1、 For office use only T1 _ T2 _ T3 _ T4 _ Team Control Number 32150 Problem Chosen AFor office use only F1 _ F2 _ F3 _ F4 _ 2015 Mathematical Contest in Modeling (MCM/ICM) Summary Sheet How to Eradicate Ebola? The breakout of Ebola in 2014 triggered global panic. How to control and eradicate Ebola ha
2、s become a universal concern ever since. Firstly, we build up an epidemic model SEIHCR (CT) which takes the special features of Ebola into consideration. These are treatment from hospital, infectious corpses and intensified contact tracing. This model is developed from the traditional SEIR model. Th
3、e models results (Fig.4,5,6), whose parameters are decided using computer simulation, match perfectly with the data reported by WHO, suggesting the validity of our improved model. Secondly, pharmaceutical intervention is studied thoroughly. The total quantity of the medicine needed is based on the c
4、umulative number of individuals CUM (Fig.7). Results calculated from the WHO statistics and from the SEIHCR (CT) model show only minor discrepancy, further indicating the feasibility of our model. In designing the delivery system, we apply the weighted Fuzzy -c Means Clustering Algorithm and select
5、6 locations (Fig.10, Table.2) that should serve as the delivery centers for other cities. We optimize the delivery locations by each citys location and needed medicine. The percentage each location shares is also figured out to facilitate future allocation (Table.3,4). The average speed of manufactu
6、ring should be no less than 106.2 unit dose per day and an increase in the manufacturing speed and the efficacy of medicine will reinforce the intervention effect. Thirdly, other critical factors besides those discussed early in the model, safer treatment of corpses, and earlier identification/isola
7、tion also prove to be relevant. By varying the value of parameters, we can project the future CUM . Results (Fig.12,13) show that these interventions will help reduce CUM to a lower plateau at a faster speed. We then analyze the factors for controlling and the time of eradication of Ebola. For examp
8、le, when the rate of the infectious being isolated is 33% - 40%, the disease can be successfully controlled (Table.5). When the introduction time for treatment decreases from 210 to 145 days, the eradication of Ebola arrives over 200 days earlier. Finally, we select three parameters: the transmissio
9、n rate, the incubation period and the fatality rate for sensitivity analysis. Key words: Ebola, epidemic model, cumulative cases, Clustering Algorithm Contents 1. Introduction 1 1.1. Problem Background . 1 1.2. Previous Research 2 1.3. Our Work 2 2. General Assumptions . 3 3. Notations and Symbol De
10、scription 4 3.1. Notations 4 3.2. Symbol Description . 4 4. Spread of Ebola 5 4.1. Traditional Epidemic Model 5 4.1.1. The SEIR Model 5 4.1.2. Outbreak Data 6 4.1.3. Results of the SEIR Model 7 4.2. Improved Model. 8 4.2.1. The SEIHCR (CT) Model . 8 4.2.2. Choosing Parameters . 10 2.1.1. Results of
11、the SEIHCR (CT) Model 11 5. Pharmaceutical Intervention 13 5.1. Total Quantity of the Medicine 13 5.1.1. Results from WHO Statistics. 13 5.1.2. Results from the SEIHCR (CT) Model . 15 5.2. Delivery System . 15 5.2.1. Locations of Delivery 16 5.2.2. Amount of Delivery . 19 5.3. Speed of Manufacturing
12、 . 20 5.4. Medicine Efficacy 21 6. Other Important Interventions 21 6.1. Safer Treatment of Corpses 21 6.2. Intensified Contact Tracing and Earlier Isolation 22 6.3. Conclusion . 24 7. Control and Eradication of Ebola 24 7.1. How Ebola Can Be Controlled 24 7.2. When Ebola Will Be Eradicated 26 8. Se
13、nsitivity Analysis 27 8.1. Impact of Transmission Rate I 27 8.2. Impact of the Incubation Period 1/ 28 8.3. Fluctuation of H . 28 9. Strengths and Weaknesses 29 9.1. Strengths 29 9.2. Weaknesses 30 9.3. Future Work . 30 Letter to the World Medical Association . 31 References 33 Team #32150 page 1 of
14、 34 1. Introduction 1.1. Problem Background Ebola virus disease (EVD), formerly known as Ebola haemorrhagic fever, is a severe, often fatal illness in humans. The current outbreak in West Africa, (first cases notified in March 2014), is the largest and most complex Ebola outbreak since the Ebola vir
15、us was first discovered in 1976. There have been more cases and deaths in this outbreak than all others combined. It started in Guinea and later spread across land borders to Sierra Leone and Liberia1. The current situation in the most affected countries can be seen clearly by the latest outbreak si
16、tuation graph released by the World Health Organization (WHO). Figure 1. Ebola Outbreak Distribution Map in West Africa on 4th Feb, 2015 http:/www.cdc.gov/vhf/ebola/outbreaks/2014-west-africa/distribution-map.html Ebola was first transmitted from fruit bats to the human population. It can now spread
