2015年美模A题M奖论文An Optimal Strategy to Eradicate Ebola.docx

Team # 37090Page 33 of 34For office use onlyT1_T2_T3_T4_Problem ChosenAFor office use onlyF1_F2_F3_F4_2015 Mathematical Contest in Modeling (MCM) Summary Sheet The advent of licensed Ebola vaccines and drugs delights the whole world while also posing a dilemma of how to allocate the needed quantity among all Ebola outbreaks and deliver them with effectiveness and efficiency. We establish comprehensive Ebola response models in three most suffering countries (Guinea, Liberia and Sierra Leone) including a prediction model generating short-term estimates of the Ebola transmission situations, an allocation-and-delivery model planning the needed quantity of medicines and the optimal delivery route, and a cellular automaton model measuring the effect of effective isolation and treatment. Besides, we also give policy making suggestions to prevent international spread to some unaffected countries.Based on the special characteristic of Ebola, we create a modified SEIR epidemic model with an added intervention factor to stand for the effect of some forms of interventions other than vaccines and drugs. We predict the potential number of future Ebola cases with or without the use of effective medicine and the result also shows that if the transmission trends continue without effective interventions, countries will undergo worse and worse situations In the next model, we first classify all outbreaks into five levels due to the different Ebola case numbers. Then we apply minimum spanning tree method, Monte Carlo method and 0-1 programming to our model to locate an optimal number of medical center and sub-centers in each country aiming to eradicate Ebola. We set one medial center in each country and one more sub-center in Guinea, three more sub-centers in Liberia and four more sub-centers in Sierra Leone. The model also calculates the minimal needed number of vaccines and drugs in every manufacturing cycle. Then, we discuss the effect of isolation and treatment by cellular automaton model and find out that if only effective isolation is conducted, the retarding effect is limited. We present a comprehensive strategy to eradicate Ebola by conducting dynamic models and as time passes, we can update the statistic data to reality which adds accuracy to our models and optimal results. An Optimal Strategy to Eradicate EbolaIntroductionEbola virus disease (EVD) is a severe, often fatal illness in humans. It has become one of the most prevalent and devastating threat for its intense transmission. Since first cases of the current West African epidemic of Ebola virus disease were reported on March 22, 2014, over 20000 new cases have been found and about 9000 patients have died from it. The western Africa areas-Guinea, Liberia and Sierra Leone in particular-are outbreaks that have suffered most 1.With the help of licensed vaccines and drugs, we aim to stop Ebola transmission in affected countries within a short period and prevent international spread. Our objectives are:l to achieve full and fast coverage with vaccines for susceptible individuals and drugs for infectious individuals among three most suffering countries (Guinea, Liberia and Sierra Leone);l to ensure emergency and immediate application of comprehensive Ebola response interventions in countries with an initial case or with localized transmission;l to strengthen preparedness of all countries to rapidly detect and response to an Ebola exposure，especially those sharing land borders with an intense transmission area and those with international transportation hubs1. For the first objective, we create a comprehensive Ebola response models in those three countries including a prediction model of Ebola transmission, an allocation-and-delivery model for vaccines and drugs used and a cellular automaton model measuring the effect of some crucial interventions. The last two objectives are closely related to policy making and in the following part of our paper we just present detailed information of our models.Basic Assumptions1. A patient can only progress forward through the four states and can never regress (e.g. go from the incubating to the susceptible) or skip a state (e.g. go from the incubating to the recovered state, skipping the infectious state).2. Once recovered from Ebola, an individual will not be infected again in a short time.3. Populations of each country remain the same over the prediction period.4. In absence of licensed vaccines or drugs, some other interventions are used, such as effective isolation for Ebola patients and safe burial protocol.5. When vaccines and drugs are introduced to the prediction model, the incubation period and the effect of interventions other than medicine will not change.6. Building a medical center is at a high cost (e.g. storage facilities of medicines, etc.) and every medical center are capable of delivering all needed medicines.7. We ignore the potential damage to medicines when delivering.8. We calculate the distance between two sites by measuring the spherical distance and ignore the actual traffic situation.9. Once received treatment with licensed drugs, patients will no longer be infectious individuals, which also means that we do not take the needed recovery period into account. 