model is not suitable for the fact for the following reasons. (1) The medicine can only cure the patients whose disease is not advanced and vaccine acts on susceptibles.(2) There is a period of time for patients to recover after taking medicine, so the medicine should be supported continuously.(3) The cured people are immune for some time which is different in different people, and the number of cured person is small, so there is no statistical law.(4) Some patients can recover without medicine, and some people have genetous immunization to Ebola, that is, they have no symptoms after being infected. [11](5) The dead bodies are still contagious within a certain time, and the local funeral customs such as no cremation and contacting bodies aggravate the spread of the epidemic.(6) The production of the medicine is limited, so all of patients can’t take medicine.Taking these factors into account, we modify and extend the model so as to get themedicine model and medicine-vaccine model.3.2The Medicine ModelSince the infectious time of bodies, the treatment time and the latent period can’t be ignored, we use an iterative method to simulate the transmission of the epidemic. 3.2.1Partition of PeopleFor the medicine can only cure the patients with early disease and there is latent period, we divide all the people into the following kinds.Table 1 The situation of infection and medicine given on different kinds of people3.2.2AssumptionsIn order to simply the model, the following assumptions are made in this model. (1) Only patients with early disease are given medicine and they are given enough medicine the first time, that is, they will not be given medicine.(2) Patients with early disease not given medicine may change to patients with advanced disease at any time, which obeys a constant proportion. Besides, they may die at any time, which also obey a constant proportion.(3) Healthy people are considered to be susceptible. According to the data, more than 99 percent of people are without Ebola while patients with early disease take less than 0.1 percent of population. Additionally, the number of medicines is limited, soonly a small number of patients are cured which is less than 0.5 percent of susceptibles adding up the data of a month. Therefore, the number of recovered people is neglected when calculating the number of new infections every day, that is, recovered people are considered as susceptibles.(4) The growth rate of population is neglected so that the number of susceptibles is constant, because the death rate for Ebola is approximately equal to the growth rate.(5) Some patients may recover without medicine, the proportion of which is pretty small for a patient. Howeve r, the proportion can’t be neglected when considering a large number of people. We can get the proportion by researching the recovery situation of patients with early disease without medicine and patients with advanced disease. We assume that the proportion of recovery without medicine is the same for patients with early disease without medicine, patients with advanced disease and patients during latent period.3.2.3Establishment of the ModelWe consider the each district separately and iterate the equations with day as unit. Let X(i,t)denote the situation of i tℎdistrict on t tℎday, so the number of a kind of people can be represented by the number of the former day as follows.X(i,t)=X(i,t−1)+ΔX(i,t) (2) In order to determine the number of people in each category, we plot Figure 2, showing a conversion between each category.Figure 2: Conversions between each category in medicine modelEach conversion process and the number of conversion are explained as follows, where N i denotes the number of conversion, b denotes the total number of medicine, δdenotes the daily infection rate, Q(t−1)denotes the total number of patients, p denotes death rate for Ebola, q denotes the average transformation rate from patients with early disease to patients with advanced disease, m denotes the average death rate of patients with advanced disease and w is the proportion of cremation. t1is the time when bodies is infectious, t2is the time for cure, t3is the time of latent period, t4is the time for the early disease lasting and t5is the time for the advanced disease lasting.①: Patients during latent period recover without medicine,N1=I3(i,t−1)1−p t4+t5②: Susceptibles are infected,N2=S(i,t−1)βN[I1(i,t−t3)+I2(i,t−t3)+R(i,t−t3)]③: Onset occur to patients during latent period,N3=S(i,t−t3)(1−t31−pt4+t5)βN[I1(i,t−t3)+I2(i,t−t3)+R(i,t−t3)]④: Patients with early disease recover without medicine,N4=I1(i,t−1)1−p t4+t5⑤: Patients with advanced disease recover without medicine,N5=I2(i,t−1)1−p t4+t5⑥: Patients with early disease without medicine become patients with advanced disease,N6=qI1(i,t−1)⑦: Patients with early disease without medicine are given medicine,N7=bQ t−1[I1(i,t−1)+I2(i,t−1)]⑧: Recovered people become susceptibles,N8=bQ(t−1)[I1(i,t−t2)+I2(i,t−t2)]⑨: Patients with advanced disease die and become bodies,N 9=I 2(i,t −1)m(1−w)⑩: Bodies are cremated.N 10=I 2(i,t −t 1)mwAccording to the equations, we can get the iterative equations as follows.