leqcia 10
5 maTematikuri modelebis Sesaxeb biologiaSi, qimiaSi, medicinasa da ekologiaSi
5.1 balansis meTodi
sanam biologiaSi, medicinasa da ekologiaSi zogierTi maTematikuri modelis Seswavlaze gadavidodeT, ganvixiloT maTematikuri modelebis agebis erT-erTi efeqturi meTodi, romelic balansis meTodis saxelwodebiT aris cnobili.
davuSvaT, zedapiriT SemosazRvrul moculobaSi mimdinare raime procesi an movlena drois t momentSi aditiuri sididis mniSvnelobiT xasiaTdeba. G sididis aditiuroba Semdegnairad unda gavigoT: Tu moculobas gavyofT or nawilad da TiToeul nawilSi mimdinare igive procesi an movlena drois t momentSi xasiaTdeba da sidideebis mniSvnelobebiT, maSin . am sidides SeiZleba hqondes masis, energiis, impulsis, raime regionSi mosaxleobis raodenobis, sawarmoo fondebis da a. S. Sinaarsi. cxadia, rom Tu aRniSnavs raime regionSi mosaxleobis raodenobas, maSin warmoadgens am regionis Sesabamis zedapirs, xolo -s qveS am zedapiris SemomsazRvreli wiri igulisxmeba.
Cveni mizania, gavarkvioT, Tu ram SeiZleba gamoiwvios droTa ganmavlobaSi sididis cvlileba raime aTvlis sididis sistemis mimarT fiqsirebul (ucvlel) moculobaSi da rogor SeiZleba am cvlilebis ricxviTi maxasiaTeblebis gamoTvla.
ganvixiloT konkretuli magaliTi. vTqvaT, raime dasaxlebuli punqtia, ki _ misi sazRvari. maSin am dasaxlebul punqtSi mosaxleobis raodenobis Secvlis ori ZiriTadi, erTmaneTisgan gansxvavebuli tipis mozezi arsebobs.
pirveli mizezia dasaxlebuli punqtidan mosaxleobis gamgzavreba anu misi sazRvris gadalaxva Signidan da dasaxlebul punqtSi mosaxleobis Camosvla, anu sazRvris gadalaxva garedan. sxva sityvebiT rom vTqvaT, dasaxlebul punqtSi mosaxleobis raodenobis cvlilebis erTi mizezia am punqtis SemomsazRvreli wiris gamWoli mosaxleobis “nakadebis” arseboba.
mosaxleobis raodenobis cvlilebis meore mizezia dasaxlebul punqtSi adamianebis dabadeba da sikvdili. aqac, Tu sxva sityvebiT davaxasiaTebT am process, SeiZleba vTqvaT, rom -s SigniT mosaxleobis “wyaroebis” cnebis qveS vgulisxmobT sididis rogorc “gaCenas”, ase “gaqrobasac”. am “gaCenisa” da “gaqrobis” konkretuli mizezebi SeiZleba sxvadasxva iyos, magram am mizezebis detaluri ganxilva Cvens mizans ar warmoadgens.
amrigad, zedapiriT SemosazRvrul moculobaSi sididis cvlilebas ori saxis procesi ganapirobebs: zedapiris gamWoli sididis “nakadi” da TviT moculobaSi sididis “wyaroebis” moqmedeba.
rogorc aRvniSneT, sazogadod SesaZlebelia arsebobdes rogorc “nakadebis”, aseve “wyaroebis” sxvadasxva saxeoba. kerZod, zemoT ganxilul magaliTSi mosaxleobis raodenobis cvlilebis Sesaxeb rogorc “nakadis”, aseve “wyaroebis” or-ori saxeoba gvqonda: “gasvla” da “Semosvla”, “dabadeba” da “sikvdili”. am SemTxvevaSi maTematikuri modelis Sedgenisas gaTvaliswinebuli unda iqnes rogorc nakadebis, aseve wyaroebis yvela saxeobis erToblivi moqmedebis Sedegebi.
gadavideT axla raodenobrivi Tanafardobebis dadgenaze.
Tu vigulisxmebT, rom funqcia uwyveti da warmoebadia, maSin warmoebuli moculobaSi G sididis cvlilebis siCqares, anu moculobaSi drois erTeulSi G sididis cvlilebas warmoadgens.
sididis im raodenobas, romelic nakadebis yvela saxeobis erToblivi moqmedebis Sedegad gamWoli moZraobiT gaivlis zedapirs drois erTeulSi, vuwodoT “G sididis nakadi” da aRvniSnoT igi Q simboloTi, xolo G sididis im raodenobas, romelic wyaroebis yvela saxeobis erToblivi moqmedebis Sedegad Cndeba moculobaSi drois erTeulSi, vuwodoT “G sididis wyaroebis simZlavre” da aRvniSnoT igi F-iT.