17、 from human to human via direct contact with the blood, secretions, organs or other bodily fluids of infected people, and with surfaces and materials contaminated with these fluids. Burial ceremonies can also play a role in the transmission of Ebola because the virus can also be transmitted through
18、the body of the deceased person1. Control of outbreaks requires coordinated medical services, alongside a certain level of community engagement. The medical services include rapid detection of cases of disease, contact tracing of those who have come into contact with infected individuals, quick acce
19、ss to laboratory services, proper healthcare for those who are Team #32150 page 2 of 34 infected, and proper disposal of the dead through cremation or burial1,2. There are lots of different experimental vaccines and drug treatments for Ebola under development, tested both in the lab and in animal po
20、pulations, but they have not yet been fully tested for safety or effectiveness3, 4. In the summer of 2014, the World Health Organization claimed fast-tracking testing was ethical in light of the epidemic4. In fact, the first batch of an experimental vaccine against Ebola have already been sent to Li
21、beria in January 2015. According to the Dr Moncef Slaoui of British production company GlaxoSmithKline, the initial phase is encouraging and encourages them to progress to the next phases of clinical testing5. 1.2. Previous Research The analysis of the spread of epidemic has been a universal concern
22、. Moreover, there have been a lot of research into the studies of the 2014 Ebola epidemic in West Africa6. For the spread of the disease, which is considered as an important factor in eradicating Ebola, there are a lot of previous research that can facilitate our understanding of the disease. For ex
23、ample, Fisman et al. used a two-parameter mathematical model to describe the epidemic growth and control7. Gomes et al. researched into the risk of Ebolas international spread along with its mobility data. His research reached a conclusion that the risk of Ebolas international spread to outside Afri
24、ca is relatively small compared to its expansion among the West African countries8. The Centers for Disease Control and Prevention used the traditional SIR model to extrapolate the Ebola epidemic and projected that Liberia and Sierra Leone will have had 1.4 million Ebola cases by Jan. 20, 20159. Cho
25、well et al. used the SEIR model and studied the effect of Ebola outbreaks in 1995 in Congo and in 2000 in Uganda10. The SEIR model takes the state of exposure into consideration, which is a special feature of Ebola because exposure to the virus will make individuals a lot more easily to be infected.
26、 Based on the SEIR model of Chowell et al10, Althaus developed a model where the reproduction number is dependent on the time11. Valdez et al. developed a model and found that reducing population mobility had little effect on geographical containment of the disease and a rapid and early intervention
27、 that increases the hospitalization and reduces the disease transmission in hospitals and at funerals is the most important response to any possible re-emerging Ebola epidemic12. 1.3. Our Work We are asked to build a realistic, sensible, and useful model to optimize the eradication of Ebola or at le
28、ast its current strain. Our model should not only consider Team #32150 page 3 of 34 the spread of the disease, the quantity of the medicine needed, possible feasible delivery systems, locations of delivery, speed of manufacturing of the vaccine or drug, but also other critical factors that we consid
29、er to be necessary. To begin with, we searched a large number of papers that discuss the spread of Ebola to help us deepen the understanding of the problem. Chowell et al. provided a large amount of background information and their work6 served as an important introduction. We found that a few of th
30、e papers used the traditional epidemic model to predict the transmission of the disease such as the SEIR model used by Althaus11 to estimate the reproduction number of the virus during the 2014 outbreak. Therefore, we also applied the SEIR model in the early stage to predict the spread of Ebola. Lat
31、er, we found out that the Ebola virus has some specific feathers that also needed to be considered and that are, the potential transmission threat posed by the highly infectious corpses, the improved infection control and reduced transmission rate if patients can be treated in hospitals, and the pow
32、erful intervention method: contact tracing. After taking all these critical factors into consideration, we improved our original epidemic model. Next, we deeply analyzed the pharmaceutical intervention. We first used the statistics provided by WHO24 to help figure out the quantity of the needed medi
33、cine. We also used our improved model to predict the number of the patients and needed medicine in Guinea, Sierra Leone and Liberia. Two parameters: 1) the current cumulative number of patients pCUM and 2) the increasing rate of the disease to decide the quantity of the medicine are set up as criter