{ I 1(i,t )=I 1(i,t −1)(1−1−p t 4+t 5) +S (i,t −t 3)βN [I 1(i,t −t 3)+I 2(i,t −t 3)+R (i,t −t 3)](1−t 31−p t 4+t 5) −b (t )Q t −1[I 1(i,t −1)+I 2(i,t −1)]−qI 1(i,t −1)I 2(i,t )=I 2(i,t −1)(1−m −1−p t 4+t 5)+qI 1(i,t −1)I 3(i,t )=I 3(i,t −1)(1−1−p t 4+t 5)+S (i,t −1)βN [I 1(i,t −1)+I 2(i,t −1)+R (i,t −1)] −S (i,t −t 3)(1−t 31−p t 4+t 5)βN[I 1(i,t −t 3)+I 2(i,t −t 3)+R (i,t −t 3)]R (i,t )=R (i,t −1)+b (t )Q (t −1)[I 1(i,t −1)+I 2(i,t −1)] −b (t −t 2)Q (t −1)[I 1(i,t −t 2)+I 2(i,t −t 2)]S (i,t )=S (i,t −1)D (i,t )=D (i,t −1)+m (1−w )[I 2(i,t −1)−I 2(i,t −t 1)]Q (t )=∑[I 1(i,t )+I 2(i,t )]i(3)3.2.4 Parameter of the ModelBy fitting a straight line on the total data, we use the least squares method to obtain the daily contact rate βN .βN =5.82×10−10According to the articles of studying Ebola[8] [9] and reports [2], we set theparameters as follows.t 1=20/day t 2=20/day t 3=10/day t 4=100/day t 5=4/dayp =0.8 q =0.1 m =p/4 w =0.73.2.5Result of the ModelWe iterate the equations (3) for 100 days with 3 February as the first day, so we obtain the following results of the whole situation of epidemic.Figure 3: The total number of death changes over time under different maximum of medicine given every day. We test when the maximum is 0, 10, 100, 500, 1000, 2000 and 5000.According to Figure 3, when given medicine, the number of death increases more and more slowly, and finally becomes constant, that is, the epidemic is controlled. When increasing the maximum of medicine, the total number of death decreases, and the time of epidemic decreases. When the maximum of medicine is 2000, the total number of death become approximately least, and increasing the maximum of medicine has little influence on the control of epidemic. Consequently, when the maximum of medicine given every day is 2000, the epidemic will be controlled best. On this condition, the epidemic is controlled after about 20 days, and the total number of death is 1100 or so.When the maximum of medicine given every day is 2000, the number of different kinds of people and the number of medicine given every day vary as Figure 4 shows.Figure 4: When the maximum of medicine given every day is 2000, the number of patients with early disease, patients with advanced disease, patients during latent period and all the patients changes over time as Figure a, b, c, d show. The total number of death varies over time as Figure e shows. The number of medicine given every day varies over time as Figure f shows. The abscissa of the figures is time whose unit is day.In Figure d, the curve changes suddenly at two points because patients given medicine recover during the time between the two points and the total number of patients decreases rapidly.3.3The Medicine-vaccine ModelWe add the effect of vaccine on base of the medicine model, so we get themedicine-vaccine model.3.3.1Assumptions(1) The people injected with vaccine will be permanently immune to Ebola.(2) The distribution of vaccine for districts obeys the proportion of patients’number.(3) Only people without symptoms of Ebola are injected with vaccine, including patients during latent period. Besides, patients during latent period will recover after being injected with vaccine.3.3.2Establishment of the ModelWe refine Figure 2 and then get the conversion between each category as Figure 5 shows.Figure 5: Conversions between each category in medicine-vaccine modelLet A denote people injected with vaccine, n denote the total number of vaccine, so we get the additional conversions as follows.⑪: Susceptibles won’t be infected after being injected with vaccine,N11=I1(i,t−1)+I2(i,t−1)Q(t−1)n⑫: Patients during latent period will recover after being injected with vaccine and they won’t be infected.N12=I1(i,t−1)+I2(i,t−1)Q(t−1)n∑S(i,t−1)it3N(t3+t4+t5)Therefore, we get the iterative equations as follows.{I 1(i,t )=I 1(i,t −1)(1−1−pt 4+t 5) +S (i,t −t 3)βN [I 1(i,t −t 3)+I 2(i,t −t 3)+R (i,t −t 3)](1−t 31−p t 4+t 5)−b (t )Q (t −1)[I 1(i,t −1)+I 2(i,t −1)]−qI 1(i,t −1)I 2(i,t )=I 2(i,t −1)(1−m −1−pt 4+t 5)+qI 1(i,t −1)I 3(i,t )=I 3(i,t −1)(1−1−p t 4+t 5)+S (i,t −1)βN [(i,t −1)+I 2(i,t −1)+R (i,t −1)] −S (i,t −t 3)(1−t 31−p t 4+t 5)βN[(i,t −t 3)+I 2(i,t −t 3)+R (i,t −t 3)] −I 1(i,t −1)+I 2(i,t −1)Q (t −1)n∑S (i,t −1)i t 3N (t 3+t 4+t 5)R (i,t )=R (i,t −1)+b (t )Q (t −1)[I 1(i,t −1)+I 2(i,t −1)] −b (t −t 2)Q (t −1)[I 1(i,t −t 2)+I 2(i,t −t 2)]S (i,t )=S (i,t −1)−I 1(i,t −1)+I 2(i,t −1)Q (t −1)n D (i,t )=D (i,t −1)+I 2(i,t −1)p t 5w −I 2(i,t −t 1)p t 5w Q (t )=∑[I 1(i,t )+I 2(i,t )]i (4)3.3.3 Result of the ModelWe iterate the equations (4) for 100 days with 3 February as the first day, and then we get the following results the whole situation of epidemic.Firstly, we test the different daily maximum of vaccine delivered under the situations that medicine isn’t given and that the maximum of medicine given every day is 2000, and we get the results as Figure 6 and Figure 7 show.Figure 6: The three curves show that the total number of death varies over time