axla ukve sirTules aRar warmoadgens, SevadginoT “balansi” _ moculobaSi G sididis Semcvelobis cvlileba drois erTeulSi gavutoloT G sididis nakadisa da wyaroebis simZlavris jams:
. (5.1.1)
am gantolebas balansis gantolebas vuwodebT. swored is warmoadgens balansis meTodis maTematikur formulirebas.
ganvixiloT ekologiis zogierTi maTematikuri modeli, romelSic diferencialuri gantolebebi iqneba gamoyenebuli. am SemTxvevaSi drois cvlileba uwyvetad xdeba da maTematikuri modelis saSualebiT drois nebismier momentSi ama Tu im populaciis Sesaxeb daskvnis gakeTebis saSualeba gveqneba. SevniSnoT, rom maTematikuri modelebis SedgenisTvis am SemTxvevaSi arsebiTad Cvens mier aRwerili balansis meTodi iqneba gamoyenebuli.
5.2 populaciis malTusis diferencialuri modeli
malTusis models safuZvlad udevs martivi debuleba _ populaciaSi individebis raodenobis cvlilebis siCqare t momentSi individebis raodenobis pirdapirproporciulia.
vTqvaT, raime populaciaSi individebis raodenobas warmoadgens t momentSi. Tu A im individebis raodenobaa, romlebic drois erTeulSi ibadebian, xolo B _ im individebis raodenoba, romlebic drois erTeulSi kvdebian, maSin balansis meTodis gaTvaliswinebiT SegviZlia davaskvnaT, rom arsebobs sakmaod seriozuli safuZveli imisTvis, raTa sididis cvlilebis siCqare ganvsazRvroT Semdegi diferencialuri gantolebis saSualebiT
, (5.2.1)
sadac , , xolo , Sesabamisad dabadebisa da sikvdilianobis koeficientebs warmoadgenen. (5.2.1) gantoleba SeiZleba ase gadaiweros:
. (5.2.2)
Tu , , e. i., Tu da koeficientebi mxolod t drois cvladzea damokidebuli, maSin advilad SevamowmebT, rom (5.2.2) gantolebis amonaxsns aqvs
(5.2.3)
saxe, sadac populaciaSi individebis raodenobas warmoadgens sawyis momentSi.
nax. 5.2.1
naxazze (ix. nax. 5.2.1) mocemulia funqciis grafikebi, rodesac , , sadac da mudmivebia (erTmaneTis msgavs mrudeebs sawyisi momentis sxvadasxva mniSvneloba Seesabameba). gantolebis amonaxsns
funqcia warmoadgens.
Tu , maSin populaciaSi individebis raodenoba mudmivia.
Tu , maSin eqsponencialurad, rodesac .
Tu , maSin aseve eqsponencialurad, rodesac .
am garemoebam malTuss saSualeba misca, gamoeTqva eWvi imis Sesaxeb, rom dedamiwaze mosaxleobis mkveTri zrda moxdeba aqedan gamomdinare yvela SedegiT.
Tumca moyvanili maTematikuri modeli uaRresi gamartivebis gamo ararealisturia, miuxedavad amisa, is kargad aRwers baqteriebis koloniebSi individTa raodenobis zrdis dinamikas sakvebi garemos gamofitvamde. amrigad, Cvens mier ganxiluli maTematikuri modeli samarTliania garkveul pirobebSi _ drois mcire monakveTSi. am daskvnas zogierTi tipis baqteriebze Catarebuli realuri eqsperimentebis Sedegebi adasturebs.
amasTan dakavSirebiT moviyvanoT maTematikuri modelis erTi saintereso magaliTi. mkvlevarebis mizani iyo, SeeswavlaT dedamiwaze mosaxleobis raodenobis zrdis dinamika.
vTqvaT, mosaxleobis zrdis siCqare mamakacebisa da qalebis raodenobis namravlis proporciulia (es hipoTezaa, romelic bevr praqtikulad saintereso SemTxvevaSi dasturdeba), e. i., dedamiwaze mosaxleobis raodenobis zrdis dinamika aRvweroT Semdegi diferencialuri gantolebiT:
, (5.2.4)
sadac dedamiwaze mosaxleobis raodenobaa drois t momentSi, da _ Sesabamisad mamakacebisa da qalebis raodenoba, _ proporciulobis garkveuli koeficienti.
sakmarisad didi sizustiT SeiZleba CavTvaloT, rom
,
maSin (5.2.4) miiRebs
(5.2.5)
saxes.
vTqvaT, momentSi mosaxleobis raodenobaa , maSin (5.2.5) gantolebis amonaxsni am sawyisi monacemiT gamoisaxeba Semdegi formulis saSualebiT:
(5.2.6)
sadac
.
marTlac, SemoviRoT
(5.2.7)
aRniSvna. maSin, radganac
,
(5.2.5) gantoleba Semdegi saxiT Caiwereba:
.
saidanac -dan t-mde integrebiT miviRebT, rom
,
e. i., (5.2.7)-is gaTvaliswinebiT,
.
