埃博拉病毒传播分析与数学建模word精品
美赛 数学建模 埃博拉
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 .。
埃博拉病毒的根除数学建模论文
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埃博拉病毒的传播
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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。
2015年数学建模美赛埃博拉病毒
The model of eradicating EbolaIn order to construct an effective model, the following factors need to be taken into consideration: the propagation of the disease, the demand of drugs, the transportation system and the producing speed of the drug. We first applied the SIR model to stimulate how the disease propagated and by assuming different cure rates, we got the demand of the drug when the disease propagated. Then, considering the situations of the epidemic, population and traffic of three countries in West Africa, we made suitable delivering systems for these countries.In terms of the propagation of the disease, we searched for statistics and applied the SIR model to calculate the daily contact rate. Also, considering that the producing speed of drugs would increase due to economies of scale, we started from the current situation of the epidemic and increased the cure rate gradually. According to the calculation of MATLAB, we stimulated how the disease would propagate after effective drugs came out and the result was that the disease would be under control after fifty days.With respect to the demand of the drug, because the supply of the drug would gradually increase as the production capacity strengthening, the cure rate would also increase. Combining the effect of the decrease in the number of patients and that of the increase in cure rate, the demand of the drug is supposed to go up and then go down.As for the delivery system and the place of delivery, we took Sierra Leone as an example. Taking the population distribution, the situation of the epidemic and the traffic situation into consideration, we set the capital city Freetown as the chief trading place. After drugs were delivered to Freetown, most drugs would be distributed to those western coastal areas with a dense population and severe epidemic situations and the remaining drugs would be delivered to other places of the country. Key words:Ebola SIR Model Drug Delivery Demand of the DrugNon-technical LetterAccording to the statistics of WHO, the cumulative cases has been reaching 22460 in the three countries in west Africa (Guinea, Liberia, Sierra Leone) until February 1, 2015. And the death toll has been reaching 8968 people. The morbidity increases nearly a week. Although only a few countries and regions suffer the disease, the world medical association has developed a new medication could stop Ebola and cure patients whose disease is not advanced to response the serious disease.From March 21, 2014, Guinea, Liberia, Sierra Leone suffered from the disease one by one. Under the effective prevention interventions, the daily contact number of infected people has decreased. However the number of infected people is increasing.To optimize the eradication of Ebola, or at least its current strain, we estimate that the recovery rate every day will reach 0.2 from 0.05 considering the speed of manufacturing of the vaccine and the treatment level. The total quantity of the medicine needed is close to 35000. Under this speed of medicine supply and treatment, the disease will be controlled in 50 days.Focus on Sierra Leone for example, since the western area based on the capital Freetown is overcrowded and the disease of the western area is serious, the medicine provided by world medical association will be delivered in Freetown. Then the medicine will be allotted again based on