1、For office use onlyT1 _T2 _T3 _T4 _ Team Control Number41961Problem ChosenA For office use onlyF1 _F2 _F3 _F4 _2015 Mathematical Contest in Modeling (MCM) Summary SheetAbstract:WiththecontinuousspreadofEbola,currently,theworldisfacingan“unprecedented“publichealthcrisis.Byestablishingthemodels,westud
2、iedthetransmissionofthedisease,drugsindemand,drugsmanufacturing,transportationlocation,transportationsystemandotherkeyfactors,achievingthecorrespondedresults,whichisbeneficialtothecontrollingoftheepidemic.Initially, for the transmission of the disease, we do simulation and the stabilityanalysisofGui
3、neatotheepidemicofthediseasebyestablishingtheSIRmodel.Andthe results is kindly fitted with the actual status. Then we do prediction on thedevelopment of the epidemic;As for the quantity of the drugs demand, we get thedemand of the emergency medical resources by analyzing the changes of states ofinfe
4、cted patients, we establish the requirement analysis model, introducing theconcept of the isolation. Through the requirement analysis model above, takingGuinea as example, we firstly can get the minimum of the demand of the new drugwith tas the unit of the time.Then we calculate the sum of the new d
5、rug in Guineafrom the beginning of the vaccines production to the epidemic is controlled.Eventually, we quantitatively calculate the sum of drugs demand by using therelationship between total number of infected and the number of the drugs demandfromtheoutbreakstotheepidemicidcontrolled.Forthespeedof
6、manufacturingthedrugs and the vaccine, we utilize its relationship with the drug demand,with theanalysisofthe fact, todocorrect and therestrain ontheproducing speedcalculated.For the selection of the transport location and the transportation system, we selecttransportbyairandbyseaasthetransportation
7、tobechosen.Wemakethehardesthitof Ebola including Guinea, Sierra Leone, and Liberia as schematic layer, and thefactors including transportation costs, transportation system, the situation of theepidemicandthetransportationdistanceasrulehierarchy.Webuildanalytichierarchymodel, gaining Guinea as prefer
8、red transport location, transport by air as the maintransportation.At last, we consider governments support, coercive measures and other keyfactors which can release the current stress, raising proposals and writing anon-technical letter to appeal all over the world get together to against the Ebola
9、epidemic.Key words: SIR model Stability analysis Requirement analysismodel Isolation Analytic hierarchy modelTeam#41961 Page 1 of 26Problem Restatement: Theworldmedicalassociationhasannouncedthattheirnew medication could stop Ebola and cure patients whose disease is not advanced.Buildarealistic,sens
10、ible,anduseful modelthat considersnotonlythespreadofthedisease, the quantity of the medicine needed, possible feasible delivery systems(sending the medicine to where it is needed), (geographical) locations of delivery,speedofmanufacturingofthevaccineordrug,butalsoanyothercriticalfactorsyourteamconsi
11、dersnecessaryaspartofthemodeltooptimizetheeradicationofEbola,orat least its current strain. In addition to your modeling approach for the contest,prepare a 1-2 page non-technical letter for the world medical association to use intheirannouncement.1.The Spread of the EbolaAsearlyas1996,therehasbeenma
12、thematicalmodelforthestudyofEbolavirusindocument2whichisontheuseofSIRandSEIRmodel,simulatingtwoperiodsofZaireEbolaoutbreak:976YambukuoutbreakandKikwitepidemicoutbreakof1995They concluded that when fundamental regeneration rate meet 1.72 0R 8.60,itmeans the infectivity of Ebola is not as serious as b