amrigad, momentSi dedamiwaze macxovrebelTa raodenoba usasrulo gaxdeba. “samyaros aRsasrulis” droa. dedamiwis mosaxleobis statistikuri monacemebis gaTvaliswinebiT gamoTvales koeficientis mniSvneloba da daadgines, rom katastrofa moxdeba paraskevs, 2026 wlis 13 noembers. magram katastrofa ar moxdeba, radgan ufro zusti meTodebiT dadgenilia, rom am droisTvis dedamiwis mosaxleoba 10 miliards miaRwevs da ara -s.
rogorc aRvniSneT, (5.2.2) gantoleba kargad aRwers zogierT populaciaSi individTa raodenobis zrdis dinamikas drois mcire monakveTSi. amis Semdeg am procesze gavlenas axdens sakvebi resursebis nakleboba, rac maTematikur models SesamCnevad cvlis. am SemTxvevaSi SeiZleba populaciaSi garkveul doneze individTa raodenobis stabilizacia moxdes, an es raodenoba SeiZleba regularul an araregularul fluqtuaciebs ganicdides, an es raodenoba SeiZleba Semcirdes.
iseTi populaciis yofaqceva, romelSic individTa raodenoba garkveul mdgrad doneze stabilizirdeba, xSirad aRiwereba Semdegi logisturi gantolebis saSualebiT:
, (5.2.8)
sadac populaciaSi individTa raodenobas warmoadgens t momentSi, xolo , garkveuli mudmivebia.
(5.2.8) gantoleba warmoadgens umartives diferencialur gantolebas, romelsac Semdegi ori mniSvnelovani Tviseba gaaCnia:
1) x-s mcire mniSvnelobebisTvis (5.2.8) gantolebis amonaxsni uaxlovdeba (5.2.2) gantolebis amonaxsns da aqvs eqsponencialuri xasiaTi.
2) t-s zrdasTan erTad monotonurad uaxlovdeba garkveul mudmiv mniSvnelobas, rac populaciaSi individTa raodenobis stabilizacias asaxavs.
ganvixiloT koSis amocana (5.2.8) gantolebisTvis
, (5.2.9)
sadac populaciaSi individTa raodenobas warmoadgens momentSi.
advilad SeiZleba SevamowmoT, rom (5.2.8), (5.2.9) amocanis amonaxsns aqvs Semdegi saxe:
. (5.2.10)
t
0
nax. 5.2.2
aqedan Cans, rom, rodesac , maSin populaciaSi individTa raodenoba . amasTan SesaZlebelia ori SemTxveva: da . gansxvaveba am or SemTxvevas Soris kargad Cans nax. 5.2.2-ze (aq ganxilulia SemTxveva, rodesac ).
SevniSnoT, rom (5.2.8) gantoleba da, maSasadame, (5.2.10) funqcia sakmaod kargad aRwers zogierTi tipis baqteriis populaciaSi individebis raodenobis cvlis dinamikas sawyis etapze (mag., kulturaSi safuaris ujredebis cvlilebis dinamikas).
5.3 “mtacebeli _ msxverplis” maTematikuri modeli
5.2 paragrafSi arsebiT cvladad miviReT ama Tu im populaciaSi individebis raodenoba da SevecadeT, Segveqmna romelime erTi calkeuli populaciis ganviTarebis matematikuri modeli im pirobiT, rom populacia izolirebulia.
axla gadavdgaT Semdegi nabiji da ufro metad davuaxlovdeT realur situacias.
ganvixiloT ori saxeobis urTierTqmedeba. SeviswavloT ori “izolirebuli” populaciis ganviTarebis dinamika sxvadasxva faqtorebis gaTvaliswinebiT.
sxvadasxva saxeobis or populacias Soris urTierTqmedebis meqanizmebi SeiZleba sam kategoriad davyoT:
a) konkurecia, rodesac erTi saxeobis ganviTareba meoris ganviTarebaze damTrgunvel zegavlenas axdens.
b) komensalizmi, rodesac erTi saxeoba meoris ganviTarebis stimulirebas axdens.
g) mtacebloba, rodesac erTi saxeoba (“mtacebeli”) meore saxeobiT (“msxverpliT”) ikvebeba da, maSasadame, misi raodenobis Semcirebas iwvevs, xolo “msxverpli” xels uwyobs “mtaceblebis” raodenobis zrdas.
Cvens mizans ar Seadgens saxeobaTa urTierTqmedebis am meqanizmebis detaluri ganxilva. SevecdebiT, maTematikuri modelis saSualebiT aRvweroT iseTi ori saxeobis populaciis ganviTarebis dinamika, romlebic erTmaneTTan “mtacebeli _ msxverplis” principiT urTierTqmedeben.
maTematikuri modelis Sedgenisas vigulisxmebT, rom msxverpls yovelTvis aqvs saSualeba, ipovos sakvebi, xolo yoveli Sexvedrisas mtacebeli aucileblad klavs msxverpls, romelic misi erTaderTi sakvebia. cxadia, rom am daSvebis Sedegad miviRebT sakmaod “idealizebul” models, romlis gamoyenebac mxolod zogierT SemTxvevaSi SeiZleba, Tumca am modelis saSualebiT bevri saintereso, praqtikisTvis mniSvnelovani daskvnis gakeTebaa SesaZlebeli.