population density and the condition of disease.In fact, the most effective way to prevent the further spread of Ebola is effective isolation of the infected people, reducing the probability that susceptible populations contact with patients. The transmission speed, the range, the strength of Ebola depend on quantities of the infected persons and susceptible persons, and effective contact between the two parts which could be affected by exposure level, pathogenic species, the quantity of excreted pathogens, resistance of the susceptible.Ebola virus is the most serious outbreak of the last 40 years, is a common challenge all the world as well. Currently, the fight situation against the epidemic is still very grim. As we all know, economic level and medical standard of the West African region is behindhand, the current state has overstepped their capabilities to fight with Ebola alone. We appeal to all countries for assisting the countries that are under the attack of Ebola, help them strengthen health systems and auxiliary infrastructure further.Contents:1. Introduction (1)1.1. Background (1)2. Model 1—Ebola Virus Propagation Model (1)2.1 Model Assumptions (1)2.2 Model Constitution (1)2.3 Numeric Calculation (2)2.4 Analysis of Phase Trajectory Figure (4)2.5 Conclusion (5)3. Model 2—Medicine Supply Model (6)3.1 The demand of medicine (6)3.2 Medicine Delivery System (7)3.2.1Drug distribution proportion (7)3.2.2Methods of the Medicine Delivery (7)4. Sensitivity Analysis and Improvements (8)4.1 Sensitivity Analysis (8)4.2 Improvements (8)5. Model Evaluation (9)5.1 strengths (9)5.2 weaknesses (9)6. Reference (9)1. Introduction1.1BackgroundWest Africa is hit by the most unprecedented outbreak of Ebola virus caused by the most lethal strain from Ebola virus family. The World Health Organization (WHO) declared it as an International Medical Emergency. As of 31st August 2014, the numbers of Ebola cases are 3685 with 1841 deaths reported from Liberia, Guinea, Senegal, Sierra Leona and Nigeria. WHO director general said that the actual numbers of cases are more than the reported cases.Guinea, Sierra Leone and Liberia are the poorest countries of the world with scruffy healthcare system. Even their hospitals lack the basic public health facilities and they are facing the terrible Ebola epidemic.2. Model 1—Ebola Virus Propagation ModelEbola virus is mainly spread through blood and excreta of patients, and we build our model according to its propagation mechanism. Once cured, Ebola virus patient has a strong immunity, so, the people who have recovered are neither healthy (susceptible) nor patient (infective), they have dropped out infected system. In this case, situation is much complicated, so the following will be a detailed analysis of the modeling progress.2.1 Model Assumptions:● The total number of population N is constant. We don ’t consider the born anddeath of people, and no migration of population, so people can be divided into three parts: the healthy, the infected and the removed recovered from Ebola disease. At time t , the proportion of three parts in the total population N were referred as s(t), i(t) and r(t).● The average number of a patient contacts per day is a constant effective λ, λ iscalled daily contact rate. When patients have effective contact with the healthy, it makes the healthy people infect and become patients.● The proportion of cured patients everyday in total number of patients is a constanteffective μ, μ is called daily cured rate. Patients cured and recovered from the disease will not infect Ebola again.