埃博拉病毒传播分析与数学建模

For office use only T1________________ T2________________ T3________________ T4________________Team Control Number39595Problem ChosenAFor office use onlyF1________________F2________________F3________________F4________________2015 Mathematical Contest in Modeling (MCM) Summary SheetEradicating EbolaSummaryWith a high risk of death, Ebola virus disease (EDV), or simple Ebola, is a horrible disease which has caused great amount of death. In this paper, we mainly build two mathematical model to help eradicate Ebola, including a Virus Propagation Model based on BA scale-free network and SIRED, a delivery system model base on local optimization.For the former part, we firstly establish a BA scale-free network to simulate the realistic interpersonal network. Basing on this network, we set up a series rules to describe the procedure of Ebola propagation, which can be refined as the “Susceptible-Exposed- Infective-Removal-Death” (SEIRD) model. By combining this two model toget her via stimulation, we, using the variation of infective number and death number to reflect the procedure of Ebola spread, successfully restore the propagation of Ebola and predict the variation trend of them. Both the infective number and death number have a high agreement with the report from WHO. Basing on the infective number curve, we easily gain the quantity of the medicine needed and the speed of manufacturing of the vaccine and drug.For the latter part, we use a local optimization method to establish a feasible delivery system. Firstly, we choose Representative points in the map and make clustering analysis based on Euclidean distance, to classify points into three area parts. Then, we select delivery centers based on Analytic Hierarchy Process (AHP) and Principal Component Analysis (PCA) in each part. Besides, routes are designed according to prim algorithm, aiming at minimum the cost in every part. In this way, we build a delivery system. By comparing the results with treatment Centers distribution which has been built, the effectiveness of the model could be examined.Besides, we also discuss other critical factors, such as isolation measures, in the further discussion part. We conclude that isolation measures play a significant role thought the entire process of eradicating Ebola.Above all, our models are both scientific and reliable. They can be applied further to other relative problems.Key Words:SIRED, Complex Network, Cluster Analysis, Analytic Hierarchy Process (AHP) Delivery Systems Model (DSM), Principal Component Analysis (PCA)Table of Content1.Introduction (1)1.1.Background (1)1.2.Restatement of the Problem (1)2.Assumptions and Notions (1)2.1.Assumptions and Justifications (1)2.2.Notions (2)3.The Virus Propagation Model Based on Complex Networks and SEIRD Model (3)3.1.Model Overview (3)plex Network Model (3)3.2.1.Small-World Network Model (3)3.2.2.BA Scale-Free Network Model (4)3.3.SIR-Based SEIRD Model (5)3.3.1.SIR Model (5)3.3.2.SEIRD Model (6)3.4.The Study of Infection Rate, Recovery Rate and Death Rate Based on the LeastSquare Method (6)3.4.1.The Relevant Calculation about Infection Rate (6)3.4.2.The Relevant Calculation about the Recovery Rate (7)3.4.3.The Relevant Calculation of Death Rate (7)3.5.The Simulation of the Transmission of Ebola Virus (8)3.5.1.The Simulation of Complex Network Model (9)3.5.2.The Simulation of Virus Transmission (10)3.6.Results and Result Analysis (11)3.6.1. A Complex Network Simulation Results Model (11)3.6.2.The Spread of the Virus the Simulation Results (12)4.Delivery Systems Model(DSM) Based on Local Optimization (13)4.1.Model Overview (13)4.2.Cluster Division Based on Cluster Analysis (14)4.3.Delivery Centers and Routes Planning Based on AHP and PCA (17)4.3.1The Three-hierarchy Structure (18)4.3.2Analytic Hierarchy Process and Principal Component Analysis for DSM .. 194.3.3Obtain the Centers (21)4.3.4Obtain the Routes (22)4.4.Results and Analysis (23)5.Other Critical Factors for Eradicating Ebola (24)5.1The Effect of the Time to Isolate Ebola on Fighting against Ebola (24)5.2The Effect of Timely Medical Treatment to Isolate Ebola on Fighting against Ebola (25)6.Results and results analysis (26)6.1.The virus propagation model based on complex networks (26)6.1.1.The contrast and analysis concerning the results of simulation and thereality (26)6.1.2.Forecast for the future (28)6.2.Delivery Systems Model Based on Local Optimization (28)7.Strengths and Weaknesses (29)7.1.Strengths (29)7.2.Weaknesses (29)8.Conclusion (30)9.Reference (30)10.Appendix (1)1.Introduction1.1.BackgroundWith a high risk of death, Ebola virus disease (EDV), or simple Ebola, is a disease of humans and other primates. Since its first outbreak in March 2014, over 8000 people have lost their lives. And till 3 February 2015, 22,495 suspected cases and 8,981 deaths had been reported. [1] However, this disease spreads only by direct contact with the bold or body fluids of a person who has developed symptoms of the disease. Following infection, patients will typically remain asymptomatic for a period of 2-21 days. During this time, tests for the virus will be negative, and patients are not infectious, posing no public health risk.[2] And recently, the world medicine association has announced that their new medication could stop Ebola and cure patients whose disease is not advanced. Thus, a feasible delivery system is in great demand and measures to eradicating Ebola should be taken immediately.1.2.Restatement of the ProblemWith the background mentioned above, we are required to build a model to help eradicate Ebola, which can be decomposed as:●Build a model, which can estimate the suspects number, exposed number,infect number, death number and recover numbers, to describe the spreadprocedure of the Ebola from its very beginning to the future.