Tu am situacias ganvixilavT, cxadi gaxdeba, rom mtaceblebis raodenoba manam imatebs, sanam maT sakmarisad aqvT sakvebi, e. i. sanam msxverpli sakmarisi raodenobiTaa. bolos da bolos dadgeba momenti, rodesac mtaceblebis zegavleniT msxverplis raodenoba sakmarisad Semcirdeba, am dros mtaceblebs sakvebi ar eyofaT da daiwyeba maTi raodenobis Semcireba. es iqamde migviyvans, rom mtaceblebis raodenobis Semcirebis gamo daiwyeba msxverplis raodenobis mateba. es kvlav miscems stimuls mtaceblebis raodenobis zrdas da a. S. cikli kvlav ganmeordeba. “mtacebeli _ msxverplis” tipis urTierTqmedeba sakmaod xSirad gvxvdeba sxvadasxva praqtikuli amocanis Seswavlisas. problemis aqtualurobis gamo misi Seswavla bolo periodSi rogorc ekologiis, ise sxva dargis mecnierebis, maT Soris maTematikosTa, yuradRebis centrSi moeqca.
aRvniSnoT da -Ti Sesabamisad mtaceblisa da msxverplis raodenoba t momentSi. imisTvis, rom CamovayaliboT maTematikuri modeli, romelic garkveul miaxloebaSi populaciaSi individTa raodenobis cvlilebis dinamikas aRwers, gavakeToT ramdenime daSveba, rac amocanas gaamartives. jer erTi, davuSvaT, rom rom im SemTxvevaTa raodenoba, rodesac mtacebeli msxverpls klavs, damokidebulia maT SexvedraTa sixSireze. CavTvaloT, rom es sidide xy namravlis proporciulia. meore, ugulebelvyoT is dro, romelic mtacebels msxverplis SesaWmelad sWirdeba. rac Seexeba bunebrivi Sobadobisa da sikvdilis pirobebSi populaciis raodenobis cvlilebas, is (5.2.8) saxis logisturi gantolebebis saSualebiT aRvweroT.
(5.2.2) saxis logisturi gantolebis saSualebiT miviRebT, rom orive populaciaSi individTa raodenobis cvlileba aRiwereba Semdegi pirveli rigis diferencialur gantolebaTa sistemis saSualebiT:
, (5.3.1)
(5.3.2)
sadac a, b, c da d garkveuli dadebiTi mudmivebia.
(5.3.1), (5.3.2) gantolebebi pirvelad gamoyvanili iqna 1925 w. da cnobilia lotka-volteras gantolebebis saxelwodebiT.
aq gaTvaliswinebulia is garemoeba, rom msxverplis ararsebobis SemTxvevaSi mtacebelTa raodenoba bunebrivi sikvdilis gamo iklebs (maT sakvebi ar aqvT) da amitom individTa raodenobis cvlilebis siCqare x sididis proporciulia proporciulobis uaryofiTi koeficientiT. msxverplis sakmao raodenobiT arsebobis SemTxvevaSi , mtacebelTa raodenoba izrdeba. analogiurad, mtaceblebis ararsebobis SemTxvevaSi msxverplTa raodenoba imatebs da am populaciaSi individTa raodenobis cvlilebis siCqare y sididis proporciulia proporciulobis dadebiTi koeficientiT (aq, iseve rogorc adre, vgulisxmobT, rom msxverplis sakvebis raodenoba SemosazRvruli ar aris). amave dros, Tu arseboben sakmao raodenobis mtaceblebi , maSin msxverplis raodenoba iklebs.
amocana jer-jerobiT mTlianad dasmuli ar aris, unda iyos cnobili sawyis momentSi TiToeul populaciaSi individebis raodenoba. amrigad, (5.3.1), (5.3.2) gantolebebs unda daematos e. w. sawyisi pirobebi
, , (5.3.3)
sadac da mocemuli dadebiTi ricxvebia.
(5.3.1)-(5.3.3) amocana warmoadgens koSis amocanas pirveli rigis diferencialur gantolebaTa sistemisTvis. SevniSnoT, rom gansxvavebiT zemoT ganxiluli koSis amocanisgan ar arsebobs am amocanis amoxsnis analizuri warmodgena. am amocanis amonaxsni SeiZleba miRebuli iqnes mxolod ricxviTi meTodebis gamoyenebiT. rac Seexeba (5.3.1)-(5.3.3) koSis amocanis amonaxsnis arsebobas da erTaderTobas, gamomdinareobs Teorema 3.7.5-dan, radgan (5.3.1), (5.3.2) sistemis marjvena mxareebi y-is mimarT lipSicis pirobas akmayofilebs.