● The contact number in the infectious period σ=λ/μ.2.2 Model ConstitutionAccording to assumption one, it is apparent that:s(t)+i(t)+r(t)=1 (1)According to assumption two and three, a patient can make λs(t) healthy people into patients per day, because the number of patients is Ni(t), so the number of infected healthy people is λN s(t)i(t), and λNsi is the increasing rate of patients, hence we have:Ni Nsi dtdi N μλ-= (2) As to the recovered and immune people, we haveNi dtdr Nμ= (3) We assume that at the beginning, proportion of the healthy and patients are s 0(s 0>0) and i 0(i 0>0) (we might as well define r 0=0), and according to equations (1), (2) and (3), we get the equation of SIR model: ()()⎪⎪⎩⎪⎪⎨⎧=-==-=000,0s s si dtds i i i si dt di λμλ， (4) The analytical solution of equation (4) can ’t be obtained, so we first make numerical calculation.2.3 Numeric CalculationIn order to work out the analytical solution of equation (4), we have to figure out precise value of each parameter. Firstly, we need to figure out the value of daily contact rate λ. We find some necessary data in WHO official website, which include the population of Ebola severest three countries: Guinea, Sierra Leone and Liberia is 21.6 million and the number of infected people every once in a while. So we can get the proportion of the healthy and the infected, the number of new infected per day and the daily contact rate in the table below:Table 1timeintervalproportion of health/% proportion of infection/% new infection per day Contact number/λ 00.9999963 0.0000037 0.0 — 130.99999338 0.00000662 4.8 0.0339 190.9999888 0.0000112 5.2 0.0215 620.99997227 0.00002773 5.8 0.0096 320.99993875 0.00006125 22.6 0.0171 110.99992079 0.00007921 35.3 0.0206 90.99990153 0.00009847 46.2 0.0217 70.99981741 0.00018259 127.4 0.0323 100.99972949 0.00027027 147.7 0.0253 90.99966866 0.00033134 146.0 0.0204 120.99954116 0.00045884 127.9 0.0129 90.99937315 0.00062685 436.6 0.0323 120.9993487 0.0006513 44.0 0.0031 70.99926384 0.00073616 150.9 0.0095 50.99920782 0.00079218 188.4 0.0110 60.99917093 0.00082907 132.8 0.0074 70.99904111 0.00095889 78.4 0.0038 70.99899588 0.00100412 61.1 0.0028 70.99897884 0.00102116 52.6 0.0024 7 0.99896019 0.00103981 57.6 0.0026According to the actual data, we can getλat different time periods. As it clearly shown in the table, when the Ebola just out-broke, we figure out thatλ=0.0339, and it obtained its maximum value. With time pasted, the value of λis becoming smaller and smaller, which indicates the patient contacted less people every day and less people were infected. This might because governments took some effective action to control and prevent the spread of the disease, the patients were separated from the healthy people. To simplify our model, we don’t take separation of patients into our consideration, and we ignore the condition that patients died of the virus. So, we determineλ=0.0339, because at the beginning of the virus out-broke, no external constrains obstruct the spread of the disease, it is an ideal environment for the virus, we takingλ=0.0339 will make our model more accurate.Then we have to determine values of other parameters. Ebola is a very dangerous virus and it has caused devastating destruction. So we assume that at the beginning, Ebola disease can’t be cured, no effective measures can be taken to control the virus. In this conditionμ=0, Ebola virus will spread at a high speed.Figure 2Combined with the actual situation, we assume that at different time, the value of λis changing. We make every five days as an interval, and redefine the value of λevery five days. In the first five days, the disease was just out-broken, people had less conscience of the seriousness, and they lacked necessary action to protect themselves. What’s