13、efore and it can reduce thepotentialdeaths.Inrecentyears,therearesomedocuments(see3-6)doingresearchinEbolavirus.Basedonthesedocuments,weestablishthemathematicalmodeloftheinfectedquantityoftheEbolavirus(1)Model Assumptions Regard the object of study as ideal population and remain the total number ofp
14、eopleinafixedlevelN.Thereisnomortalitycausedbymovingto,movingoutothercauses.AssumebothofthepeoplewhoisinjectedbyEbolavaccineandthepeoplewhoiscuredaftersufferingfrominfectiousdiseases,havealong-termimmunity.Atthesametime,assumetheincubationperiodofinfectiousdiseasesissoshortthatcould be negligible, t
15、hat is any person will become contagious at the moment ofgettingsickbecauseofEbola.Accordingtothecurrentstatusofpeopleshealth,dividethepeopleintofiveparts:susceptible people S,that ispeople inhealth; infect people I, namely who isinfectedbyEbolavirus;leaversRincludingimmunepeopleaR,curedpeopleaRandt
16、hedeath.Andarepresentsimmunizationrate whichistheproportionofthevaccinatedpeopleintotalpopulation.Basically,underthegovernmentscoercivemeasures,thepopulationdoesnotflow,so no pathogens flows in and out, avoiding cross-infection, reducing the cardinalnumberofinfection. Cut off contact between people
17、in isolation and the outside world, to ensure thattheyarenotcontagious.Team#41961 Page 2 of 26（2）Establish the Mathematical ModelDefinitionsofparametersUnderthiscircumstance,let (t)s ,i(t),r(t)betheinhabitantsofsusceptiblepeopleS,infectpeopleIandleaversRandkeepthesumofthesethreeinaconstant N,whichme
18、ans (t) i(t) r(t) Ns （1.1）.Inthisformulaistherateofdailycontactwiththepatients;istherateofdailycureandthe meansthecontactednumbersininfectiousperiod.ModelconstitutionAccordingtotheepidemicmodel: 0,0, (0) s, (0) i (1.2) ids si sdtdi sdr idt i idt Wecangetthefollowingresultbysolvingthe(1.1)and(1.2):0(
19、t) s rs e （1.3）AftertakingthefirstthreeitemsoftheTaylorexpansionof re ,andtakingoutoftherateofthechangeinpopulation,itisapproximatelyequalto:20 1(1 ( ) )2dr r rN r sdt (1.4)Undertheinitialvalue 0 0r ,Thesolutioniscumulativelyremovednumber:2 00 1(t) 1 tanh( ) ,2sr ts Amongthem, 1220 0 0 022( 1) , tan
20、hs s i s Thus,the(1.4)canbetransformedinto:Team#41961 Page 3 of 262 2 20 1 ,2 ( )2dr tdt s ch (1.5)Wecanconclude 2 1 1( )2tch from 2( ) 12tch According to(1.5) , we know if and only if 02t , that is when 2t ,drdt cangetthemaximumwhichmeanstomoveoutthebiggestnumberofpeople.Inthefollowingpart,letsanal
21、ysisthechangesofs(t),i(t)andr(t).Thereisnorelationbetweenther(t)andthefirsttwoequations,sowecansolvetherelationbetweeni(t)ands(t)fromthefirsttwoequations.Thatis,firstlyletstakeconsiderationof 0 0, 0 (1.6), 0 ds si s sdtdi si i i idt Aftereliminatingdt,wecanhaveafirst-orderequation:0 01 , s sdi i ids
22、 s 丨Bysolvingthisequation,wecanhave:0 0 0(s) +s ln (1.7) si i s sMark ,lim lim ,lim ,t t ts t s i t i r t r wegettheconclusions:a.Nomatterhowtheinitialconditions 0s ， 0i are, 0i ,whichmeanspatientswilleventuallybecomecuredordeath.b.The percentage of the people who is eventually non-infected is s . I
23、n (1.7), letsformulate (s) 0i ,thesolution sistherootoftheequation 0 0 0ln 0si s s s ,intheinterval (0, )c. If 0s , then i(t) increases, when s , i(t)can reach its maximum:Team#41961 Page 4 of 2600 0 1 lnm si s i , theni(t) decreases until tends to 0 and (t)s decreasesmonotonicallytill s.Thatistosay