34、ia. After the quantity of the medicine had been calculated, we sorted the affected cities of the three countries into several groups to determine the location of delivery. We applied the Fuzzy -c Means Clustering Algorithm and eventually selected 6 delivery points. These delivery points will be in c
35、harge of storing the medicine and transporting them to the set of cities surrounding them. Later, we took the speed of manufacturing and the efficacy of medicine into consideration and tested how these parameters could affect intervention. We then considered other important factors that would help t
36、he eradication of Ebola: 1) safer treatment of corpses and 2) intensified contact tracing and earlier isolation. After finishing analyzing all the relevant factors, we reached our final conclusion and predicted a time for Ebolas eradication. In the last stage, we provide a non-technical letter for t
37、he world medical association to use in their announcement of inventing a new drug to stop Ebola and treat patients whose disease is not advanced. 2. General Assumptions The population of the new-born are not counted in the total population. An individual who is exposed enters the incubation period a
38、nd is not yet Team #32150 page 4 of 34 infectious. Recovered patients will not be infected again. Medicine are only provided in hospitals. Every location inside Guinea, Sierra Leone and Liberia can be selected as the delivery center and delivery routes between cities are straight lines. Medicine and
39、 vaccines can be delivered across borders. 3. Notations and Symbol Description 3.1. Notations Susceptible individual12: A person with a clinical illness compatible with EVD and within 21 days before onset of illness, either: a history of travel to the affected areas OR direct contact with a probable
40、 or confirmed case OR exposure to EVD-infected blood or other body fluids or tissues OR direct handling of bats, rodents or primates, from Ebola-affected countries OR preparation or consumption of bush meat from Ebola-affected countries. Exposed individual14: A person who has been infected by Ebola
41、virus but are not yet infectious or symptomatic. Basic reproduction number6: The average number of secondary cases caused by a typical infected individual through its entire course of infection in a completely susceptible population and in the absence of control interventions. Contact tracing15: Fin
42、d everyone who comes in direct contact with a sick Ebola patient. Contacts are watched for illness for 21 days from the last day they came in contact with the Ebola patient. If the contact develops a fever or other Ebola symptoms, they are immediately isolated, tested, provided care, and the cycle s
43、tarts again-all of the new patients contacts are found and watched for 21 days. 3.2. Symbol Description Symbol Description 0R Basic reproduction number N Size of total population ()St Number of suspected individuals at time t ()Et Number of exposed individuals at time t ()It Number of infectious ind
44、ividuals outside hospital at time t ()Ht Number of hospitalized individuals at time t ()Ct Number of contaminated deceased at time t Team #32150 page 5 of 34 ()Rt Number of removed individuals at time t ()CUMt Cumulative number of individuals at time t ,I H C Transmission rate (outside hospital, ins
45、ide hospital, by corpses) 1/ Average duration of incubation Rate of infectious individuals to be identified/isolated ,IH Average time from symptoms onset to recovery (outside hospital, inside hospital) ,IH Average time from symptoms onset to death (outside hospital, inside hospital) ,IH Fatality rat
46、e (outside hospital, inside hospital) 1/ Average time until the deceased is properly handled Average number of contacts traced per identified/isolated infectious individual ,EI Probability a contact traced individual (exposed, infectious) is isolated without causing a new case ,EI Ratio of probabili
47、ty that contact traced individual is (exposed, infectious) at time of originating case identification to the probability a random individual in the population is (exposed, infectious) 4. Spread of Ebola 4.1. Traditional Epidemic Model 4.1.1. The SEIR Model The transmission of EBOV follows SEIR (susc
48、eptible-exposed-infectious-recovered) dynamics and can be described by the following set of ordinary differential equations10: ( ) ( ) ( ) / ,( ) ( ) ( ) / ( ) ,( ) ( ) ( ) ,( ) ( ) ,( ) ( ) ,S t S t I t NE t S t I t N E tI t E t I tR t I tCU M t E t (4.1) Team #32150 page 6 of 34 where: ()St is the
49、 number of susceptible individuals at time t, ()Et is the number of exposed individuals at time t, ()It is the number of infectious individuals at time t, ()Rt is the number of removed individuals at time t, ()CUMt is the cumulative number of Ebola cases from symptoms onset, N is the size of total population, 1/
