发布时间 : 星期一 文章数学建模 美赛 2015 A题更新完毕开始阅读80d6420fa58da0116d1749ca
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fatality rate for infected individuals while δ2 is for quarantined individuals); is the fraction of infected quarantined. Note that λ is a composite of all β transmission terms.
Table 2: Explanation for Parameter in Process
Process Number Parameter Explanation
1 infection rate of S ? 2 average latent period 1a 3 θ fraction of infected quarantined 4 funeral infection rate of Q ?2??DQ duration of traditional funeral 1?Q 6 δ1 fatality rate of I 7 δ1(1-θ)γD funeral infection rate of I 8 δ2 fatality rate of Q 9 βQ contact rate between S and Q 10 βF contact rate between S and F 11 βI contact rate between S and I
Along with the function (1), table 2 gives the explanations of parameters or combinations of them governing each process.
3.3.1 Ascertainment of the Parameters
One of the essential parts of the model is confirming parameters, which will affect the type and the trend of the epidemic process curve considerably. Theoretically speaking, using the provided data from the World Health Organization, parameters mentioned in formula (1) can be obtained by data fitting.
Implementing the morbidity data of Ebola cases into the least square method, we can reach to the estimated values of contact rates mentioned above.
The differences between each real value and the value from the differential equation at the same time is noted as ft, according to the least square method, our object function is:
5
min F(t)??ft2
(2)
In F(t), actually, there are several unknown numbers involving all contact rates. The solution is a set of parameters, which can produce the best fit curve of real data. We call the solution as estimated parameters.
We now get the best fit for real data, but the problem is that any combos of parameters may fit the same curve. Generally, the epidemic model yields a curve of infected individuals at the time that remains low and steady at the prophase, then soars to the summit in short period, and decreases continuously approaching to a small value. Based on the curve we get at present, we can adjust our estimated parameters by comparing with the general regulations of the epidemic curves.
If the later trend of the curve deviates from normal epidemic curves, then another fit is needed.
min G(t)??ft2?F(t1)
F(t1) is the best fit we got before.
(3)
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Therefore we acquire the second best fit parameters. In this way, the suitable set will be eventually reached after rounds of search. 3.3.2 Solution and Result
Taking Sierra Leone for study case, we use MATLAB to solve the problem. When deal with the calculation of parameters, the results are hard to reach perfection. Therefore, the assistance of relevant researches is generously adopted.
Referring to the research of Virginia Tech university[3], we ascertain the proper set as follows:
Table 3: Parameter Set
Parameter Value
α 0.1 βF 0.107743 βQ 0.076871 βI 0.121417 θ 0.265867 γI 0.05 γD 0.077336 γF 0.233802 γQ 0.242025 γDQ 0.113652 γIQ 0.063019 δ1 0.55 δ2 0.55
The values of parameters are mainly unchanged, while the values of γDQ and γDQ are calculated by
11?DQ??IQ?
1111and ???D?Q?I?Qrespectively.
Then we generate the epidemic simulation curve, which is purely based on natural situation without any intervention from human.
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Fig 2: Real case & Simulated case
The real data we acquire is until 01/02/2015, which should be fit by our model with proper parameters. In our model, I + Q should fits the real infected cases, and we can see an accordance between the real total cases and simulated I + Q cases in early time.
Later on, the simulated curve appears to climb up faster than the real one, which is probably because that intervention methods are taken afterwards so as to ease the epidemic aggravation to some extent.
Fig 3: Future epidemic simulation
The figure above is the simulation of the epidemic situation from 03/2014 to 08/2015, which period lasts long enough to allow us to observe the whole trend of the curve. There is an absence of the compartment S. It is because that the quantity of S is much larger than the other’s, the presence of S curve conceals the main features of other curves. And satisfactorily, the
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simulated situations of all other compartments fit the general regulations well. Thus the model is valid enough to yield reasonable results. 3.3.3 Analysis of the Result
Inspecting the results yields from the model, we find them matching our expectation. At the prophase, the curve simulated can fit the real one, and the later trend of it is in accordance of general epidemic curves. To be specific:
The curve for E: the peak value appears first, for it is the source of the transition to other compartments. After the relevant period, people exposed will gradually become infected.
The curve for I: following the transition sequence, I peaks right after E. There is a dip of cases from E to I, which is due to the other transition traces of I.
The curve for Q: as the next stage of transition, the peak value of Q approximately halves of that of I. The other half died for the fatality of quarantined is set as 0.55.
The curve for F: the value of F fluctuates with its source, so it has a similar trend and a lag from other curves. ? Stability analysis
A key parameters describing the natural spread of an infection is the basic reproduction number R0, which are defined as the number of secondary infections generated by an infected index case in the absence of control interventions.[6][7]
If R0 drops below unity, the epidemic eventually stops, otherwise the epidemic will maintain at certain district (developing as an endemia), which we do not wish. As the parameters are determined, R0 can also be calculated by:
???QR0?R0I?R0Q?R0F=where Δ is confirmed as:
?I???DQ??2??IQ(1??2)??Q??1?F ?F(4)
?=???Q+?D(1??)?1??I(1??)(1??1)
So we get the basic reproduction number of Sierra Leone is R0 = 1.89, which exceeds unity. The result infers that without any intervention, the spread of Ebola epidemic will not stop by itself.
3.4 Medication Distribution Optimization Model
In the epidemic model, as long as the parameters are determined, then the later spread situation of Ebola virus can be estimated. To control the aggravation of the epidemic, medication should be provided properly. This model solves how the materials should be distributed before delivery.
First, the immune population is calculated by the vaccinated population and the successful vaccination rate. Then we define the demand of medication, which should be offered through