24、,ifonlytheproportionofinfectedindividualsi(t)has a period of growth can be considered in the spread of infectious diseases, then isathreshold， when 0s , infectiousdiseasewillspread.d.If 0s ,theni(t)decreases monotonicallytill0，(t)s decreases monotonicallytill s . Reduce the number of contacts in inf
25、ectious period , which results ininfectious disease will not spread. We can find that with the increase of peoplesawarenessandthehigherlevelinhealth,wewillhavethelowerdailycontactrate,thehigher the medical level and the greater daily cure rate. Therefore, improving thehealth and medical standards is
26、 an effective way to control the spread of infectiousdiseases.analyzethestabilityandthesensitivitythroughexamplesThrough long-term study, it is found that the Ebola virus is mainly spreadthrough the patients blood, saliva, sweat and secretions and other approaches.Routine inspection found the severe
27、 reduction of platelet , decrease of normallymphocytes , some increase of serum amylase , and sometimes the virus can beobservedbyelectronmicroscopyintheliverslices.TheincubationperiodofEbolais2-21 days of infection. Infected person initially showed a sudden high fever, sorethroat, muscle pain, head
28、ache, and general weakness. Then the symptom turns intoabdominal pain, vomiting, diarrhea. Within two weeks after the onset, the virus willspill, resultinginbleeding,bloodclottingbothinsideandoutsidethebody.Necrosisand systemic blood quickly pass various organs, and ultimately patient nasal, oral,an
29、albleedingandothersymptoms,severepatientscandiewithin24hours,however,mortified blood will spread to various body organs rapidly.And finally it will occurbleeding and other symptoms in patientsnasal, oral and anal , some severe patientseven can die within 24 hours. The most effective way to prevent t
30、he Ebola isimplementationoftheisolationwhichaimsatpreventingcontactwithillpeople.Thetransmissionspeed,strengthandbreadthofthepopulationofEbolavirusdependsonthe number of infected and susceptible persons and also the effective contactbetweenthem.Theso-calledeffectivecontact meansthatthepathogenfrom i
31、nfectedbodytothesusceptiblebody,resultingininfectioneventheonset.Itissubjecttothedegree of contact, the type of pathogen, the discharged condition of pathogen,resistance of susceptible people and other factors. After the establishment ofepidemiological mathematical model, the solution obtained from
32、the model must beput in a fitting inspection with the actual data. If agreement can be reached, we caninitiallyconsideredthemodelasareasonableone,anduseitinpracticetoapplyandfor further verification according to scope of practice to apply and for furtherverification within its scope of application.
33、If not, assumptions and mathematicalTeam#41961 Page 5 of 26formulaofthemodel shouldbecarefullychecked andchangesshouldbe madeuntilthe fitting results are basically satisfied. Here, according to the World HealthOrganization announced in Guinea since the Ebola virus disease outbreak occurredMarch 26 t
34、o July 17 , the evolution of the data ( Table 1) we draw scatter plot ofnumbers of daily increasing and deaths ( Figure 1). The total number of cases canchange due to the reclassification laboratory data, retrospective survey, mergers,consolidationandstrengtheningofmonitoringactivities.Table.12014Gu
35、ineacumulativecasesanddeathnumberscausedbyEbolavirus.Date 3.26 3.27 3.28 3.31 4.1 4.4 4.7Cumulativecases 86 103 112 122 127 143 151Deaths 62 66 70 80 83 86 95Date 4.9 4.14 4.16 4.2 4.23 4.26 5.1Cumulativecases 158 168 197 208 2218 224 226Deaths 101 108 122 136 141 143 149Date 5.3 5.5 5.7 5.1 5.12 5.