more, at the beginning, no effective drugs to treat patient, and the supply of medicine was not sufficient, so we define the daily cured rate μ=0.05. The next five days, people gradually realized that in their country has broken out a dangerous disease, they improved the awareness to protect themselves, and avoided contacting with the patient, so at this timeμ=0.1. In the third five days, government put a lot of effort to control and prevent the disease, patients were separated from the public, a lot of effort was put to research the virus and produce effective drugs, soμ=0.15. In the twentieth day and after, the whole world paid much attention to this disease and gave a lot of assistance to Africa countries, the disease was under control, the supply ofmedicine was sufficient, so more patients were cured, so we define the value of daily cured rate is constantμ=0.2.We can get i0and s0 from Table 1: i0=0.00104, s0=0.99896. Then we use MATLAB to draw curves of i(t) and s(t).Table 3t 0 1 2 3 4 5 6i(t) 0.0010398 0.0010232 0.0010068 0.0009907 0.0009748 0.0009592 0.0008978 s(t) 0.9989602 0.9989253 0.9988909 0.9988571 0.9988238 0.9987911 0.9987596 t 7 8 9 10 11 12 13i(t) 0.0008403 0.0007866 0.0007362 0.0006891 0.0006135 0.0005463 0.0004864 s(t) 0.9987302 0.9987027 0.9986769 0.9986528 0.9986308 0.9986112 0.9985937 t 14 15 16 17 18 19 20i(t) 0.0004330 0.0003855 0.0003265 0.0002765 0.0002342 0.0001984 0.0001680 s(t) 0.9985782 0.9985643 0.9985523 0.9985421 0.9985335 0.9985262 0.9985200Figure 4s(t) i(t)From Figure 4 we can see that values of s(t) and i(t) change a little, s(t) changes from 0.99896 to 0.99849, and finally tends to a constant value. This means at last nobody was infected Ebola any more. This is conformed to real situation. After finding several people were died of Ebola virus, western Africa countries and WHO had paid highly attention to this problem. They took effective and instant action to control the spread of the disease. They separated the patient from the public and used the most advanced methods to treat the patient. In addition to this, these countries conducted a large investigation among suspicious people, making sure that the disease wouldn’t be infected in a large scale. Under these powerful action, Ebola virus didn’t spread, the number of infected people was increasing slowly, so the change of daily infected rateλis very small, Analogously, the change of i(t) is small because of the same reason. Finally, the value of i(t) equals to zero, which indicates that Ebola virus was wiped out. But this is the result of our model, it is an ideal situation, and now this situation has not happened.2.4 Analysis of Phase Trajectory FigureIn order to analyze the general variation of s(t) and i(t), we need to draw i~s relationship figure. This i~s figure is called phase trajectory figure.Based on the numeric calculation and observation of the figure, we can use phase trajectory line to analyze the character of s(t) and i(t).The s~i plane is called phase plane, the domain of definition of phase trajectory line in the phase plane (s, i) ∈D is:(){}1,0,0,≤+≥≥=i s i s i s DWe erasure dt in the equation (4), and noticing that σ=λ/μ, we can get00,11i i sds di s s =-==σ (5) We can easily figure out that the answer of equation (5) is:()000ln 1s s s i s i σ+-+= (6) In the domain of definition D, the line that equation (6) displays is phase trajectory line, we can get the changes of s(t), i(t) and r(t).