The model of eradicating Ebola
In 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 Drug
Non-technical Letter
According 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
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1. Introduction
1.1Background
West 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 Model
Ebola 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 and death 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 is called 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 constant effectiveμ, 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 Constitution
According 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 isλN s(t)i(t), and λNsi is the increasing rate of patients, hence we have:
diN??Nsi??Ni (2) dt As to the recovered and immune people, we have
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dr??Ni (3) dt We assume that at the beginning, proportion of the healthy and patients are s0(s0>0) and i0(i0>0) (we might as well define r0=0), and according to equations (1), (2) and (3), we get the equation of SIR model:
N?di??si??i，i?0??i0??dt (4) ?ds????si,s?0??s0??dt The analytical solution of equation (4) can’t be obtained, so we first make
numerical calculation. 2.3 Numeric Calculation
In 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 1
time interval 0 13 19 62 32 11 9 7 10 9 12 9 12 7 5 6 7 7 7 7
proportion of health/% 0.9999963 0.99999338 0.9999888 0.99997227 0.99993875 0.99992079 0.99990153 0.99981741 0.99972949 0.99966866 0.99954116 0.99937315 0.9993487 0.99926384 0.99920782 0.99917093 0.99904111 0.99899588 0.99897884 0.99896019
proportion of infection/% 0.0000037 0.00000662 0.0000112 0.00002773 0.00006125 0.00007921 0.00009847 0.00018259 0.00027027 0.00033134 0.00045884 0.00062685 0.0006513 0.00073616 0.00079218 0.00082907 0.00095889 0.00100412 0.00102116 0.00103981
new infection per day
0.0 4.8 5.2 5.8 22.6 35.3 46.2 127.4 147.7 146.0 127.9 436.6 44.0 150.9 188.4 132.8 78.4 61.1 52.6 57.6
Contact number/λ — 0.0339 0.0215 0.0096 0.0171 0.0206 0.0217 0.0323 0.0253 0.0204 0.0129 0.0323 0.0031 0.0095 0.0110 0.0074 0.0038 0.0028 0.0024 0.0026