●Build an optimized model to help establish a possible and feasible deliverysystem including selecting delivery location and delivery system networkdesign.●Estimate of the quantity of the needed medicine and manufacturing speed ofvaccine or drug, based on the results of our models.●Discuss other critical factors which help eradicate Ebola.2.Assumptions and Notions2.1.Assumptions and JustificationsTo simplify the problem, we make the following basic assumptions, each of which is properly justified.●Assume that there is no people flow between countries after outbreak ofdisease in the country.After the outbreak, countries usually will ban thecontact between locals and foreigners to minimize the incoming of the virus.●Assuming that virus infection rate and fatality rate will not change bythe change of regions.Virus infection rate and fatality rate are largelydetermined by the nature of the virus itself. The different between differentregions just have a little effect and it will be ignored.●Assume that there are only rail, road and aircraft for transportation. Inthe West Africa, waterage is rare. Rail, and road are for nearby transportationwhile aircraft is for faraway.2.2. NotionsAll the variables used in this paper are listed in Table 2.1 and Table 2.2.Table 2.1 Symbols for Virus Propagation Model(VPM)SymbolDefinition Units βInfection Rate for the Susceptible in SIR Model or SEIRD Model unitless γRecovery Rate for the Infective in SIR Model unitless rRateRecovery Rate for the Infective in SEIRD Model unitless dRateDeath Rate for the Infective in the SIR Model or SEIRD Model unitless ∆N iNew Patients on a Daily Basis person N i−1The Total Number of Patients in the Previous Day person nThe Average Degree of Each Node in the Network unitless tTime S ∆D iThe New Death Toll on a Daily Basis person D i−1The Total Number of Patients in the Previous Day person moThe Initial Number of Nodes in the BA Scale-Free Network node mThe Number of Added Sides from One New Node in the BA Scale-Free Network side ∏iThe Probability for the Connection between New Nodes and the Existing Node I in the BA Scale-Free Network unitless NThe Total Number of Nodes in the BA Scale-Free Network node k iThe Degree of Node I in the BA Scale-Free Network unitless eThe Number of Sides in the BA Scale-Free Network side n SThe Number of the Susceptible in the SEIRD Model person n EThe Number of the Exposed in the SEIRD Model person n IThe Number of the Infective in the SEIRD Model person n RThe Number of the Removal in the SEIRD Model person n DThe Number of the Dead in the SEIRD Model person RThe Probability of Virus Propagation from the Recovered unitless RandomE The Number of the Exposed Who Have Reached the Exposed Time Limit Ranging from 2 to 21 Days at the Current Momentperson n E(一天)t The Number of the Exposed Who Have Reached the Final Day of the ExposedTime Limit yet not quarantine personTable 2.2 Symbols for Delivery Systems Model(DSM)SymbolDefinition Units Athe judging matrix unitless a ijThe element of judging matrix unitless λmaxthe greatest eigenvalue of matrix A unitless CI the indicator of consistency check unitlessCR the consistency ratio unitlessRI the random consistency index unitlessCW the weight vector for criteria level unitlessAW the weight vector for alternatives level unitlessY the evaluation grade unitlessV A set of points unitlessV i,V j the point of V unitlessE A set of edges unitlessDis ij The real distance of i and j unitless Arrive ij Judging for whether there is an side between i and j unitless3.The Virus Propagation Model Based on Complex Networks and SEIRD Model3.1.Model OverviewWe aim to build a Susceptible-Exposed-Infective-Removal-Death (SEIRD) virus propagation model which is based on Susceptible-Infective-Removal (SIR) model. The aimed model is featured by complex networks, which exhibit two statistical characteristics, including the Small-World Effect and the Scale-Free Effect. These characteristics could produce relatively real person-to-person and region-to-region networks. Through the statistics of the existing patients and deaths, we will try to find the relationship among the infection rate, the recovery rate and the death rate with the change of time. Then, with the help of SEIRD model, these statistics would be used to simulate the current situation concerning the number of the susceptible, the exposed, the infective, the recovered and the dead and conduct the prediction of the future.plex Network ModelResearches have shown that the person-to-person networks in real life exhibit the Small-World Effect and Scale-Free Effect. Here we will introduce the Small-World Network Model by Watts and Strogatz, and the Scale-Free Network Model by Barabdsi and Albert [3].3.2.1.Small-World Network ModelSince random network and regular network could neither properly present some important characteristics of real network, Watts and Strogatz proposed a new network model between the random network and regular network in 1998, namely WS Small-World Network Model, the construction algorithm of which is as follows.Start from a regular network: consider a regular network which contains N nodes, and these nodes form a ring. Each node is linked with its adjacent nodes, the number of which is K/Z on both left side and right side. Also, K