● At any case, the patient will disappear at last, that is i ∞=0● The final proportion of the uninfected healthy people is s ∞, we define i=0 in theequation (6), so s ∞is the answer of equation0ln 100=+-+∞∞s s s i s σ ● If s 0<1/σ, i(t) is monotone decrease and finally drops to 0, s(t) is monotonedecrease to its minimum s ∞ .2.5 ConclusionAccording to our analysis and calculation, the number of patients is constantly decrease, and in the fiftieth day after the virus broke out, we figure out that the patients will drop to 24 people, compared with the beginning of the disease, the number of patients is more than 20000, so we can draw the conclusion that we havealready successfully controlled the disease, the virus didn’t spread in a large range. If we continue conducting some effective action, such as separating the patient, propagating the information about avoid the virus, Ebola disease will be completely wiped out in the next few weeks.3. Model 2—Medicine Supply Model3.1 The demand of medicineThe world medicine association has announced that their new medication could stop Ebola and cure patients whose disease is not advanced. That is to say, as long as the supply of medicine is sufficient, all the patients could be cured. Next step our goal is to figure out the demand of medicine every day.To solve this question, we assume that one unit medicine could cure one patient, and according to model one, we have calculated the number of patients and daily cured rate, so the quantity of daily demand medicine can be calculated. We define that daily needed quantity of medicine is Q.Q=N*s(t)*μThe result is shown in the Table 5 below.Figure 5The changes of medicine demand are shown in the figure 5.We can see that in the first four days, demands of medicine slightly declined. Because at the beginning, the effective medicine has been developed, and the daily cured rate was relatively low, the virus wasn’t wide spread, the number of infected people was small, so the demand wasn’t very high. However, in the next few days, the demand was abruptly increased, almost doubled the demand in the fourth day. With more people were infected the Ebola virus, they needed more medicine to treat their diseases. Another reason is that the daily cured rate increased, which means more medicine was used to mitigate patients’ symptom and pains. The next six days, the demand of medicine was showing a wavy change, because of the intervention of governments. They put more effort to control the virus in case it was spread in a large range, and more medicine was manufactured, hospitals tried their best to treat the Ebola patient. After the fifteenth day, the demand of medicine was constantly dropping, and finally dropped to nearzero, which indicates the virus was under control, more and more people were cured, and very few people were infected Ebola virus, and at last Ebola virus was wiped out, medicine was scarcely needed.3.2 Medicine Delivery System3.2.1Drug distribution proportionGood delivery system is based on the distribution of population, the condition of disease, the condition of traffic. There we take Sierra Leone for example. We find the data about the population of every province and the number of infected people. According to the initial data, we calculate the rate of infections, rate of population density. Combined with this two rates, we get the final drug distribution proportion. This table is the result about drug distribution proportion.Table 6district rate of infections rate of population density rate of medicine Northern Province 37.57% 13.27% 25.42% Eastern Province 16.30% 20.91% 18.61% Southern Province 6.89% 14.82% 10.85% Western Area 39.24% 51.01% 45.12% Rate of infections means the rate of infected people in every district to the total infected people in the country. Rate of population density means the population density in every district compared to the