SevecadoT, gamovikvlioT (5.3.1)-(5.3.3) amocana da davadginoT kavSiri da funqciebs Soris. am mizniT SemoviRoT Semdegi aRniSvnebi:
, , , .
am aRniSvnebis Semdeg (5.3.1)-(5.3.2) diferencialuri gantolebebi miiReben Semdeg saxes:
, , (5.3.4)
sadac .
am diferencialur gantolebaTa sistemas daemateba
, (5.3.5)
sawyisi pirobebi, sadac
, , .
dasmuli amocanebis fizikuri Sinaarsidan gamomdinare, SemdgomSi Cven ganvixilavT (5.3.4), (5.3.5) amocanis dadebiT amonaxsnebs.
(5.3.4) gantolebaTa sistema Semdegi saxiT gadavweroT:
, . (5.3.6)
(5.3.6) sistemis meore gantoleba gavamravloT -ze da Semdeg es gantolebebi SevkriboT:
. (5.3.7)
amis Semdeg (5.3.6) gantolebaTa sistema Semdegi saxiT gadavweroT:
,
an
, . (5.3.8)
(5.3.8) sistemis meore gantoleba gavamravloT -ze da mivumatoT pirvels, miviRebT:
.
es ukanaskneli gamovakloT (5.3.7) gantolebas, ris Semdegadac gveqneba:
.
vaintegroT es gantoleba -dan -mde, miviRebT:
, (5.3.9)
sadac
.
(5.3.9) saSualebas iZleva, avagoT misi Sesabamisi wirebi H-is sxvadasxva mniSvnelobisTvis: ,, (ix. nax. 5.3.1).
0
nax. 5.3.1 nax. 5.3.2
rogorc vxedavT, sibrtyeze miviRebT Caketil wirTa ojaxs. davuSvaT, rom sawyisi monacemebi mocemulia A wertilis traeqtoriaze, romelic mniSvnelobas Seesabameba. vTqvaT, A wertili Seesabameba sawyis mniSvnelobebs, amitom da . (5.3.4) sistemis pirveli gantoleba gviCvenebs, rom sawyis etapze U cvladi iklebs. analogiuri Tviseba gaaCnia V cvladsac. Semdeg, rodesac U miiRebs mniSvnelobas , maSin . Semdeg, -s cvlilebis garkveul SualedSi, V cvladi zrdas iwyebs, xolo U cvladi klebas agrZelebs. rodesac toloba gveqneba, maSin da am momentidan daiwyebs zrdas U cvladic da a. S. radgan wertili Caketil traeqtoriaze moZraobs, es imas niSnavs, rom gantolebaTa sistemis da amonaxsni periodul funqciebs warmoadgens, amasTan rxeva orive populaciaSi ( da funqciebis saSualebiT aRiwereba populaciebSi individTa raodenoba) sxvadasxva fazaSi xdeba. moviyvanoT da funqciebis tipiuri grafiki im SemTxvevaSi, rodesac , , (ix. nax. 5.3.2).
SevniSnoT, rom radgan mtaceblebis sakvebis raodenoba SemosazRvrulia (es raodenoba msxverplis raodenobiT ganisazRvrvreba), amitom mtaceblebis populaciaSi individebis raodenobis aRwerisTvis mizanSewonilia (5.2.8) logisturi gantolebis gaTvaliswineba. maSin ori populaciis individTa raodenobis cvlileba “mtacebeli _ msxverplis” tipis urTierTqmedebis SemTxvevaSi aRiwereba Semdeg diferencialur gantolebaTa sistemis saSualebiT:
, ,
sadac a, b, c, d, e garkveuli dadebiTi mudmivebia, xolo da kvlav Sesabamisad mtaceblisa da msxverplis raodenobas gamosaxavs t momentSi.
cxadia, rom Cvens mier moyvanili maTematikuri modelebi ori populaciis urTierTqmedebis meqanizms garkveul, sakmarisad mkacri SezRudvebis pirobebSi aRweren. ufro zusti maTematikuri modelis Sesaqmnelad mravali sxva faqtori (mag., populaciis individTa asaki, gamravlebis sezonuroba, is, rom mtacebeli yoveli Sexvedrisas ver axerxebs msxverplis daWeras da a. S.) unda iqnas gaTvaliswinebuli. cxadia, am faqtorebis gaTvaliswineba maTematikuri models sakmaod gaarTulebs, magram amis safasurad ufro kargad aRwers realur situacias.