is an even number.Randomized re-connection: the probability P will witness a random re connection with each side in the network. In other words, an endpoint of a certain side will remain unchanged and the other endpoint would be the node in the random selection.There are two rules. The first is two different nodes will at most have one side. The second is every node cannot have a side which is connected with this node [4].Randomized re-connection in construction algorithm of the WS Small-World Model may damage connectivity of network, so Newman and Watts improved this model in 1999. The new one is called NW Small-World Model, the construction algorithm of which is as follows.Start from a regular network: consider a regular network which contains N nodes, and these nodes form a ring. Each node is linked with its adjacent nodes, the number of which is K/Z on both left side and right side. Also, K is an even number.Randomized addition of sides: the probability P will witness the random selection of two nodes and the subsequent addition of a side between these two nodes. There are two rules. The first is two different nodes will at most have one side. The second is every node cannot have a side which is connected with this node[4].The network constructed by the two models are shown in Fig 3.1.Fig 3. 1 WS Small-World network and NW Small-World network3.2.2.BA Scale-Free Network ModelIn October, 1999, Barabdsi and Albert published article in Science called "Emergence of Scaling in Random Networks" [5], which proposed an important discovery that the distribution function of connectivity for many complex networks exhibit a form of power laws. Since no obvious length characteristics of connectivity could be seen among nodes in these networks, so they are called scale-free networks.As for the cause of power laws distribution, Barabasi and Albert believe that many previous network models did not take into account two important characteristics of actual networks: the consistent expanding of network and the nature of new nodes’ prior connection in the network. These characteristics will not only make node degrees which are relatively larger increase much faster, but also produce more new nodes, thus node degrees will become even larger. Then we could see the Matthew Effect. [4]Based on the Scale-Free Network, Barabasi and Albert proposed a scale-free network model, called BA Model, the construction algorithm of which is as followsi.The expanding of network: start from a network which has Mo nodes, thenintroduce a new node after each time interval and connect this node with mnodes. The prerequisite is m≤m0.ii.Prior connection: the probability between a new node and an existing node iis ∏i, the node degree of i is k i, and the node degree of j is k j. These threefactors should satisfy the following equation.∏i=k i/∑k jj(3-1) After t steps，this algorithm could lead to a network featured by m0+t nodes and m×t sides.The network of BA Model is shown in Fig 3.2.Fig 3.2 BA Scale-Free Network3.3.SIR-Based SEIRD Model3.3.1.SIR ModelSIR is the most classic model in the epidemic models, in which S represents susceptible, I represents infective and R represents removal. Specifically, the susceptible are those who are not infected, yet vulnerable to be infected after contact with the confirmed patients. The infective are those who have got the disease and could pass it to the susceptible. As for the removal, it refers to those who are quarantined or immune to a certain disease after they have recovered.In the disease propagation, SIR Model is built with the infection rate as β, the recovery rate as γ, which is shown in Fig 3.3:Fig 3.3 SIR propagation modelThis model is suitable epidemics which have the following features: no latency, only propagated by the patients, difficult to cause death, patients are immune to this disease after recovery once and for all. As for the Ebola virus, this model is insufficient to present the propagation process. Therefore, we propose the SEIRD model based on the SIR model and overcome the defects of the SIR model, thus making the SEIRD model more suitable for the research of Ebola virus.3.3.2.SEIRD ModelThe characteristics of Ebola virus are shown in Table 3.1:Table 3.1 The characteristics of Ebola virus characteristicsDetailsLatency Exposed period ranging from 2 to 21days with no infectivity during this stage[2]Retention After the recovery, there is still a certain chance of propagation[6]Immunity Recovery is accompanied with lifelong immunityTherefore, it is needed to add E (the exposed) and D (the dead) in the SIR model.E represents those who have been infected, have no symptoms, and not contagious. But within 2-21 days, the exposed will become contagious. D represents those who are dead and not contagious.From Table 3.1, we know that Ebola virus has the feature of retention, because even when they are in the state of removal, it is still possible these recovered will be infectious.In the process of disease propagation, SEIRD has witnessed the infection rate as β and the recovery rate as rRate and dRate，which is shown as follows.Fig 3.4SEIRD propagation model3.4.The Study of Infection Rate, Recovery Rate and Death RateBased on the Least Square MethodAs for the calculation of infection rate, recovery rate and death rate, we could make use of the least square method to match the daily confirmed patients and the dead toll, thus getting the function about the relationship with the passage of time.And we choose the relevant data from Guinea since it is in severely hit by the Ebola outbreak in West Africa. The variance regarding the total number of patients and the total death toll could be seen in Appendix 11.1.3.4.1.The Relevant Calculation about Infection RateThe infection rate refers to the probability that the susceptible are in contact (here it refers to the contact with body fluids) with un-isolated patients and infected with the virus. For each infective patient, the number of side is the node degree, namely the number whom he or she could infect. Therefore, the infection rate could be calculatedin this way: the number of new patients each day divides the possible number of whom each confirmed patient could infect. The number of new patients is ∆N i. The total number of patients in the previous day is N i−1. The average node degree in the network is n. The equation is as follows.