under different conditions that the daily maximum of vaccine delivered is 0, 1000000 or 8612862 when the medicine isn’t given.Figure 7: The three curves show that the total number of death varies over time under different conditions that the daily maximum of vaccine delivered is 0, 1000000 or 8612862 when the daily maximum of medicine delivered is 2000.According to Figure 6 and Figure 7, it can be concluded that vaccine have little influence on the control of epidemic. We infer that the reason is that the contact rate βN is extremely small because we use the data of the latest 20 days when some measureshaven been taken in the three countries and the chance of infection is small. Since the chance of infection is small, the effect of vaccine is little. In order to prove the inference, we change the contact rate and get the following results.We change the contact rate to 10 times the size of the calculated contact rate and the result is showed in Figure 8 and Figure 9.Figure 8: The curves show that the total number of death varies over time under different conditions that the daily maximum of vaccine delivered is 0 or 1000000 when the medicine isn’t given.Figure 9: The curves show that the total number of death varies over time under different conditions that the daily maximum of vaccine delivered is 0 or 1000000 when the daily maximum of medicine delivered is 2000.We change the contact rate to 100 times the size of the calculated contact rate and the result is showed in Figure 10 and Figure 11.Figure 10: The curves show that the total number of death varies over time under different conditions that the daily maximum of vaccine delivered is 0, 100000 or 1000000 when the medicine isn’t given.Figure 11: The curves show that the total number of death varies over time underdifferent conditions that the daily maximum of vaccine delivered is 0 or 100000 when the daily maximum of medicine delivered is 2000.According to above figures, it can be concluded that the more infectious the disease is, the larger effect the vaccine has. When the contact rate is small, medicine is more effective than vaccine. Therefore, in current epidemic, the medicine should be delivered as early as possible to the districts and the vaccine isn’t so urgent.4The Delivery SystemsAccording to the medicine-vaccine model, the medicine should be delivered against time while the vaccine is not, so we set up two different delivery systems for medicine and vaccine.4.1Collection of DataAccording to the reported data, [10]the current epidemic concentrates on some districts in the three countries. We count the number of patients in the 21 districts with most intense transmission of Ebola on 3 February as follows. [10]Table 2 The number of patients in the 21 districts on 3 February4.2The Delivery Systems for Medicine4.2.1AssumptionsAccording the fact, we make the following assumptions.(1) The medicines are produced in other countries and delivered to the three countries by plane. Considering the location of districts, we choose BelleYella Airport as the location of delivery. When delivered to BelleYella Airport, the medicines are delivered to districts by truck every day.(2) Only 21 districts with most intense transmission of Ebola are considered.(3) The medicines are distributed on proportion of patients’ number in different districts on 3 February.(4) The medicines can be delivered to the districts within one day by truck.(5) The total number of medicine is less the maximum loading capacity of a truck.(6) There are enough trucks in BelleYella Airport.4.2.2Establishment of the Delivery systemsAccording to the transmission model, the optimal maximum of medicine delivered every day is 2000 and the medicines are required to be delivered to districts as soon as possible. Therefore, we need to choose an optimal route that the medicines are delivered as soon as possible.There are many algorithms for obtaining the optimal route, such as genetic algorithm, simulated annealing algorithm and ant colony optimization which obtain the optimal route with distance and time considered. However, in the delivery systems of medicine for Ebola, the most important factor is time. Besides, the price of medicine is much higher than transportation cost. Therefore, the transportation cost isthe secondary factor.We alter Dijkstra algorithm [13] to obtain the optimal routes because the traditional Dijkstra algorithm only considers the shortest time without considering the transportation cost.First, we add distances between adjacent districts which are directly connected by road to an adjacency matrix. Then, we use Dijkstra algorithm to obtain the shorted distance and route between each district and BelleYella Airport. The medicine for each district is transported by truck separately. Next, we use the following method to optimize the algorithm.Let T i denote the ordered set of the districts on the i tℎroute.If T i⊆T j(1≤i,j≤31 & i≠j & i,j∈Z),delete T i.Finally, we get the optimal routes. Each route has a truck running along them and the loading capacity of a truck is the sum of the medicine for districts on the route. 