10. The needed vaccines or drug for an individual is one unit.11. AllthedatasearchedfromtheInternetareoftrustworthinessandreliability. Model 1: Prediction ModelWe create a modified SEIR model 2 to estimate the potential number of future Ebola cases in countries with intense and widespread transmission- Guinea, Liberia and Sierra Leone. Not only useful in predicting future situation in absence of any licensed vaccine or drug, the modeling tool also can be used to estimate how control and prevention medicine can slow and eventually stop the epidemic. Terminology and definitionsTable 1 Parameters and definitionsParameterDefinitionTransmission rate per infectious individual per dayRecovery rate per infectious individual per dayReciprocal of average incubation periodFatality rate per infectious individuals per dayIntervention factorS(t) or SPercentage of susceptible individuals in total populationE(t) or EPercentage of exposed individuals in total populationI(t) or IPercentage of infectious individuals in total populationR(t) or RPercentage of removed and dead individuals in total populationA probability distribution which captures the likelihood of incubating a given number of days is used from previous study. The resulting distribution has a mean incubation period of 6.3 days 3 and therefore, in our prediction model, patients are assumed to be infectious after a 6.3-days incubation period. Besides, in absence of licensed vaccines or drugs, Ebola is a disease with few cases of recovery. Thus, under this situation, we assume the recovery rate is 0.001, which is very close to zero.Method A frightening characteristic of Ebola virus disease is that it has an incubation period ranging from 2 to 21 days before an individual exposed to the virus who finally become infectious. Thus, we create a SEIR epidemic model tracking individuals through the following four states: susceptible (at risk of contracting the disease), exposed (infected but not yet infectious), infectious (capable of transmitting the disease) and removed (recovered from the disease or dead).Moreover, based on Assumption 4, some forms of interventions other than vaccines and drugs may also reduce the spread of Ebola and death numbers, and therefore we introduce an intervention factor as a parameter to measure the effect. In those three intense-transmission countries(Guinea, Liberia and Sierra Leone),at least 20% of new Ebola infections occur during traditional burials of deceased Ebola patients when family and community members directly touching or washing the body. By conducting safe burial practice, the number of new Ebola cases may drop remarkably. Moreover, effective isolation with in-time treatment is also of significant importance in reducing transmission and deaths.In our modified SEIR model, we describe the flow of individuals between epidemiological classes as follows. Figure 1 A schematic representation of the flow of individuals between epidemiological classesSusceptible individuals in class S in contact with the virus enter the exposed class E at the per-capita rate (-), where is transmission rate per infectious individual per day and is the intervention factor serves to retard the transmission. After undergoing an average incubation period of 1/ days, exposed individuals progress to the infectious class I. Infectious individuals (I) move to the R-class either recover or die at rate (+), where b stands for the recovery rate and d represents the fatality rate. Besides,The transmission process above is modeled by the following differential equation set: WemodifySEIRmodelbyaddinginterventionfactor.Algorithm1. With known values of parameter and , we solve the differential equation (1.1) by assigning certain value ranges and step values to parameter , , and . 2. We get the predicted numbers of exposed, infectious and dead individuals and these numbers can be fitted to real data by using the least square method to get the residual errors of each times loop iteration. 3. By comparison every residual error, we find the least one and we use the corresponding values of parameters in our prediction for further prediction.Result Via MATLAB programming, we obtain the optimal values for parameters,and(Table 1)and then get the estimated cumulative number of cases in Guinea, Liberia and Sierra Leone separately(Figure 2, 3 and 4). The result shows that if Ebola transmission trends continue without effective drugs and vaccines, countries will undergo worse and worse situations.Table 2 The optimal values for parameters , , , and CountryGuinea0.0650.0010.15870.020.02Liberia0.0340.0010.15870.020.00 Sierra Leone0.1010.0010.15870.030.02Figure 2 Cumulative numbers of cases in LiberiaFigure 3 Cumulative numbers of cases in GuineaFigure 4 Cumulative numbers of cases in Sierra LeoneStability test Definition of stability An aggregation of all possible parameters values resulting in a downwards trend of the total number of exposed individuals and infectious individuals are defined as the stability range in our model4.Stability rangeFirst, we draw two equations from the differential equation set (1.1):As and is relatively small, we assume that. Then, we sum the two equations up and get: In order to prevent the spread of Ebola, the total percentage of E(t) and I(t) has to present a decline trend from the first day of taking action with the licensed medicine, which also means .When I(t)=I(0),the inequality is equivalent to As, the relationship of parameters , , and are To conclude, the stability range for model one is. When parameters values satisfy this inequality, the model is of stability.Model 2: Allocation-and-delivery Model We create an allocation-and-delivery model for vaccines and drugs used in three most suffering countries (Guinea, Liberia and Sierra Leone) and the optimal strategy is assumed to have significant effect of eradicating Ebola in 180 days.In our allocation-and-delivery model, we set medical centers and sub-centers, which serve to treat Ebola patients, inject vaccines to susceptible individuals and also store needed amount of