β=∆N i/(N i−1×n)(3-2) The β could be calculated based on the total number o f patients in Guinea (see Appendix 11.1). Then with time data and the method of least square method, we could do data fitting and calculate the time-dependent equation. The fitting image is listed in Fig 3.5.Fig 3.5 The fitting result image of β with the passage of timeThe result of fitting curve is：β=0.0367×t−0.3189/n(3-3) And n is the average degree of person-to-person network in the process of simulation.3.4.2.The Relevant Calculation about the Recovery RateThe recovery rate refers to the probability of recovery for those who have been infected. Since at present, few instances of recovery from Ebola disease could be witness in the world (probability is almost close to zero), so the rRate here is set to be 0.001.3.4.3.The Relevant Calculation of Death RateThe death rate refers to the probability that patients become dead in process of treatment. And the death rate is calculated in the following way: the total number of new deaths every day divides the total number of patients in the previous day. The total number of new deaths in a new day is ∆D i. The total number of patients in the previous day is D i−1. The equation isdRate=∆D i/D i−1(3-4) The dRate could be calculated based on the total number of dead patients and the total number of patients in Guinea (see Appendix 11.1). Then with time data and the method of least square method, we could do data fitting and calculate the time-dependent equation. The fitting image is listed in Fig 3.6.Fig 3.6 The fitting result image of dRate with the passage of time The result of fitting curve is:dRate=(−6.119e−07)×t2 −0.0001562×t + 0.01558(3-5) 3.5.The Simulation of the Transmission of Ebola VirusThe simulation for Ebola will mainly be divided into two aspects, namely the simulation of complex network model and that of virus spread. The related flow chart will be shown in Fig 3.7.Fig 3.7 Flow chat of Stimulation of Ebola Virus Transmission3.5.1.The Simulation of Complex Network ModelFrom the previous introduction about complex network model, BA Scale-Free Network Model has displayed the Matthew Effect, which means the stronger would be much stronger and the weaker would be much weaker. In social networks, this effect is also widely seen. Take one person who just joins in a group for an example, he would normally contact with those who have the largest circle of friends. Therefore, those who get the least friends can hardly know more new friends. This finally leadsto a phenomenon that the person who is most acquainted will have more and more friends and vice versa.Based on this, the BA Scale-Free Network Model is apparently superior to that of Small-World Model. As a result of that,we would use the former to simulate the interpersonal network.According to the rules of BA network model, we should start from a network which has Mo nodes, then introduce a new node after each time interval and connect this node with m nodes. The prerequisite is m≤m0.During the connection process, the probability between a new node and an existing node i is ∏i, the node degree of i k i, and the node degree of j is k j. These three factors should satisfy the following equation.∏i=k i/∑k jj(3-6) Specifically, when the existing nodes have larger node degrees, it would be much more easier for the new ones to connect with the existing ones.After t steps, there would be a BA Scale-Free Network Model. The number of its nodes is expressed as N and the number of its sides is expressed as e:N=m0+t(3-7)e=m×t(3-8) The population of Sierra Leone now is 6.1 million and we would use this datum to produce its interpersonal network. For more details, please refer to Appendix 11.2.3.5.2.The Simulation of Virus TransmissionIn the transmission process, we assume the infection rate is β, the recovery rate is rRate, and the death rate is dRate. Based on the fitting results we previously get, we can simulate the virus transmission situation as time goes.Here comes the details.In the first place, there would be one patient who initiates the epidemic. Every single day, the virus would transmit to others among the main network and the probability of one-time propagation is β. Also, the patients would have rRate of recovery and dRate of death. Meanwhile, if the patient has been infected for 30 days, he or she would die anyway. The exposed would be in a latent period, during which they are not infectious and asymptomatic. In 2 to 21 days, these exposed ones would become infectious.n S、n E、n I、n R、n D represent the 5 different numbers of people in the SEIRD Model. t means time step (or a day)，R represents the probability that those who have recovered patients would infect others. RandomE denotes the