4.2.3Result of the ModelAll in all, there are 11 trucks for 11 routes and the routes are showed as follows. The total distance of route is 6886 km.Table 3 The optimal delivery system consisting of 11 routesIn Table 3, the number 1~21 is the number of districts in Table 2, and the number 22~31 denotes the nodes of roads. The locations of districts and nodes are showed in Figure 12.Figure 12: The locations of districts and node are showed on the map and the curve denotes road.4.3The Delivery Systems for Vaccine4.3.1AssumptionsAccording the fact, we make the following assumptions.(1) The vaccines are produced in other countries and delivered to the three countries by plane. Considering the location of districts, we choose BelleYella Airport as the location of delivery. When delivered to BelleYella Airport, the vaccines are delivered to districts by truck every day.(2) Only 21 districts with most intense transmission of Ebola are considered.(3) The vaccines are distributed on proportion of population in different districts on3 February.(4) The vaccines will be delivered to the districts for 30 days so that everyone can be infected with vaccine in the three countries.(5) There are enough trucks in BelleYella Airport.4.3.2Establishment of the Delivery SystemsAccording to the medicine-vaccine model, we find that the vaccine has little effecton the current epidemic situation. However, if the management measures are improper, the contact rate will rise rapidly and the effect of vaccine will be apparent at this time. What’s more, the injection with vaccine has a large role in preventing Ebola in the future. Thus, it’s important to tra nsport vaccine to the districts and the distribution of vaccine obeys the proportion of population in different districts.Compared with medicine, the delivery systems for vaccine focus on not the shortest time but the shortest distance for decreasing the transportation cost. Besides, the number of vaccine is much larger than that of medicine and there are ten thousands of vaccine needed every day.Therefore, we improve the Saving Algorithm [14] so as to get the delivery systems with shortest distance and the improvements are showed as follows.(1) In the Saving Algorithm, each two user points are connected with a straight line while the districts are connected with a road. We combine the Dijkstra Algorithm with Saving Algorithm and let the shortest distance between districts calculated by Dijkstra Algorithm take place of the linear distance in the Saving Algorithm.(2) In the Saving Algorithm, the delivery point is different from user points. In fact, the airport is closed to the 13th district, so it can be considered to be in the 13th district, that is, the delivery point (the airport) is in the web of user points (the districts).(3) In the Saving Algorithm, when the two user points are not directly connected, the saved distance may be negative (often making the negative distanced zero represents not saving distance). But in the improved algorithm, the saved distance will be positive, so the total distance saved will be longer, which proves that the improved algorithm is better.We list the steps of calculation in the improved algorithm.(1) Calculate the shortest distances between each two adjacent districts by using Dijkstra Algorithm and make it represented by A(i,j)where i,j is the number of the districts. Thus, we can get the matrix A which consists of all the shortest distances between each two adjacent districts.(2) Calculate the shortest distance between a district and the airport by using Dijkstra Algorithm and make it denoted by L(i)where i is the number of thedistricts. Thus, we can get the matrix L which consists of all the shortest distances between a district and the airport.(3) Calculate the saved distances between the districts according to the basic formula of the Saving Algorithm.d(i,j)=L(i)+L(j)−A(i,j) (5) d(i,j)denotes the saved distance.(4) List the saved distances from the maximum to the minimum.(5) Optimize the routes in order of the saved distances when the load of transport truck satisfied. When the routes overlap, delete the overlapped route and only hold one route connecting the districts.Let Q(i)denote the number of vaccine for the i tℎdistrict, M denote the load, and M0denote the maximal load of truck.The order of the saved distances is d(i,j), d(j,k)….When M+Q(i)+Q(j)≤M0The route is optimized to the airport→the i tℎdistrict→the j tℎdistrict, and delete d(i,j).(6) Calculate the optimal route by computer.According to the load of truck and the results of the medicine-vaccine model, we set the load of truck as 5 tons, the weight of a piece of vaccine as 100 g, so each truck can transport 50000 pieces of vaccine. The vaccines are delivered for 30 days and the total population of the three countries is about 8.6 million, so 0.3 million pieces of vaccine are transported every day.4.3.3Result of the ModelWe get the optimal routes and distances as follows.Table 4 The optimal delivery system consisting of 7 routes。