sum of population density. Rate of medicine equals to the average of rate of infections and rate of population density. That is the proportion of drug distribution.As we can see from the table, the western area where the capital is needs 45.12% of the drug. However, western area covers the area of less than 10% compare to the total land area. And the condition of disease is as follow:Figure 73.2.2Methods of the Medicine DeliveryBecause the railway and road system of Sierra Leone are defective and out-dated, the medicine was firstly delivered to the capital city Freetown, then by the air transportation delivering to other airports. Afterwards, the medicine was delivered tothe medical center in the Ebola serious area. The situation of airports in Sierra Leone is shown in the table below:4. Sensitivity Analysis and Improvements4.1 Sensitivity AnalysisModel one we use SIR model to simulate the development of Ebola virus, and through two variables λ(daily infected rate) and μ(daily cured rate) to describe the situation of the disease. We can figure out the number of healthy people and infected people every day, there are many factors affecting the changes of healthy and infected number. For the convenience of calculation, we ignore the death of patients, and assume the total population is constant, and the daily cured rate is defined by ourselves, and the disparity between real value and the calculated one may be obvious. What’s more, we assume the medicine is very effective, all of the patients take the medicine will be cured, and we don’t consider the time of treatment, these problems could make our model have some drawbacks and less accuracy.4.2 ImprovementsFor model one, according to analysis before, two measures can be taken to restrain the spread of the disease, one is improving health and medical level, in the other words is dropping the daily contact rate λand improving daily cured rate μ; the other method is herd immunity, that is improving the original rate of the removed r0. We can see that in the SIR model, σ=λ/μis a very important parameter. In reality, the value of λand μis difficult to estimate, but when an infectious disease is over,we can get the value of s 0 and s ∞, and we neglect the parameter i 0because of its small value, so we can have the result of σ:∞∞--=s s s s 00ln ln σ (7)We can use the change of σ to analyze the development of the disease instead of λ and μ.5. Model Evaluation5.1 Strengths● Model is simple and easy to understand, and we innovatively define the suspicious degree, making us to carry out a quantitative analysis of suspects.●Processes the data and make a variety of charts, simple and intuitive shortcuts●Model established in this paper and the actual closely, give full consideration to the different stages of the reality of the situation, so that the model is more realistic 5.2 Weaknesses● In order to make the calculation is simple in the model, so that the results obtained are more ideal, ignoring the minor factors.●For some data, we carried out a number of necessary treatment, which will bring some errors.6. Reference:[1]. Xuan Zhou, Junquan Song, Xuejun Wu. Introduction and Improvement of Mathematical Contest in Modeling [M]. Zhejiang:ZHEJIANG UNIVERSITY PRESS.2012.Page 201 to 205.[2]. Qiyuan Jiang, Jinxing Xie, Jun Ye. Mathematical Model [M].Beijing:HIGHER EDUCATION PRESS.2011.Page 136 to 145.[3]. Ebola Situation Report[DB/OL],http://apps.who.int/ebola/ [4]. Sierra Leone —Provinces anddistricts[EB/OL]./wp/s/Sierra Leone.htm[5].HongqingZhou,ZhuanXu. A Mathematical Model of Ebola Virus Infection Numbers [A].[6].Themap of Sierra Leone[Z/OL]./maps/HTML/49248.htmlAppendix: BasicMatlabprogram：>>function y=ill(t,x)a=0.0339;b=0.2;y=[a*x(1)*x(2)-b*x(1),-a*x(1)*x(2)]'; ts=0:100x0=[0.00103981,0.99896019];[t,x]=ode45('ill',ts,x0); [t,x]plot(t,x(:,1)),grid on;plot(t,x(:,2)),grid on;plot(x(:,1),(:,2)),grid on;Data recording:。