number of exposed patients who have reached the period of 2 to 21 days at the current moment.Here is the formula showing the changes in the numbers of those five types of people.n S t+1=n S t−n I t×n×β−n R t×n×β×R(3-9) n E t+1=n E t+n I t×n×β+n R t×n×β×R−RandomE(3-10) n I t+1=n I t+RandomE−n I t×rRate−n I t×dRate(3-11)n R t+1=n R t+n I t×rRate(3-12)n D t+1=n D t+n I t×dRate(3-13) After that, when the transmission has reached a certain scale (20 days after the transmission), the international organizations would adopt the measure of quarantine towards infective patients to avoid further contagion. As for the exposed patients, since they could not be quarantined immediately, so they have one day to infect others and in the next day, they would be quarantined at once.Finally, for those who have recovered, there is still a certain chance that they will propagate the disease within their networks.n Ed t on behalf of the moments lurk in reaching the last day with infectious but has not yet been isolated number. The process of five types of personnel number change:It represents the number of the exposed who have reached their last day of latency, begin to be contagious and have not yet been quarantined at the current moment. During this process, the formula exhibiting changes for these five categories of people could be listed as follows.n S t+1=n S t−n Ed t×n×β−n R t×n×β×R(3-14) n E t+1=n E t+n Ed t×n×β+n R t×n×β×R−RandomE(3-15)n I t+1=n I t+RandomE−n I t×rRate−n I t×dRate(3-16)n R t+1=n R t+n I t×rRate(3-17)n D t+1=n D t+n I t×dRate(3-18)3.6.Results and Result Analysis3.6.1. A Complex Network Simulation Results ModelPersonnel network is illustrated as Fig 3.8. Because of the population is too large so it is difficult to figure out. We use a red point to represent 10000 persons.Fig 3.8 Personnel relation network diagramThe probability distribution of nodes in a network of degrees is illustrated as Fig 3.9.The same node degrees set in 2-4, in it with degree of 4most.On behalf of each person every day in the network average fluid contact with 2-4 people.Fig 3.9 The probability of the node degree distribution map network3.6.2.The Spread of the Virus the Simulation ResultsThe number of every kind of the curveof the change over time in SEIRD model is illustrated in Fig 3.10. Due to the large population, the graph is local amplification. It is unable to find the number of susceptible people in the picture. For the rest of the curve, black represents the exposed, red represents the sufferer and pink for the removed.Fig 3.10 The number of SEIR model with the change of time4.Delivery Systems Model(DSM) Based on Local Optimization4.1.Model OverviewOptimized distributing is the most significant problem while building Delivery Systems, and it is a NP (nondeterministic polynomial) problem. In order to studied the problem, Li Zong-yong, Li Yue and Wang Zhi-xue organized an optimized distributing algorithm based on genetic algorithm in 2006[7]. In addition, Liu Hai-yan, Li Zong-ping, Ye Huai-zhen[8]discussed logistics distribution center allocation problem based on optimization method. With the help of current literature, we build a Delivery Systems Model (DSM) for drug and vaccine delivery, based on Local Optimization.In this topic, in order to establish the feasible delivery systems for WesternAfrica, we take Sierra Leone as an example. There is 14 Districts in Sierra Leone.In this model, we choose points based on Sierra Leone politics. Representative point in every District is selected. The points are located by longitude and latitude. We use Euclidean distance-based clustering analysis to process the data, so that the point set will be classified into three sub-set. Every sub-set is a part. Then, one point in every sub-set will be selected as a delivery center based on Analytic Hierarchy Process (AHP) and Principal Component Analysis (PCA). Besides, we will design the routes based on Prim algorithm, aiming at minimum the cost in every sub-set. In this way, we will build a delivery system.In addition, we will compare the results with Treatment Centers distribution which has been built, to analyze the model.4.2. Cluster Division Based on Cluster AnalysisThere are 14 districts in Sierra Leone. In every district, we choose the center position as a point. In this model, the first step is to cluster. Cluster Analysis is based on similarity. In this model, the similarity could be measured by geography distance. There is different method for cluster.Suppose there are n variable, and the objects are x and y1212(,,,),(,,,),n n x x x x y y y y ==By Euclidean distance, the distance can be calculated by (4-1)(,)d x y =(4-1)By Cosine Similarity distance, the distance can be calculated by (4-2)2(,)ni ix yd x y =∑(4-2)In this model, we use Euclidean distance. We cluster 14 districts into three Parts. First of all, we need to know about the distances between districts. The 14 Districts of Sierra Leone are located by longitude and latitude. Establish a right angle coordinate system. Set the longitude as the abscissa and latitude as ordinate. Set the Greenwich meridian and the Equator as 0 degree. The West and the South are regarded as negative while the East and the North are regarded as positive. In addition, the data related to the latitude and longitude would be converted to standard decimal form. For example, point (1330',830'W N ︒︒) is located as (-13.5, 8.5).According to World Health Organization （WHO ）[9]statistics data, the basic data of Sierra Leone’ Districts are obtained, which is shown in Table 4.1 .。