5.4. epidemiologiis umartivesi maTematikuri modelebi
istoriidan cnobilia faqtebi, rodesac sxvadasxva epidemiuri daavadebisgan didi raodenobiT adamianebi iRupebodnen. qolera, Savi Wiri, gripi da sxva xSirad iwvevs adamianTa mniSvnelovani raodenobis daavadebas. imisTvis, rom rom efeqturad vebrZoloT am daavadebaTa gavrcelebas, saWiroa, winaswar ganisazRvros, Tu ra Sedegi moyveba daavadebis sawinaaRmdego RonisZiebebs, e. i., saWiroa moxdes sxvadasxva RonisZiebis Catarebis Sedegad avadmyofTa raodenobis dinamikis prognozireba. aqedan gamomdinare mivdivarT iseTi maTematikuri modelis Seqmnis aucileblobamde, romelic daavadebis gavrcelebis garkveuli prognozirebis saSualebas iZleva.
simartivisTvis jer ganvixiloT situacia, rodesac araferi ar keTdeba ama Tu im epidemiis gavrcelebis winaaRmdeg, e. i., movaxdinoT epidemiis gavrcelebis bunebrivi procesis prognozireba.
cxadia, rom maTematikuri modeli epidemiis gavrcelebaze sxvadasxva faqtoris gavlenas unda iTvaliswinebdes. ase, magaliTad, gaTvaliswinebuli unda iqnes is kanonebi, romlis mixedviTac xdeba ama Tu im virusis gamravleba, calkeuli adamianis imuniteti ama Tu im daavadebis mimarT, infeqciis matarebeli adamianebis Sexvedris albaToba janmrTel adamianebTan, da mravali sxva faqtori. sxva sityvebiT rom vTqvaT, epidemiis ase Tu ise sruli modeli unda Seicavdes im kvlevis Sedegebs, romelsac mecnierebis sul cota oTxi dargi mainc awarmoebs, kerZod, mikrobiologia, medicina, farmakologia da socialuri fsiqologia.
radgan Cveni mizani mxolod sailustrcio modelis Sedgenaa, amitom maTematikuri modelis Sedgenisas bevr faqtors ar gaviTvaliswinebT. amis miuxedavad, aseTi uxeSi modelis saSualebiTac ki SeiZleba epidemiis gavrcelebis meqanizmis aRwera mis garkveul etapze.
amrigad, ganvixiloT adamianebis jgufi, romelic individisgan Sedgeba. vTqvaT, momentSi am jgufSi moxvda avadmyofi adamiani (infeqciis wyaro). vigulisxmoT, rom am jgufisgan arc erTi avadmyofis CamoSoreba (mag., karantinis saSualebiT) ar xdeba, aseve ar aris arc gamojanmrTelebisa da arc sikvdilis SemTxvevebi. aseTi daSvebebi sruliad bunebrivia epidemiis dawyebidan drois mcire intervalis ganmavlobaSi. CavTvaloT agreTve, rom nebismieri adamiani infeqciis wyarod iTvleba maSinve, rodesac is daavaddeba.
aRvniSnoT t momentSi daavadebuli adamianebis raodenoba simboloTi, xolo jerjerobiT janmrTeli adamianebis raodenoba _ simboloTi. cxadia, rom Cveni daSvebebis pirobebSi t-s nebismieri mniSvnelobisTvis samarTliania
(5.4.1)
toloba. rodesac , maSin .
cxadia, rom daavadebuli adamianebis raodenobis cvlilebis siCqare damokidebulia avadmyofi da janmrTeli adamianebis Sexvedraze, e. i., SeiZleba CavTvaloT, rom es siCqare namravlis proporciulia. am daSvebis safuZvelze SegviZlia davweroT
an, Tu (5.4.1) tolobas gaviTvaliswinebT,
, (5.4.2)
sadac garkveuli mudmivia. miRebuli diferencialuri gantolebisTvis ganvixiloT koSis amocana
. (5.4.3)
(5.4.2), (5.4.3) warmoadgens epidemiis gavrcelebis umartives models, romlis saSualebiTac drois nebismier t momentSi daavadebuli adamianebis raodenobis gansazRvra SeiZleba.
amovxsnaT es amocana. am mizniT SemoviRoT
aRniSvna, saidanac miviRebT, rom
.
Tu am ukanasknel tolobas gaviTvaliswinebT, (5.4.2) gantoleba SeiZleba Semdegi saxiT CavweroT:
, (5.4.4)
radgan , amitom .
advilad SeiZleba Semowmeba, rom sawyisi pirobebis gaTvaliswinebiT (5.4.4) gantolebis amonaxsns warmoadgens Semdegi funqcia:
,
saidanac miviRebT, rom
. (5.4.5)
gavaanalizoT miRebuli formula. t-s zrdasTan erTad wiladis mniSvneli izrdeba, e. i., izrdeba. es Seesabameba Cvens varauds, rom avadmyofTa raodenoba SeiZleba mxolod gaizardos.
sainteresoa, gamovikvlioT, Tu rogor icvleba daavadebuli adamianebis raodenobis zrdis siCqare. am sakiTxis Sesaswavlad unda gamovikvlioT sidide.
