Team#37026 page 8 of 9
the medical center in the Ebola serious area. The situation of airports in Sierra Leone is shown in the table below:
Table 8 1 2 3 4 5 6 7 8 9 10 Airport Bo Airport Bonthe Airport Daru Airport Gbangbatok Airport Hastings Airport Kabala Airport Kenema Airport Lungi International Airport Sierra Leone Airport Yengema Airport Location Bo Bonthe Daru Gbangbatok Freetown Kabala Kenema Freetown Sierra Leone Yengema Size Small Small Small Small Small Small Small Medium Small Small 4. Sensitivity Analysis and Improvements
4.1 Sensitivity Analysis
Model one we use SIR model to simulate the development of Ebola virus, and through two variables λ(daily infected rate) and μ(daily cured rate) to describe the situation of the disease. We can figure out the number of healthy people and infected people every day, there are many factors affecting the changes of healthy and infected number. For the convenience of calculation, we ignore the death of patients, and assume the total population is constant, and the daily cured rate is defined by ourselves, and the disparity between real value and the calculated one may be obvious. What’s more, we assume the medicine is very effective, all of the patients take the medicine will be cured, and we don’t consider the time of treatment, these problems could make our model have some drawbacks and less accuracy. 4.2 Improvements
For model one, according to analysis before, two measures can be taken to restrain the spread of the disease, one is improving health and medical level, in the other words is dropping the daily contact rate λ and improving daily cured rate μ; the other method is herd immunity, that is improving the original rate of the removed r0. We can see that in the SIR model, σ=λ/μ is a very important parameter. In reality, the value of λ and μ is difficult to estimate, but when an infectious disease is over,
Team#37026 page 9 of 9
we can get the value of s0 and s∞, and we neglect the parameter i0because of its small value, so we can have the result of σ:
??lns0?lns? (7)
s0?s? We can use the change of σ to analyze the development of the disease instead of λ and μ.
5. Model Evaluation
5.1 Strengths
● Model is simple and easy to understand, and we innovatively define the suspicious degree, making us to carry out a quantitative analysis of suspects.
●Processes the data and make a variety of charts, simple and intuitive shortcuts
●Model established in this paper and the actual closely, give full consideration to the different stages of the reality of the situation, so that the model is more realistic 5.2 Weaknesses
● In order to make the calculation is simple in the model, so that the results obtained are more ideal, ignoring the minor factors.
●For some data, we carried out a number of necessary treatment, which will bring some errors.
6. Reference:
[1]. Xuan Zhou, Junquan Song, Xuejun Wu. Introduction and Improvement of Mathematical Contest in Modeling[M]. Zhejiang:ZHEJIANG UNIVERSITY PRESS.2012.Page 201 to 205.
[2]. Qiyuan Jiang, Jinxing Xie, Jun Ye. Mathematical Model[M].Beijing:HIGHER EDUCATION PRESS.2011.Page 136 to 145.
[3]. Ebola Situation Report[DB/OL],http://apps.who.int/ebola/ [4]. Sierra Leone—Provinces and
districts[EB/OL].http://schools-wikipedia.org/wp/s/Sierra Leone.htm
[5].HongqingZhou,ZhuanXu. A Mathematical Model of Ebola Virus Infection Numbers [A].