drugs and vaccines. Besides, countries manufacturing medicines (e.g., America, Canada, etc.) are not where in need of medicines, so we set one medical center to receive drugs and vaccines from the manufacturing country and then delivers the needed amount to every sub-center once a month. For sake of the inconvenience might face when delivering medicines across borders, we model three countries desperately. In another word, we set one medical center in Guinea, one in Liberia and one in Sierra Leone respectively and drugs and vaccines are delivered from every center to the sub-centers within borders. The Figure 5 below demonstrates the model with a hypothetical scenario. The dotted arrow lines show that individuals from every Ebola outbreak (E) will go to the nearest medical center (MC) or sub-center (MSC) for treatment or injection, while the solid arrow lines represent the delivery process of medicines from manufacturing county to each medical center and then to sub-centers. Figure 5 The allocation-and-delivery modelInstead of building new treating places, we locate our medical centers and sub-centers in some existing Ebola Treating Units (ETUs) 1. The model shows how we choose from current ETUs, including deciding the optimal number and location.Table 3 existing ETUs their locationTerminology and definitionsTable 4 Parameters and definitionsParameterDefinitionxiThe ith Medical sub-centeryiThe ith outbreaksDijThe distance between xi and xjdijThe distance between xi and yjDIiAdded infectious individuals in day iDDiAdded dead individuals in day iDEiAdded exposed individuals in day iAIiAccumulative infectious individuals in day iADiAccumulative dead individuals in day iAEiAccumulative exposed individuals in day iCNTotal delivering distance of N center and sub-centers AVNAverage workload of N center and sub-centersSVNWorkloads Standard Variance of N center and sub-centersCVN（CVN=SVNAVN）Workloads coefficient of variation of N center and sub-centersdsuccessful immune ratek1Percentage of injected individuals in total populationrrecovery rate when drugs are usedk2Percentage of drug used individuals in total infectious individualsGoalWe determine the number and location of medical center and sub-centers on the basis ofl Minimizing the total time-cost that an infectious individual from one outbreak spends on the way to the corresponding medical center or sub-center, while locating those center and sub-centers as few as possible, also meansl Minimizing the total distance among one medical center to other sub-centers, also meansl Averaging the workloads of medical center and sub-centers, also meansAlgorithmFigure 6 the flow chart for model 2Initialize parameters in previous prediction modell We do not change the value of and used in Model 1.l We have deduced the relationship of parameters , , and in the stability test of model 1.Estimate daily added number of infectious individualsWe use the prediction model to simulate the situation of daily added number of infectious individuals DIi in 6 months(180 days) for 10 times and choose the worst case(maximal numbers) as the final estimation of daily added number.Build geographical distribution of new added infectious individualsWe categorize all outbreaks into five levels as level I, II, III, IV and V according to the number of confirmed cases and then calculate each levels probability of a new occurring case. According to the number of new added infectious individuals and the probability of occurring in every outbreak, we build geographical distribution among all outbreaks of new added infectious individuals.Table 5 Outbreaks and classificationSet n from 1 to k We set n from 1 to k to conduct the process for k times and compare each optimal result as N changes.Locate sub-centers randomlyWe locate sub-centers randomly and for each sub-center, the corresponding outbreaks represent all those outbreaks with a nearer distance to this sub-center compared to others.Calculate total time-cost We define the time-cost as the period that an infectious individual from one outbreak spends on the way to the corresponding medical center or sub-center, and we add up the corresponding distance as the measurement of the time-cost. When calculating the total time-cost, the number of all potential patients is taken into account.Make comparisonWe compare the total time-cost calculated in 400 times loop and choose the minimal one as the optimal result.Output optimal n, Cn, AVn, CVnLocate medical center We calculate the total distance of every medical sub-center to others and locate the one with minimal total distance as the medical center which serve to receive all needed medicine from manufacturing country and deliver the required amount to every sub-center 5. Resultl We locate medical centers and sub-centers separately in three countries as shown in Table 7 and Figure7. We get the different values of indicators (shown in Table 6) and taking total distance and margin distance into account, we choose the optimal number and location of medical sub-centers Table 6 Values of indicatorsTable 7 Location of medical center and sub-centers and their corresponding outbreaksFigure 7 Locations of medical center and sub-centers and the routesl We determine the needed amount of vaccines and drugs.We assume that the successful immune rate is 90%, the recovery rate when drugs are used is 60% and the manufacturing cycle of the licensed drug is 30 days. These rates and cycle-days can be adjusted according to reality.VaccinesIndividuals having received vaccine injection can be protected from being infectious. The larger proportion of population being injected, the lower the transmission rate is. This relat