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After predicting the trends of Deaths and Cases, we predict changes in the amount of funding, according to the logic of the value of deaths, cases and funding. In this polynomial the number of Deaths and Cases is dependent variables and USD is independent variables.
By applying SPSS regression analysis tools, predictive values about USD contributed and relative error is given.
The data in Table 10 is transformed into Figure 9.
Figure 9. Forecast chart about USD contributed.
As shown in Figure 9, from left to right, the first forecasts on USD based on Deaths and Cases about Guinea; the second forecasts on USD based on Deaths and Cases about Sierra Leone; the last forecasts on USD based on Deaths and Cases about Liberia.
According to Figure 9, predictive value of USD was significantly lower than the actual value at the beginning of the epidemic outbreak, which indicates that the model underestimated the epidemic in Africa.
Interestingly, the model predictions are always lower than the actual value of the rescue funds in Guinea epidemic, which indicates the epidemic outbreak in Guinea is a sudden-onset disasters. And that’s what our polynomial model cannot follow up in real time.
In Sierra Leone region, predictive value of USD was lower than the actual value before December 29, 2014. With the development of the epidemic, predictable funding has getting close to the actual requirements, which reflects the rule of epidemic situation. And that’s the perfect cases fitting our control model.
In Liberia region, forecast is higher than actual funds before December 29, 2014. The predict trend curve of model are agreed with the data in practice, showing the fragility of local economy.
5.1.4 Strengths and Weaknesses
● Strengths: The model is simple and efficient, but yet precise. Although the model is based on
USD, it can also help predict the needs of a variety of resources.
According to USD contributed, we analyze several of important turning points in the curve, and predict the trend curve of the situation in February and March 2015. And the results are closer to the actual data.
● Weaknesses: Some certain errors are existing due to our model only define the number of Deaths and Cases. As the saying goes, you can't have your cake and eat it too. Our objective is to get to zero cases, and sacrifice bit of accuracy for efficiency and simplicity.
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5.2 Allocation Strategy
5.2.1 Allocation Formula
Under conditions of limited medical resources provided by international relief organizations, we need to be the most efficient, the most equitable distribution of resources.
Considering Epidemic Level (EL) of the model one, we give a formula which shows
Si?ELi?Stotal??specialj?,
(5.2.1)
where: i represents the country category; ELi represents the epidemic level value of the country; Stotal represents the total amount of the resource; specialj represents the special resource requirements in other countries.
Table 11. Simulation/Actual resources’ distribution in Dec 31, 2014.
Si Guinea Liberia ELi 0.2163 0.3921 Personal Protective 669.4485/481 1213.5495/2200 Equipment (PPE)(tons) Medical 24.8745 45.0915 supplies (tons) requirements 213.66114/194 387.31638/473 (million) burial kit 4321.674/2520 7834.158/13560 phone 1094.478/840 1984.026/2920 ambulance 9.7335/6 17.6445/26 motorbike 275.9988/212 500.3196/634 car 69.8649/52 126.6483/128 car4*4 68.3508/51 123.9036/179 Note: the values of Stotal come from WHO.
Sierra Leone 0.2156 667.282/618 Mali 0.1187 367.3765 Nigeria 0.1187 367.3765 Stotal 1.0 3095 24.794 212.96968/220 4307.688/3900 1090.936/1300 9.702/13 275.1056/330 69.6388/88 68.1296/87 13.6505 117.25186 2371.626 600.622 5.3415 151.4612 38.3401 37.5092 13.6505 117.25186 2371.626 600.622 5.3415 151.4612 38.3401 37.5092 115 987.8 19980 5060 45 1276 323 316 Data in Table 11 are \resources’ distribution\of medical resources. Although limited to official statistics, we do our best efforts to gather reliable, available data and there must be other statistics of indicators out of our consideration and search scopes.
5.2.2 Analysis of Result
As shown in Table 11, our allocation scheme match up the realistic situations in most cases. Efficiency of resource distribution has been highly improved due to the efficient and simple program. Besides, unfair treatment of affected countries by related resources management departments can be avoided as well. Therefore, the model needs to take more specific factors into consideration, which is the blind spot of our model. And the imperfection of our model is the existence of some error (marked with red) of the allocation scheme.