埃博拉病毒的传播预测与控制摘要2014年非洲爆发了历史上最为严重的病毒疫情--埃博拉。

据科学研究报道，这个病毒一旦感染人体，将有着高达90%以上的死亡率，这是一种世上最厉害的感染病毒（生物安全等级为4级），如何消灭埃博拉成为当前的首要任务。

当然，疾病的传播、患病人口的预测、药物的生产和运输，都是消灭埃博拉必须考虑的因素。

根据病毒传播率、感染者人数的预测、药物的合理分配和隔离人数的比重等因素，本文运用随机微分方程、产销平衡和最优控制三种算法分别建立了随机微分方程模型、线性规划模型和最优隔离控制模型。

这三个模型分别解决了埃博拉病毒的传播规律、感染者人数的预测问题、药物的运输问题和以隔离控制为决定性作用因素的优化问题。

针对模型一：将环境因素作为随机变量，结合病毒传播率，本文建立了随机微分方程模型，对以后10个月的患病人口总数进行了预测。

利用数值解方法,对埃博拉病毒感染者人数进行预测,并通过仿真过程验证了疾病传播率的一个临界值，得出能使埃博拉传播速度降低直至消亡的一个条件。

针对模型二：假设几内亚、利比里亚和塞拉利昂为需求地，美国、中国、日本、俄罗斯、法国以及瑞士为药物生产地。

利用产销平衡原理，建立了时间优化模型，求得产地与需求地之间的最短运输时间为15.8小时。

针对模型三：本模型基于SIR传染病模型，利用极值原理给出了最优控制的设计方案，通过仿真，验证了最优控制方案的优越性。

同时，由协态方程得到当系统控制变量为0.50时，隔离效果最佳，也证明了隔离是控制疾病继续传播最有效的控制措施。

本文三个模型均使用的官方数据，而且内容上层层优化，互相补充，使文章所述更为具体，更为实用，为埃博拉病毒问题的解决提供了一份可靠地，可行的，可依赖的数学模型。

关键词：埃博拉病毒预测随机微分方程优化问题最优隔离控制1.问题重述—不用翻译1995年5月14日，扎伊尔发现罕见传染病埃博拉。

2014年，埃博拉病毒首次爆发就夺走了近300人的生命,2014年再度爆发,大约4000人命丧黄泉。

关于埃博拉病毒传播的分析与预测模型作者：梁晴宋彦辰来源：《科学与财富》2018年第33期摘要：埃博拉病毒所致疾病传播速度远远高于其控制的速度，本文通过分析其传播过程建立了埃博拉病毒传播模型，并使用拉丁超立方抽样对疾病传播进行预测估计。

首先，根据Sir Epidemic动态系统-seiifr模型建立了一个模型，在这个模型中，考虑了暴露于人群中的易感人群、埃博拉病人以及患者的自愈恢复。

为了确定最优参数，构造了最小二乘法优化方案。

结果显示该模型短期预测效果较好。

关键词：埃博拉病毒传播拉丁超立方抽样最小二乘法引言2014年西非全面爆发的埃博拉疫情（EVD）是世界近年来见证的规模空前、规模巨大、范围广泛和严重后果的公共卫生危机。

本轮埃博拉疫情主要突击西非几内亚、利比里亚和塞拉利昂三个国家。

这是自1976年人类首次发现并命名埃博拉病毒以来历史上最严重的公共卫生灾难，显示出与过去的许多不同特征。

1 埃博拉传播模型为了研究埃博拉病毒传播的规律性，本文建立了分阶段的埃博拉传播模型，明确纳入疾病的自然病史和动态ETU能力，同时考虑到诸如报告率、症状评分和高危人群比例问题等因素。

可以使用本文模型来评估预测病例的不确定性、死亡和床位预测，估计疾病的每个阶段对传播的影响，并检查ETU能力供应，减少传染病对EVD爆发的影响动态的关系。

从图1所示的结构模型可看出，该模型包括分阶段感染过程，以反映随着感染进展而增加的感染症状和传播。

埃博拉经常经历多个阶段的疾病：第一阶段是感染的初始阶段，并且症状趋于轻微，并且常常发展成腹泻和呕吐；第二阶段是更严重的阶段，伴有更严重的症状（例如出血和多器官衰竭），典型的进展约5-7天。

受感染的人最有可能在第一阶段复原，而第二阶段通常是致命的。

图1的相应等式：其中S代表易感人群比例，E代表暴露人口比例，R代表最近恢复的人口F代表已经死亡和正在被埋葬的人的比例。

最后，将模型与世卫组织提供的关于累计病例和死亡的数据联系起来。

****大学数学建模竞赛承诺书我们仔细阅读了****大学数学建模竞赛的参赛规则与竞赛纪律。

我们完全明白，在竞赛开始后参赛队员不能以任何方式（包括电话、电子邮件、网上咨询等）与队外的任何人研究、讨论与赛题有关的问题。

我们知道，抄袭别人的成果是违反竞赛纪律的，如果引用别人的成果或其他公开的资料（包括网上查到的资料），必须按照规定的参考文献的表述方式在正文引用处和参考文献中明确列出。