埃博拉病毒的传播分析
=P N (t)， 其中 P 是常数。但 应是一个有上限的值， 不
能无限增长。 对于模型中的 M， 定义 10 ， (3) M= 10 I t） 考虑到感染者在不同时期的感染能力不同， 我们在（ 上 加入了影响因子 M。题中表明感染者一般 6 ～ 9 天死亡， 为 10） 简化我们统一将感染者存活时间定为 10 天， 通过 Mod （t， M 则反映了此时他的感染传 来反映感染者处于哪一感染期， 播能力。 4 问题的求解 4.1 问题一的求解 4.1.1 参数的求解 首先对 SEIR 模型中的四个参数进行求解： 对于潜伏者转换为感染者的转化率 1， 在网上查阅相关资 料可得：
理|论|广|角
埃博拉病毒的传播分析
王
（西北工业大学
摘 要
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710068）
陕西·西安
由于猩猩的潜伏期无法忽略，因此我们建立了经典的传染病传播 SEIR 模型。但由于在发病的不同时期患
病者的感染能力不同， 因此我们在此模型的基础上对感染人数增加了一个扰动项。我们未考虑猩猩的出生率和死亡 率， 从而得到适合此问题的改进的 SEIR 模型。通过对模型中参数的求解， 我们可以对埃博拉病毒在猩猩群体的传播 进行预测。 关键词 埃博拉病毒 R332 中图分类号： 1 问题分析 1.1 问题一的分析 问题一是对埃博拉病毒在猩猩种群中传播感染规律的研 究。此问题属于传染病传播问题，解决此类问题一般建立经 典的 SIR 仓室模型进行求解。考虑到本问题中埃博拉病毒的 潜伏期对本问题的结果有不可忽略的影响，因此我们在经典 SIR 模型的基础上增添潜伏期这一仓室建立 SEIR 模型。考 虑到猩猩种群的总数较小 （3000 只） ， 并且猩猩的繁殖周期较 长 （3-4 年） ， 因此在本模型中忽略猩猩自然出生率及死亡率对 问题结果的影响。 2 模型假设 （1） 假设猩猩的自然出生率及死亡率对预测结果无影响； （2） 假设猩猩和人都处在除埃博拉疫情外正常的生活状态， 不受其它因素影响； （3） 假设感染者在自愈的同时获得免疫力， 不会再感染病 毒； 3 模型的建立 3.1 问题一模型的建立 问题一是考虑在猩猩种群中埃博拉病毒的传播情况，本 问题中埃博拉病毒的潜伏期对本问题的结果有不可忽略的影 响。猩猩种群的总个体数较小 （3000 只） ， 并且猩猩的繁殖周 期较长 （3-4 年） ， 因此在本模型中忽略猩猩自然出生率及死亡 率对问题结果的影响。同时考虑到有效传染率 、 潜伏期到患 病的转换率 、 死亡率 、 治愈率 r 等因素的影响。我们建立如 下 SEIR 模型： 改进的 SEIR 模型 最小二乘法 A 文献标识码： 们令
埃博拉病毒传播模式的研究
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【 Ｙ （ Ｏ＝ （ ｒ ） ＋ Ｄ（ ｆ ） ＋ （ ３模 型 的 求 解 ３ ． １ 控前模型求解 对 于现 有 病 人数 ｘ（ ｔ ） ， 可 以根 据 病 毒传 播 的控 ， ｔ ≤Ｔ 其中， ｆ ｒ ：Ｏ ． ５ ５
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《 Ｉ ｒ （ ０ ） ＝ １
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２ ． ２ 控 后 模 型
控后隔离强度从控前的 ０ 变为 Ｐ 。未被隔离的病人平均 每 人 单 位 时 间感 染 的人 数 ｒ 随 时 间 逐渐 变 化 ， 它 从 初 始 的 最大 值 ａ ＋ ｂ逐渐减少至最小值 ａ ， ａ 、 ｂ的值客观存在。设每个被隔 离 的病 人 单 位 时 间感 染 的 人数 为 ｒ （ ｔ ） ＝ ａ ＋ ｂ ｅ 。 其 中， 用来反映 ｒ （ ｔ ） 的变 化 快 慢 。 类似 于 控 前 模 型 ， 我 们 得 到埃 博拉 传 播 的控 后 模 型 ： ￣ 一 ｃ 其中 ’ ＝ ａ ＋ｂ ｅ
流行病学案例分析
流行病学案例分析
流行病学是研究疾病在人群中传播和影响的学科，通过分析疾病的流行特征和传播规律，可以为疾病防控提供科学依据。
本文将以埃博拉病毒疫情为例，对流行病学案例分析进行探讨。
埃博拉病毒是一种由埃博拉病毒科的病毒引起的急性出血热，最初在1976年被发现。
该病毒的传播主要通过接触感染者的体液，如血液、尿液、呕吐物等。
该病毒具有高度传染性和致死率，2024-2024年在西非国家爆发的埃博拉疫情导致了逾1.1万人死亡。
在对埃博拉疫情的流行病学分析中，可以从疫情的传播过程、暴发源和风险因素等方面进行研究。
首先，疫情的传播过程是流行病学分析的重要内容之一、通过追踪感染者的接触史和传播链，可以了解疾病在人群中的传播过程。
在埃博拉疫情中，病例的传播主要是由感染者的体液和分泌物通过直接接触传播，如接触患者的呕吐物、尸体、患者的床单和衣物等。
疫情传播的过程中，密切接触者和医护人员是高风险群体。
其次，暴发源是流行病学分析的重要内容之一、疫情的起源能够对防控措施产生重要影响。
根据流行病学调查，2024-2024年的西非埃博拉疫情的暴发源是来自几个国家的野生动物（如果蝠、猴子）与人类的接触。
由于食用野味等传统习俗，导致病毒从动物传播到人类，形成病毒的人际传播链。
因此，需要加强对野生动物的监测和禁止非法贸易，以防止类似疫情的再次发生。
总之，流行病学案例分析对了解疾病的传播过程、暴发源和风险因素至关重要。
通过对埃博拉病毒疫情的分析，可以为类似疫情的防控提供科学依据，并加强对潜在传染病的监测和预警，以保障人民的生命安全。
埃博拉病毒传播问题的数学模型汇编
-4-
现在我们根据建立的模型利用已知数据进行求解，我们将题目中给出的数 据进行整理，得出我们建模求解需要的 S、E、I、R 四个群的猩猩数量（见附录 1），并且利用整理出的已知数据对微分方程组中的参数进行求解：
1）潜伏者日接触率1 (t)：
1(t)
E / t I (t) * S(t)
(3)
潜伏人群 处于发病状态 隔离治疗 累计治愈 累计因病死亡
第 80 周 105.4
17.53
12.5
1164.4
2985.6
第 120 周 101.7
5.491
4.3
1330.5
3448.5
第 200 周 100.9
0.4208
0.3
1397.0
3621.0
问题三 针对此问中采取隔离与治愈感染者的措施后，要求预测疫情在人类中的发 展情况、并与问题二结果作比较的问题，我们利用问题二的结论排除猩猩对人 的影响后，此时病毒是在单物种内传播，适用于我们建立的单物种 SEIR 模型。 利用排除猩猩影响后得到的数据，建立微分方程解得：
2）感染者日接触率 1 (t):
1 (t )
I / t I *S
(4)
3）退出率1(t) ：
1 (t )
R(t) I (t)
(5)
4）潜伏群发病率1(t) ：
1
(t
)
=每日新确认病例数
每日疑似累计数
(6)
由我们整理出来的数据带入以上公式，估算得：
1 1.085E-4, 1 0,1 0.3018,1 0.5121
三 、 符号说明
使用符号 S E I R R1 R2
(t ) (t) (t) (t)
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****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。
关键词：非线性曲线拟合；微分方程；MATLAB数学模型1问题的重述埃博拉病毒（又译作伊波拉病毒）于1976年在苏丹南部和刚果的埃博拉河地区被发现后，引起了医学界的广泛关注和重视。
该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，其生物安全等级为4级。
埃博拉病毒有传染性，主要是通过病人的血液、唾液、汗水和分泌物等途径传播。
各种非人类灵长类动物普遍易感，经肠道、非胃肠道或鼻内途径均可造成感染，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。