The innovation of our model is the use of EL we defined.
Informed by Model one, EL is an important criterion for a comprehensive evaluation of infected areas, containing many factors of epidemic situations. Admittedly, material distribution program cannot be completely determined on single indicator. To realize a more equitable distribution of medical resources, we consider the special needs of resources in affected areas, such as inconvenient transportation of some affected areas, more need of ambulances, sudden vicious weather and emergency supplies are reflected in the formula.
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5.2.3 Strengths and Weaknesses
● Quickly allocate medical resources by EL, which is efficiency and justice.
● Evaluating the EL calls for large amounts of data by AHP, which appears cumbersome.
6 Model Four: Control the Epidemic Spread
6.1 Introduction
The mentioned above is that how to evaluate the severity of Ebola in each nations, then describe how to construct a more efficient delivery system and how to calculate quality of needed supplies. This part proposes to improve SIR model to be linked with the EL for implementing to control Ebola.
This paper put forward an optimization of SIR model after above work. Through adding the EL to the SIR model, it is easier for government or other relevant departments to make right decision. The result of our research represent that lower El can decrease the severity of spread of Ebola.
6.2 Improved SIR Model
Ebola virus disease is a disease of humans and other primates caused by Ebola viruses. Some treatment or vaccine for the virus seem available, and a number of potential treatments are being studied. People cured wouldn't infect others. So we can describe it using SIR model. S(t), I(t), R(t) respectively represent the number of susceptible, infected, recovered individuals. So, if N is the total population (7,450,000 in our example), we have
dS???SIdt, dI??SI??Idt,
dR??Idt, S?I?R?N.
(6.2.1)
(6.2.2)
(6.2.3) (6.2.4)
We design the optimization of SIR model through introducing the EL. The method is as follow
formulas.
??PELi, 1?(6.2.5) (6.2.6)
???1?ELi?P2.
Parameters?, ?, P1, P2 all is positive real number, ELi is the Epidemic Level. ?is infection
rate；? is the cure rate.
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Table 12.
Lower EL before control The Parameters of optimized SIR model
P1 1.275 P2 0.548 ELh 0.392 ?h 0.500 ?h 0.333 Higher EL after control P1 1.275 P2 0.548 ELl 0.373 ?l 0.475 ?l 0.344 We don’t know values for the parameters ? and ? yet, but we can estimate them, and then adjust them to fit the existed death and case data. We apply this optimized SIR model in Guinea, which the ? is suggested 0.500. If we guess that each infected would make a possibly infecting contact every two days, then ? would be 0.333. Then we assume that governments’ control make EL lower. We obtain the new data and SIR curves.
6.3 Analysis of Result
In our optimized model, the parameters are affected by the national EL. Therefore, the control for Ebola can be quantified to the EL. As Figure shows, that left figure is under higher initial value EL, and the right figure is under the situation that after the decrease of EL. Figure 10 shows the solution curves for these choices of ? and ?.
Obviously, the drop of EL make SIR model curve gentle, which is in favor of the control of epidemic. We also analysis the other change between results of different situation.
● The Maximum of I(t)l is lower than I(t)h, which illustrate the amount of illness people decrease.
● The slope of I(t)l is lower than I(t)h, which illustrate the spread of Ebola has been controlled. ● The Min of S(t)l is lower than S(t)h, which illustrate the main crowd get less affected.
Figure 10. Higher EL & Lower EL.
6.4 Sensitivity to Parameters
In order to measure the stability of the optimized SIR model, we select several sets of data to test our model. The data are shown in Table 13. Corresponding SIR result is shown in Figure 11. The curves in different figures are corresponded with the fundamental feature for SIR model. It proved that our optimized model has observability and practical significance.
Table 13. The Parameters of optimized SIR model
1 2 P1 P1 1.275 1.275 P2 P2 0.548 0.548 ELh ELl 0.400 0.330 ?h ?l 0.510 0.421 ??h l 0.329 0.367