发布时间 : 星期六 文章数学建模 美赛 2015 A题更新完毕开始阅读
Team Control Number For office use only T1 ________________ T2 ________________ T3 ________________ T4 ________________ 38915 Problem Chosen For office use only F1 ________________ F2 ________________ F3 ________________ F4 ________________ A 2015 Mathematical Contest in Modeling (MCM) Summary Sheet Modeling the Impact of Medication on Ebola Ebola virus disease is spreading in West African countries. As the medication for Ebola has been developed, we manage to offer an optimal plan for controlling the spread of Ebola. At first, we consider the situation without intervention. Based on SEIR model, we formulate an epidemic model (SEIQFR) with time-lag to simulate the future situation of Ebola. Referring to the data from the WHO early period, we fit the unknown parameters with the least square method. Implementing our model with the data of Sierra Leone, we give a future prediction of Ebola epidemic situation with Runge Kutta method. The result suggests that all compartments become stable in the end, which means an equilibrium point is reached. As for intervention involved situation, we build a non-liner programming model to generate a distribution plan of medication. Based on the first model, we add intervention of vaccine. Assisted with modified Particle Swarm Optimization algorithm, we reach to a solution leading to fewer infected population in Sierra Leone after 4 days with a steady manufacture speed of vaccine. We find the infected people will decrease by 80, therefore proves that vaccine is able to ease the epidemic. Then, we develop a liner programming model to provide a delivery system with the least cost. According to the results in distribution model, we get this solution system for Sierra Leone. Our sensitivity analysis considers influence of other factors. Situations with contact rate changes are tested. The results suggest that the contact rate between infected and susceptible people has the most impact. Our SEIQFR model considers the effect of time-lag, so it suits the features of Ebola better. The model is flexible in infected countries, as long as the initial data of Ebola cases are available.
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Content
I. INTRODUCTION .................................................................................................................... 1 1.1 Background .............................................................................................................................. 1 1.2 Previous work .......................................................................................................................... 1 1.3 Our work .................................................................................................................................. 2 II. THE DESCRIPTION OF THE PROBLEM ........................................................................ 2 2.1 How do we simulate future epidemic situation? ...................................................................... 2 2.2 How do we consider the influences of medication distribution? ............................................. 2 2.3 How do we analyze our results? .............................................................................................. 2 III. MODELS ............................................................................................................................... 3 3.1 Notations .................................................................................................................................. 3 3.2 Assumptions ............................................................................................................................ 3 3.3 Improved SEIR Epidemic Model ............................................................................................ 4 3.3.1 Ascertainment of the Parameters ...................................................................................... 6 3.3.2 Solution and Result ........................................................................................................... 7 3.3.3 Analysis of the Result ....................................................................................................... 9 3.4 Medication Distribution Optimization Model ......................................................................... 9 3.4.1 Ascertainment of terminology ........................................................................................ 10 3.4.2 Solution and Result ......................................................................................................... 11 3.4.3 Analysis of the Result ..................................................................................................... 13 3.5 Medication Delivery Model ................................................................................................... 13 3.5.1 Solution and Result ......................................................................................................... 14 IV. SENSITIVITY ANALYSIS ................................................................................................ 15 4.1Influence of β .......................................................................................................................... 15 4.1.1 Influence of βI ................................................................................................................. 15 4.1.2 Influence of βQ ................................................................................................................ 16 4.1.3 Influence of βF ................................................................................................................. 17 4.1.4 Analysis of results ........................................................................................................... 17 4.2 Time begin intervention ......................................................................................................... 17 V. CONCLUSIONS ................................................................................................................... 18 5.1 Conclusions of the problem ................................................................................................... 18
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5.2 Strengths and weaknesses ...................................................................................................... 18 5.2.1 Strengths.......................................................................................................................... 18 5.2.2 Weaknesses ..................................................................................................................... 19 VI. FUTURE WORK ................................................................................................................ 19 VII. REFERENCES .................................................................................................................. 20 VIII. MEMO .............................................................................................................................. 21
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I. Introduction
1.1 Background
Ebola virus disease (EVD), the disease with most fatality rate, spreads by direct contact with body fluids, such as blood, of an infected human or other animals. The current outbreak in west Africa, (first cases notified in March 2014), is the largest and most complex Ebola outbreak since the Ebola virus was first discovered in 1976. Ebola is not only fatal, but also with high risk of transmission. Even the body of Ebola patients are infectious, so improper burials may also cause infection. Another characteristic of Ebola is the latent period, which possibly varies from 4 to 6 days but can top to 29 days long. During the latent period, the infectious has little chance of transmission and appears no symptom.
Recently, drugs aiming at curing Ebola patients has been successfully developed. As a world-focusing virus, Ebola’s spread situation has been studied for a period of time. The new medicine is a great help to control the epidemic situation. Thus at present, we can assist the Ebola eradication process if optimal plan is proposed, which can be reached by building an mathematical simulation model of the epidemic situation. Furthermore, relevant factors, such as the quantity of the medicine needed, possible feasible delivery systems, locations of delivery, speed of manufacturing of the vaccine or drug and other possible ones, can also be addressed on the basis of the former model.
Among all infected countries, Guinea, Liberia and Sierra Leone are the three most affected ones. And there are still newly occurred cases daily. So every beneficial measurements counts. As long as the method is reasonable and scientific, the adoption of which can be of great importance, even save a large number of lives. 1.2 Previous work
The earliest outbreak of Ebola happened at Yambuku, Zaire in 1976, and happened again at Kiewit, Zaire in 1995. Then some relevant researches has been done in the next year. In 1996, Aimee Astacio and other scholars adopted SIR and SEIR model to simulate the spread of Ebola.[1] They fit the parameters based on data acquired from two former outbreaks. Ebola outbroke in Congo and Uganda 10 years later. J.Legrand and others brought out a new model called SEIHFR model modified from premier one.[2] They added two compartments: hospitalized and funeral, and improved the differential equations for Ebola’s features, such as the infectious patient body. The parameters are obtained by knowledge of statistics.
In 2014, Ebola appeared again in three west African countries and the relevant research became more. Caitlin.M.Rivers and others then added intervention into the SEIHFR model, and fit the parameters by former data in Sierra Leone and Libya. More analysis like the upgrade of quarantine and community isolation are also addressed.[3] Network Dynamics & Science