我们郑重承诺，严格遵守参赛规则和竞赛纪律，以保证竞赛的公正、公平性。

如有违反竞赛纪律的行为，我们将受到严肃处理。

我们授权****大学数学建模竞赛组委会，可将们的论文以任何形式进行公开展示（包括进行网上公示，在书籍、期刊和其他媒体进行正式或非正式发表等）＜日期：2015年05月04日埃博拉病毒传播分析摘要本文的研究对象为1976年在苏丹南部和刚果的埃博拉河地区发现的埃博拉病毒。

埃博拉病毒是一种生物安全等级为4级，并且能引起人类和灵长类动物产生埃博拉出血热的烈性传染病病毒，其主要是通过病人的血液、唾液、汗水和分泌物等途径传播。

其病毒的潜伏期通常只有5天至10天，感染后2〜5天出现高热，6〜9天死亡。

面对其强大的传染力和对人类健康的巨大威胁，本文通过数学建模的方法了解埃博拉病毒的传播规律，并分析隔离措施的严格执行和药物治疗效果的提高等措施对控制疫情的作用。

本文中，首先我们根据已给的信息及相关假设数据，通过对已知条件和所给表格书记的分析，我们大致明白了猩猩从潜伏到发病再到死亡或自愈的过程，因此我们采用了excel拟合曲线，分析其发病、潜伏、自愈、死亡和隔离的相应的变化曲线，估计参数，再根据其建立数学模型，并用MATLA求解方程组，调试参数，从而得到我们需要的结果。

其次通过对已经得到的数据和曲线图的分析，可以得出人类通过严格的药物控制过后，对其发病和潜伏的影响，从而能够达到对疫情的控制的作用，并且对埃博拉病毒未来发展趋势有了更深刻的了解，以为更好的控制埃博拉病毒做出贡献。

西安工业大学数学建模竞赛承诺书我们仔细阅读了中国大学生数学建模竞赛的竞赛规则.我们完全明白，在竞赛开始后参赛队员不能以任何方式（包括电话、电子邮件、网上咨询等）与队外的任何人（包括指导教师）研究、讨论与赛题有关的问题。

我们知道，抄袭别人的成果是违反竞赛规则的, 如果引用别人的成果或其他公开的资料（包括网上查到的资料），必须按照规定的参考文献的表述方式在正文引用处和参考文献中明确列出。

我们郑重承诺，严格遵守竞赛规则，以保证竞赛的公正、公平性。

如有违反竞赛规则的行为，我们将受到严肃处理。

我们参赛选择的题号是（从A/B/C中选择一项填写）： B所属学校（请填写完整的全名）：西安工业大学参赛队员(打印并签名) ：1. 陈文兴2. 闫丽萍3. 魏栩指导教师或指导教师组负责人(打印并签名)：日期：2015 年8 月1 日埃博拉病毒传播及控制分析摘要埃博拉病毒是能引起人类和灵长类动物产生埃博拉出血热的烈性传染病病毒，有很高的死亡率。

本文根据研究人员统计所给出的前四十周人类和猩猩的发病数量和死亡数量等信息，对该病毒的传播、预测与控制进行研究并建立模型，并分析了隔离措施的严格执行和药物治疗效果的提高等措施对控制疫情的作用。

针对问题一，在了解埃博拉病毒的传播情况后，根据猩猩的发病情况建立了马尔萨斯模型：()t e t x 0270.097.154=。

在此模型中，较好地描述病毒在“虚拟猩猩种群”中的传播情况；根据“虚拟猩猩种群”中的数据，用matlab 拟合出不同状态下猩猩数量的变化曲线，并以发病状态为例建立灰色预测模型()()()⎪⎪⎩⎪⎪⎨⎧-=+-=+=+-∧897947))1((10539.600.0669 0124.0)0(11e a b e a b x k x x dt dx a ，从而较准确的预测出接下来第80、120、200周的猩猩发病状态的数据。

针对问题二，为描述埃博拉病毒在“虚拟种群“中的相互传播规律及人和猩猩的疫情发展状况，建立SEIR 模型 ()()()()()()()()()()⎪⎩⎪⎨⎧⋅⋅--=⋅⋅---⋅⋅---=⋅⋅----⋅⋅=----)(111)(111)(111)(111)(331212t I e a dt dT t I e a t Q e a a dt dI t Q e a a t I r dt dQ t t t t λ 模型求解时，通过对模型的推导，我们发现不能给出每个函数的解析解，因此考虑利用matlab 中的ode45函数进行求解。