36、23 5.27Cumulativecases 231 235 236 233 248 258 281Deaths 155 157 158 157 171 174 186Date 5.28 6.1 6.3 6.5 6.16 6.2 6.25Cumulativecases 291 328 344 351 398 390 393Deaths 193 208 215 226 264 270 275Date 6.3 7.2 7.6 7.8 7.12 7.14 7.17Cumulativecases 413 412 408 409 406 411 410Deaths 303 305 307 309 304
37、 310 310Team#41961 Page 6 of 260 5 10 15 20 25 30 35 4050100150200250300350400450Figure1Thescatterplotofinfectingnumbersanddeaths.Figure2RelationshipbetweendailyinfectingnumbersandweeksLets analyze and forecast the daily infection. According to the data form (1.3)formulaandtheWorldHealthOrganization
38、,letsvalue=3.72， =0.863,wecanestimatethenumberofdailyinfections=1882ch2(0195t 602),itsgraphcanbereferred to Figure 2 . From Figure 2 above, we find that in the 32nd week or 33rdweekswilloccurthelargestnumberofdailyinfection.Thatis,ifwesetMarch26asthe starting point, the daily increase of people will
39、 reach the largest number in lateNovember 2014 or early December. Subsequently with strengthening of the activeTeam#41961 Page 7 of 26treatment, prevention and control, the number of people infected will be graduallyreduced.ItisFebruary8,2015,todayanditisfoundthattherearenewsreportsfromthe World Hea
40、lth Organization statistics, saying that the number of Ebola patientsshowsadownwardtrend.Lessthan150newcaseswerediscoveredlastweek.Ebolavirusissuspectedinhavingpassedthepeakofthecrisis.Uptonow,thetotalnumberofWestAfricanpatientshasexceeded8400.Tobemoreconvincing,accordingtotheWorld Health Organizati
41、on announced evolution of Ebola virus epidemic occurredin Guinea since December 1 to February 6 data (Table 2), we draw scatter gram ofincreasingnumbersanddeathnumber(Figure.3)Table.2 14-15GuineacumulativecasesanddeathnumberscausedbyEbolavirus.Date 12.1 12.2 12.5 12.8 12.1012.1112.1212.15 12.17 12.1
42、9 12.2212.23Cumulativecases 2155 2164 2192 2283 2292 2339 2354 2394 2416 2453 2571 2597Deaths 1312 1327 1366 1412 1428 1454 1462 1518 1525 1550 1586 1607Date 12.2612.2912.301.2 1.5 1.6 1.7 1.8 1.10 1.11 1.12 1.14Cumulativecases 2630 2695 2706 2730 2769 2769 2775 2775 2775 2775 2799 2806Deaths 1654 1
43、697 1708 1739 1767 1767 1781 1781 1781 1781 1807 1814Date 1.15 1.16 1.18 1.19 1.20 1.22 1.23 1.24 1.26 1.27 1.28 1.29Cumulativecases 2817 2825 2825 2825 2871 2873 2873 2873 2909 2920 2920 2921Deaths 1821 1829 1829 1829 1876 1879 1880 1880 1906 1910 1910 1911Date 1.30 2.2 2.3 2.4 2.5 2.6Cumulativecas
44、es 2920 2959 2975 2975 2975 2975Deaths 1913 1937 1944 1944 1944 1944Team#41961 Page 8 of 260 5 10 15 20 25 30 35 40 451200140016001800200022002400260028003000Figure.3ThescatterplotofinfectingnumbersanddeathsBy Figure 3, we can find the number of people infected becomes graduallystabilized since late
45、 January 2015, slope of graphics tends to 0. It means the dailyincreasing number of the people infected will decrease by degrees. With thepromotionofthevaccineandimprovementofmedicalstandards,weexpecttheEbolaviruswillbecontrolledinthenextshortperiodoftime.Butwiththeadventofwinterandspringashighseaso
46、nofthevirus,preventionisstillanarduoustask,wemustpaycloseattentiontopreventanoutbreakagain.Onlybytheunityofthinking,raisingofawareness, mastering the facts, scientific response, prevention and control work canweachievethefinalvictory.2. Quantity of Drugs DemandedOwing to the constant change of the e
47、mergency supplies which is caused byEbola epidemic situations diffusion, in other words, the quantity of the drugsdemanded,isadynamicprocessofchange,areasonableestimateofthechangeisthekey to the epidemic needs analysis. We do not use the cumulative number of casesand total number of deaths for direc
48、t quantitation ofthe demand of the drugs, but toestablishanewmodel-requirementanalysismodel,becausewehavenotestablishedadetailedanalysisofthepatientsisolationdetailsinSIRmodelabove.Inthismodel,weregardthisproblemasamulti-stagechangeofstatus,byanalyzingthenumberofemergencymedicalsuppliesdemandedaccordingtopatientsstatuschanges.(1)AssumptionAssumingEbolavaccinesanddrugscollectivelyknownasnewdrugsandweonlyTeam#41961 Page 9 of 26consider for usin
