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According to the actual data, we can getλ at 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 λ is 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 2
Combined with the actual situation, we assume that at different time, the value ofλ is changing. We make every five days as an interval, and redefine the value of λ every 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 of
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medicine 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 3
t 0 1 2 3 4 5 6
0.0010398 0.0010232 0.0010068 0.0009907 0.0009748 0.0009592 0.0008978 i(t)
0.9989602 0.9989253 0.9988909 0.9988571 0.9988238 0.9987911 0.9987596 s(t)
t 7 8 9 10 11 12 13
0.0008403 0.0007866 0.0007362 0.0006891 0.0006135 0.0005463 0.0004864 i(t)
0.9987302 0.9987027 0.9986769 0.9986528 0.9986308 0.9986112 0.9985937 s(t)
t 14 15 16 17 18 19 20
0.0004330 0.0003855 0.0003265 0.0002765 0.0002342 0.0001984 0.0001680 i(t)
0.9985782 0.9985643 0.9985523 0.9985421 0.9985335 0.9985262 0.9985200 s(t)
Figure 4
s(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 rateλ is 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.
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2.4 Analysis of Phase Trajectory Figure
In 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:
D???s,i?s?0,i?0,s?i?1?
We erasure dt in the equation (4), and noticing thatσ=λ/μ, we can get
di1??1,is?s0?i0 (5) ds?s We can easily figure out that the answer of equation (5) is:
i??s0?i0??s?1?lns (6) s0In 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 the equation (6), so s∞is the answer of equation
s0?i0?s??1?lns??0 s0● If s0<1/σ, i(t) is monotone decrease and finally drops to 0, s(t) is monotone decrease to its minimum s∞ . 2.5 Conclusion
According 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 have
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already 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 Model
3.1 The demand of medicine
The 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)*μ
The result is shown in the Table 5 below.
Figure 5
The 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 near
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zero, 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 System
3.2.1Drug distribution proportion
Good 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 6
district Northern Province Eastern Province Southern Province Western Area
rate of infections
37.57% 16.30% 6.89% 39.24%
rate of population density rate of medicine
13.27% 20.91% 14.82% 51.01%
25.42% 18.61% 10.85% 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 7
3.2.2Methods of the Medicine Delivery
Because 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 to