数学建模在疾病扩散中的应用新冠病毒的爆发给全球范围内带来了灾难性的破坏，各国政府和科学家都在竭力寻找有效的方法来控制疫情的扩散和降低死亡率。

其中，数学建模成为了一种重要的工具，能够对疫情的传播规律和趋势进行分析，并预测未来发展趋势，以便及时采取应对措施。

疫情建模的基本思路是把一个疾病的传播过程看作一个数学模型，然后通过建立、分析和优化模型，预测疾病传播的趋势和规律，为疫情防控工作提供科学依据。

疫情建模主要涉及到统计学、动力学和微积分等方面的知识，其中微积分是建模的重要工具。

微积分是关于导数和积分的学科，其核心是对变化的分析和对整体和局部的描述。

在疫情中，微积分可以分析和预测疫情的发展趋势，可以对疾病传播速度、感染率、康复率和死亡率等指标进行深入研究。

微积分可以从中挖掘出疫情传播的规律性和规律性，找到一些有效的控制方法，并辅助政府和医疗机构在疫情防控工作中采取相应的应对措施。

数学模型能够对疾病传染过程各个方面进行深入研究，不仅能够对流行趋势、累计感染人数和病例分布等主要指标进行分析，还能够对影响疾病传播的各种因素进行分析。

例如，通过微积分分析感染速率，可以得出一个感染速率的微分方程。

再根据传染率、病毒基因特征、传染媒介的数量、个体免疫力等因素，以及人口密度、迁移历史、社会因素等因素之间的相互作用，建立一个多因素疫情模型，从而全面、准确地描述疾病传播的一切特征。

除了微积分，疫情建模还需要动力学、统计学和计算机科学等领域的知识。

动力学是研究物体运动的学科，可以用来研究病毒传染的过程和规律。

统计学主要涉及到数据的收集、分析和预测，可以用于疾病数据的处理和分析。

计算机科学提供了调用和管理这些数据的技术，从而将这些数据应用到实际问题中。

综合运用数学模型、动力学、统计学和计算机，可以建立一个完整的疫情模型，从而深入理解疾病的传播过程，为疫情防控工作提供科学依据。

例如，在新冠疫情爆发之初，各国科学家用多因素模型进行分析，推出了一系列应对措施，包括隔离措施、口罩戴法和疫苗研发等，这些措施都对抑制疫情的蔓延起到了至关重要的作用。

剖析埃博拉病毒传播模型及规律预测1 模拟真实环境埃博拉病毒的自然宿主虽尚未最后确定，但已有多方证据表明猴子及猩猩等野生非人灵长类动物有埃博拉感染现象。

该病毒的传播途径分为人畜传播、人人传播两种。

2014年，在几内亚、塞拉利昂和利比里亚等国，许多受埃博拉病毒影响的人口都以丛林肉为重要的蛋白质和营养物质来源，与丛林中动物接触频繁。

这为人畜之间的病毒传播创造了条件。

我们现假设两个感染埃博拉病毒的虚拟种群：即某地区内的20万居民和3000只猩猩。

人能以一定的概率接触到所有的猩猩，当接触到有传播能力的猩猩后有一定概率感染病毒，而人发病之后与猩猩的接触可以忽略。

人与猩猩的潜伏期都为2周。

并在出现疫情41周后模拟外界医疗力量的介入，使得人类与猩猩不再发生接触，且隔离治疗人群的治愈率提高到80%。

模拟数据详见附录。

2 建立数学模型2.1 模型假设（1）依据人或猩猩的健康状态，将人或猩猩划分为健康者、埃博拉感染者（也称患病者）、退出者（含自愈者、死亡者）;（2）自然封闭条件下，猩猩无自然迁移，故无病源的流入、流出，种群数量不变。

人类数量庞大，在无大规模迁移的情况下，认为人类数目为一定值，保持不变;（3）健康者中不包含退出者;（4）人和猩猩自愈后二度感染的概率均为0，人被治愈后二度感染的几率为0;（5）不存在有效免疫药物可使人对埃博拉病毒产生免疫，同时猩猩对病毒也不免疫;（6）人的传染途径有人传染人、猩猩传染人两条。

两条途径的传染率并不相同，分别假设为传染率C1和传染率C2。

C1猩猩与猩猩之间传染途径只有猩猩传染猩猩一条，假设猩猩之间的传染率为C0;（7）患病人无法传染患病猩猩;（8）41周外界介入后，猩猩与人的传播途径切断，隔离患者的治愈率提高到80%，同时未被隔离的患者治愈率不变。

2.2 符号说明符号说明如表1所示：3 模型的建立与求解3.1 数据处理根据累计死亡个体数，求得每周死亡个体数。

同理，根据累计自愈个体数，求得每周自愈个体数。