(5.4.5)-is gaTvaliswinebiT miviRebT, rom
.
aqedan gamomdinareobs, rom
, rodesac ;
Tu , maSin ;
Tu , maSin .
amrigad, funqcia, romelic avadmyofTa raodenobis zrdis siCqares aRwers, momentamde izrdeba, xolo Semdeg iklebs. es Sedegi, uxeSi maTematikuri modelis miuxedavad, sakmaod kargad eTanxmeba eqsperimentul monacemebs, gansakuTrebiT epidemiis sawyis etapze.
axla ganvixiloT epidemiis sxva maTematikuri modeli, romelSic zogierTi iseTi faqtoris gaTvaliswineba xdeba, romelsac epidemiis Cvens mier ganxiluli umartivesi (da rogorc aRvniSneT, uxeSi) modeli ar iTvaliswinebda.
davuSvaT, rom raime populacia, romelic N individisgan Sedgeba, sam jgufad iyofa. pirvel jgufs mivakuTvnoT individebi, romlebic mocemul momentSi janmrTelebi arian, magram ar aqvT imuniteti garkveuli daavadebis mimarT, e. i., arsebobs imis garkveuli albaToba, rom isini daavaddebian. aseTi individebis raodenoba t momentSi aRvniSnoT simboloTi. meore jgufs mivakuTvnoT individebi, romlebic aRebul momentSi avad arian da, maSasadame, infeqciis gamavrceleblad iTvlebian. maTi raodenoba t momentSi aRvniSnoT simboloTi. da bolos, mesame jgufs mivakuTvnoT individebi, romlebic ar arian avad da amave dros aqvT imuniteti am daavadebis mimarT. maTi raodenoba aRvniSnoT simboloTi. e. i.,
. (5.4.6)
davuSvaT, rom im SemTxvevaSi, rodesac daavadebulTa ricxvi garkveul fiqsirebul ricxvs gadaaWarbebs, e. i. , iwyeba epidemiis procesi, e. i., maT SeuZliaT daaavadon is individebi, romelTac am daavadebis mimarT imuniteti ar aqvT. es ki imas niSnavs, rom garkveul momentamde SesaZlebelia daavadebul individTa izolacia (mag., karantinis saSualebiT) da rom sawyis etapze daavadeba garkveuli, mcirericxovani jgufis SigniT mimdinareobs. daavadebis am etaps maTematikur modelSi ar aRvwerT. Semdeg CavTvaloT, rom da imunitetis armqone janmrTeli individebis jgufSi daavadebis gamo individTa raodenobis cvlilebis siCqare am jgufSi myofi individebis raodenobis proporciulia. am daSvebebis Sedegad miviRebT
(5.4.7)
diferencialur gantolebas.
Semdeg davuSvaT, rom im individTa raodenobis cvlilebis siCqare, romlebic daavadebulTa jgufs miekuTvnebian, magram gamojanmrTeldnen, proporciulia am jgufSi individebis raodenobisa da -s tolia. Tu gaviTvaliswinebT agreTve, rom imunitetis armqone individi bolos da bolos avaddeba da TviTon xdeba infeqciis gamavrcelebeli (e. i., aseTi individi meore jgufSi gadadis), amitom infeqciis gamavrcelebelTa raodenobis cvlilebis siCqare damokidebulia drois erTeulSi daavadebul da gamojanmrTelebul individTa raodenobis sxvaobaze. amrigad,
(5.4.8)
(5.4.7) da (5.4.8) gantolebebSi proporciulobis da koeficientebs Sesabamisad vuwodoT daavadebisa da gamojanmrTelebis koeficientebi.
da bolos, Tu CavTvliT, rom gamojanmrTelebuli individebi imunitets iZenen, maSin janmrTeli individebis (romlebsac am daavadebis mimarT imuniteti gaaCniaT) raodenobis cvlileba SeiZleba Semdegi gantolebiT aRvweroT:
. (5.4.9)
miviReT (5.4.7)-(5.4.9) diferencialur gantolebaTa sistema. ganvixiloT koSis amocana am gantolebaTa sistemisTvis (TiToeul jgufSi individTa raodenoba sawyis momentSi):
, , . (5.4.10)
(5.4.7)-(5.4.10) amocana epidemiis gavrcelebis maTematikur models warmoadgens garkveul SezRudvebSi, romelTa Sesaxebac zemoT gvqonda laparaki. cxadia, rom am gantolebebidan SegviZlia nebismieri SevcvaloT (5.4.6) gantolebiT. kerZod, Tu amovxsniT (5.4.7) da (5.4.8) gantolebebs, SegviZlia miviRoT (5.4.6)-dan.
msjelobis simartivisTvis davuSvaT, rom sawyis etapze aRebul populaciaSi ar arian individebi, romlebsac garkveuli daavadebis mimarT imuniteti aqvT, e. i., . am daSvebis Sedegad imunitets is individebi iZenen, romlebic avadmyofobis Semdeg gamojanmrTeldebian. aseve davuSvaT, rom daavadebisa da gamojanmrTelebis koeficientebi erTmaneTis tolia: (SevniSnoT, rom Tu , es msjelobaSi raime garTulebas ar gamoiwvevs).
ganvixiloT ori SemTxveva:
I. vTqvaT, . Cveni daSvebis Tanaxmad am dros populaciaSi Semaval individebs daavadeba ar gadaecemaT. am SemTxvevaSi
,
ris safuZvelzec (5.4.6) tolobis da
pirobis gaTvaliswinebiT miviRebT:
.