发病后1〜4天直至死亡，血液都含有病毒。
埃博拉病毒感染者有很高的死亡率（在50%至90%之间），致死原因主要为中风、心肌梗塞、低血容量休克或多发性器官衰竭。
当前主流的认知是，埃博拉病毒主要通过接触传播，而非通过空气传播；只有病人在出现埃博拉症状以后才具有传染性。
在疾病的早期阶段，埃博拉病毒可能不具有高度的传染性，在此期间接触病人甚至可能不会受感染，随着疾病的进展,病人的因腹泻、呕吐和出血所排出的体液将具有高度的生物危险性；存在似乎天生就对埃博拉免疫的人，痊愈之后的人也会对入侵他们的那种埃博拉病毒有了免疫能力。
埃博拉病毒很难根除，迄今为止已有多次疫情爆发的记录。
据百度百科，最近的一次在2014年。
截至2014年9月25日，此次在西非爆发的埃博拉疫情已经导致逾3000人死亡，另有6500被确诊感染。
更为可怕的是，埃博拉病毒可能经过变异后可以通过呼吸传播！1.2问题假设某地区有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。
研究人员统计了前40周人类和猩猩的发病数量和死亡数量等信息，请你根据相关信息，研究回答以下问题：1、根据猩猩的发病数量和死亡数量，建立一个病毒传播模型，动态描述病毒在虚拟猩猩种群”中的传播，并预测接下来的在猩猩中的疫情变化，并以下述格式给出虚拟猩猩种群”在第80周、第120周、第200周的相关数据；2、建立虚拟种群”相互感染的疾病传播模型，综合描述人和猩猩疫情的发展，并预测接下来疫情在这两个群体中的发展情况，并以下述格式给出虚拟人类种群”在第80周、第120周、第200周的相关数据；3、假设在第41周，外界的专家开始介入，并立即严格控制了人类与猩猩的接触，且通过某种特效药物将隔离治疗人群的治愈率提高到了80%。
请预测接下来疫情在虚拟人类种群”的发展情况，对比第2问的预测结果说明其作用和影响，给出虚拟人类种群”在第45周、第50周、第55周的相关数据，数据格式同问题2；4、请依据前述数学模型，分析各种疫情控制措施的严格执行和药物（包括防疫药物、检疫药物和治疗药物等）效果的提高等措施对控制疫情的作用。
2问题分析2.1问题一的分析通过对已知条件的分析，并通过给出的表格数据，大致明白猩猩从潜伏到发病再到死亡或自愈。
我们通过exceI作出发病随时间的变化曲线，潜伏随时间变化曲线，估计参数。
然后通过建立数学模型用MATLAB军出方程组，调试参数使其死亡，自愈等曲线与给出表格大致相同，然后通过建立的模型求出问题一。
2.2问题二的分析同问题一分析，我们通过excel作出相应处于发病状态的曲线，自愈以及死亡和隔离的曲线，估计模型相应的参数。
然后通过建立的数学模型用MATLAB解出方程组，调试参数使其自愈，处于发病等曲线和表格给出的数据大致一致。
2.3问题三的分析同问题二分析，我们通过excel作出治愈率提高80%后相应处于发病状态的曲线，自愈以及死亡和隔离的曲线，估计模型相应的参数。
然后通过建立的数学模型用MATLAB解出方程组，调试参数使其自愈，处于发病等曲线和表格给出的数据大致一致。
2.4问题四的分析通过上术数据和曲线图的分析，可以很清楚的看出当有人类干预后即就是严格的通过药物后，发病和潜伏等都有很明显的改善。
3假设与符号3.1模型的假设：由于埃博拉病毒的传播期限不是很长，故假设不考虑这段时间内的人口出生率和自然死亡率；平均潜伏期限为6天；处于潜伏期的埃博拉病人不具有传染性。
3.2符号说明：t o 表示从最初发现埃博拉患者到卫生部门采取预防措施的时间间隔；N 表示疫区总人口数；S(t) 表示t时刻健康人数占总人口数的比例；l(t) 表示t时刻感染人数占总人口数的比例；E(t) 表示t时刻潜伏期的人口数占总人口数的比例；Q(t) 表示t时刻退出类的人数占总人数的比例；入(t) 表示日接触率，即表示每个病人平均每天有效接触的人数；N' 表示疫区总猩猩口数；S(t)' 表示t时刻健康猩猩数占总猩猩数的比例；I(t)' 表示t时刻感染猩猩数占总猩猩数的比例；E(t)' 表示t时刻潜伏期的猩猩数占总猩猩数的比例；Q(t)' 表示t时刻退出类的猩猩数占总猩猩的比例；入(t)' 表示日接触率，即表示每个病猩猩平均每天有效接触的猩猩数；入(t)''表示日接触率，即表示每个病猩猩平均每天有效接触的人数；g(t) 表示政府控制力度；f(t) 表示疫情指标。
4模型的建立与求解4.1问题一模型的构建由问题的分析，将猩猩群分为易感猩猩群S,病毒潜伏猩猩群E ,发病猩猩群I ,退出者Q 四类：易感人群S 与病毒潜伏人群E 之间的转化易感者和发病者有效接触后成为病毒潜伏者，设每个病人平均每天有效接触 的健康人数为入(t)S , NI 个病人平均每天能使入(t)SNI 个易感者成为病毒潜伏 者。
故ds'dsN ''S' N ' I ',即 ’=_「 dtdt病毒潜伏人群E 与发病人群I 间的转化潜伏人群的变化等于易感人群转入的数量减去转为发病人群的数量，即dE ' S'，(t)'l '- ；’E'。
dt发病人群I 与退出者Q 间的转化单位时间内退出者的变化等于发病人群的减少，即dQ=y ：' I ' dt——二 S'，(t)T ' — ;'E' dtdQ'二'I ' dtS' ■ E ' ■ I ' Q二s °',E(0)' =E °',l(0)' =l o ',Q(O)'=Q o '很明显从我们建立的模型是无法得到 E ' ,S ' ,I ' ,Q '的解析解的。
为了解 决这个问题，我们求助于计算机软件 MATLAB^求出它们的数值解。
我们先通过附件中给的数据算出每一天的E ' ,S ' ,I ' ,Q ',做出它们与时间的函数图象，然后画出我们通过模型解出的数值解随时间变化的图象。
对比这两组图，可以发现实际和理论存在着一定的差异。
这必然是因为我们的参数估计 不合理造成的。
所以，我们必须通过不断调整那些非计算得到的参数 (入 a ')来使实际图象和理论图象趋于一致。
'S' I 'ds' 'S' I 'dt dE 'S(0)'经过多次调试,我们发现，当入’=0.680人，£ ' =0.9, a ' =0.58时，实际图 象和理论图象有最好的符合。
而这三个值均在我们估计的范围内， 所以我们认为 这三个值的得到是合理的。
一旦参数确定，就可以通过 MATLAB^件求出该方程组在某个区间段的数值 解，从而可推算出我们所需的数值如下表所示。
在根据逻辑关系式计算可得下表的预测值表1 虚拟猩猩种群”群体数量预测结果单位：只结果分析根据上表可知，在第80周以后，处于潜伏状态的猩猩接近于 0，处于发病状态的猩猩也趋近与0,且猩猩的治愈数和因病死亡数变化不大，由该模型预测 出的结果与附件中的数据的得出的发病率和累计死亡率趋势相同。
健康人数占总数比例 （比对）1如 0O%周数图1.1 健康人数占总数比例图（参考数据）潜伏人数占总数比例（比对）周数图2.1 潜伏人数占总数的比例图（参考数据）图1.2 健康人数占总数的比例图（模拟数据）0. 30^0. 60«0, 40%0. 20«0, 00%系列1系列2图22潜伏人数占总数的比例图（模拟数据）退出人数占总数比例（比对）图3.1退出人数占总数的比例图（参考数据）MATLAB主要程序fun cti on dx=rossler(t,x,flag,a,b,c)dx=[-a*x(1)+a*x(1)*x (3)+a*x(1)*x (2)+a*x(1)*x(1);a*x(1)-a*x(1)*x(3)-a* x(1)*x(2)-a*x(1)*x(1)-b*x (2) ;c-c*x(3)-c*x(2)-c*x(1)];a=0.680;b=0.90;c=0.580;x0=[0.995 0.005 0]';[t,y]=ode45('rossler',[0 80],x0,[],a,b,c);flot(t,y);20. 00%15*00%10, 00%5. 00%0. 00%0 10 203040-- 系歹牡—系列250周数S3Figure 1Fil e Edit XXi e--w In & & rt "T og II 弓Oe p Winig-j OTA/ HI & Io匚三呻导矗良』巧倒ue Q i=ai □ D图3.2退出人数占总数的比例图（模拟数据）4.2问题二模型的构建由问题的分析，将人群分为易感人群 S,病毒潜伏人群E ,发病人群I ,退出 者Q 四类：易感人群S 与病毒潜伏人群E 之间的转化易感者和发病者有效接触后成为病毒潜伏者，设每个病人平均每天有效接触 的健康人数为入(t)S , NI 个病人平均每天能使入(t)SNI 个易感者成为病毒潜伏 者。