[6].Themap of Sierra Leone[Z/OL].http://www.onegreen.net/maps/HTML/49248.html
Appendix:
BasicMatlabprogram： >>function y=ill(t,x) a=0.0339;b=0.2;
y=[a*x(1)*x(2)-b*x(1),-a*x(1)*x(2)]';
ts=0:100
x0=[0.00103981,0.99896019]; [t,x]=ode45('ill',ts,x0); [t,x] plot(t,x(:,1)),grid on; plot(t,x(:,2)),grid on; plot(x(:,1),(:,2)),grid on;
Data recording:
Detailed data of Table 1: time proportion proportion New in fections new infections Contact interval of health ofinfection at period of time per day number /% /% /λ 0 13 19 62 32 11 9 14 7 7 10 9 9 12 9 12 7 7 0.9999963 0.9999888 0.0000037 0.0000112 0 63 99 357 724 388 416 925 892 422 1477 1314 1219 1535 3929 528 1045 1056 0.0 4.8 5.2 5.8 22.6 35.3 46.2 66.1 127.4 60.3 147.7 146.0 135.4 127.9 436.6 44.0 149.3 150.9 — 0.0339 0.0215 0.0096 0.0171 0.0206 0.0217 0.0217 0.0323 0.0138 0.0253 0.0204 0.0162 0.0129 0.0323 0.0031 0.0099 0.0095 0.99999338 0.00000662 0.99997227 0.00002773 0.99993875 0.00006125 0.99992079 0.00007921 0.99990153 0.00009847 0.9998587 0.0001413 0.99981741 0.00018259 0.99979787 0.00020213 0.99972949 0.00027027 0.99966866 0.00033134 0.99961222 0.00038778 0.99954116 0.00045884 0.99937315 0.00062685 0.9993487 0.0006513 0.99930032 0.00069968 0.99926384 0.00073616
5 6 7 7 7 7 7 7 7 7 0.99920782 0.00079218 0.99917093 0.00082907 0.99914032 0.00085968 0.99914032 0.00085968 0.99906616 0.00093384 0.99904111 0.00095889 0.99901569 0.00098431 0.99899588 0.00100412 0.99897884 0.00102116 0.99896019 0.00103981 942 797 661 801 801 549 549 428 368 403 188.4 132.8 94.4 114.4 114.4 78.4 78.4 61.1 52.6 57.6 0.0110 0.0074 0.0051 0.0062 0.0057 0.0038 0.0037 0.0028 0.0024 0.0026
Detailed data of Table 3 and Figure 4： t/day i 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0.00103981 0.00102316642901013 0.00100678807840512 0.000990670759500238 0.000974810348560671 0.000959202785833016 0.000897812263961328 0.000840349967987889 0.000786564647943243 0.000736221110535207 0.000689099194662761 0.000613536145331262 0.000546258563639528 0.000486358030159459 0.000433025708102361 0.000385541430314833 s 0.99896019 λ 0.05 0.05 x 1122.9948 1105.019743 0.998925260274163 0.998890890858391 0.99885707278534 0.998823797230521 0.998791055510049 0.998759629105913 0.998730214953287 0.998702684180303 0.998676916171759 0.998652798039722 0.998630773112218 0.998611163739899 0.998593704971626 0.998578160923393 0.998564321588228 0.05 1087.331125 0.05 1069.92442 0.05 1052.795176 0.05 2071.878017 0.1 0.1 0.1 0.1 0.1 1939.27449 1815.155931 1698.97964 1590.237599 2232.681391 0.15 1987.857111 0.15 1769.877746 0.15 1575.800018 0.15 1403.003294 0.15 1665.538979
16 17 18 19 20 21 22 23 24 25 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 0.000326522803194833 0.000276524757700488 0.000234190073204331 0.000198354202974809 0.000167990890587151 0.000142267048513734 0.000120492636797839 0.000102054102458839 0.998552297149039 0.998542110696561 0.998533485653142 0.998526184714197 0.998519998763799 0.998514758053685 0.998510321985987 0.998506565549964 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 1410.57851 1194.586953 1011.701116 856.8901569 725.7206473 614.5936496 520.528191 440.8737226 373.3654233 316.2044903 267.8184648 226.8216289 192.0891055 162.6891731 137.7933896 116.6938775 98.82841797 83.70553732 70.8921291 60.03661375 50.84778302 43.06670871 36.47214531 30.88837283 26.16178262 8.64271813147248e-05 0.99850338192991 7.31954838669551e-05 6.1995014996132e-05 5.2505006689096e-05 4.44650707065251e-05 3.76595308026266e-05 3.18966179526099e-05 2.70124716405719e-05 2.28769486031412e-05 1.93762817874903e-05 1.64102150684292e-05 1.38973642942416e-05 1.17703201425963e-05 9.96914553387301e-06 8.44262622827315e-06 7.15008630285132e-06 6.05596819973069e-06 0.998500686290367 0.998498404471604 0.99849647112175 0.998494833191077 0.998493446739497 0.998492272697839 0.998491277682991 0.998490435181157 0.998489722014915 0.998489117759465 0.998488605834785 0.998488172507912 0.998487805568192 0.99848749458205 0.998487231262836 0.998487008366639