es modeli im situacias aRwers, rodesac yvela daavadebuli individi gamojanmrTelebulia. am SemTxvevaSi (5.4.8) miiRebs
saxes, saidanac miviRebT, rom
,
.
nax. 5.4.1-ze grafikulad aris gamoxatuli individebis raodenobis cvlileba samive jgufSi.
nax. 5.4.1
II. vTqvaT, . am SemTxvevaSi iarsebebs intervali, sadac Sesrulebuli iqneba utoloba. aqedan gamomdinareobs, rom Tu , maSin avadmyofoba gavrceldeba im individebze, romlebsac am daavadebis mimarT imuniteti ar aqvT. maSin (5.4.7) gantolebidan gamomdinareobs, rom
, Tu . (5.4.11)
CavsvaT -s gamosaxuleba (5.4.8) gantolebaSi, maSin miviRebT, rom
.
am gantolebis orive mxare gavamravloT -ze:
.
am tolobis gaTvaliswinebiT SegviZlia davweroT, rom
,
sadac C nebismieri mudmivia. ukanaskneli tolobidan miviRebT, rom
.
Tu gaviTvaliswinebT sawyis pirobebs, sabolood miviRebT:
, Tu . (5.4.12)
cxadia, rom upirveles yovlisa unda davadginoT T-s mniSvneloba, agreTve _ drois is momenti, rodesac daavadebul individTa raodenoba maqsimaluria.
pirvel kiTxvaze pasuxis gacema mniSvnelovania imdenad, ramdenadac T momentSi daavadebis Semdgomi gavrceleba Sewydeba. Tu ganvixilavT (5.4.12) gantolebas, maSin Cvens mier zemoT gamoTqmuli mosazrebis gamo samarTliania
. (5.4.13)
toloba.
miviReT T-s mimarT arawrfivi gantoleba, romlis amonaxsnic SeiZleba vipovoT, magaliTad, biseqciis meTodiT.
imisTvis, rom pasuxi gavceT meore kiTxvas, e. i., davadginoT -is mniSvneloba, ganvixiloT -s (5.4.12) gamosaxuleba. radgan -s maqsimalur mniSvnelobas Seesabameba, amitom unda ganvixiloT
gantoleba, saidanac miviRebT, rom
. (5.4.14)
rodesac , infeqciis gavrceleba Sewydeba da (5.4.8) gantolebis amonaxsni sawyisi pirobiT, roca , iqneba:
.
naxazze (ix. nax. 5.4.2) grafikulad aris gadmocemuli individTa raodenobis cvlis dinamika TiToeul jgufSi.
tmax
nax. 5.4.2
cxadia, rom SeiZleba analogiurad iqnes ganxiluli SemTxvevebi, rodesac , .
SevniSnoT, rom Cvens mier ganxiluli epidemiis gavrcelebis maTematikuri modeli ramdenadme gamartivebulad aRwers realur situacias, radgan igi zogierT mniSvnelovan faqtors ar iTvaliswinebs. Tumca rogorc eqsperimentul monacemebTan Sedareba gviCvenebs, es gamartivebuli modelic sakmaod karg warmodgenas iZleva infeqciuri daavadebebis gavrcelebis meqanizmze.
5.5 naxSirbadoqsidis daJangvis maqsimaluri siCqaris gansazRvra
vTqvaT, saWiroa, davadginoT Jangbadis optimaluri koncentracia, romlis drosac airTa narevSi (naxSirJangi da Jangbadi) Semavali naxSirbadis (II) oqsidis daJangvis siCqare maqsimaluria. Sesabamisi reaqciis gantoleba ase iwereba:
. (5.5.1)
CavTvaloT, rom mocemul pirobebSi procesi Seuqcevadia. maSin, cnobilia, rom moqmed masaTa kanonis Tanaxmad, (5.5.1) reaqciis siCqare aRiwereba
(5.5.2)
formuliT, sadac x CO-s koncentraciaa, y _ -is, xolo k reaqciis siCqaris mudmivaa (igi damokidebuli araa moreagire komponentTa koncentraciaze).
gamovsaxoT reaqciis komponentebis koncentracia moculobiT procentebSi. maSin SegviZlia vTqvaT, rom
.
y-is am mniSvnelobis (5.5.2) formulaSi SetaniT miviRebT:
.
am funqciis pirveli rigis warmoebuli Semdegi sididis tolia:
. (5.5.3)
amovxsnaT
gantoleba. Tu gaviTvaliswinebT, rom , miviRebT x-is or mniSvnelobas: , . Amocanis qimiuri azridan gamomdinare saintereso es ukanasknelia. amis mkacrad dasasabuTeblad gavixsenoT, rom funqciis maqsimums Seesabameba piroba , minimums _ . (5.5.3)-is gawarmoebiT miviRebT
.
cxadia, roca , maSin
,
e. i., am dros siCqare minimaluria, Tu , maSin
,
e. i., am dros siCqare maqsimaluria.
Tu , maSin Jangbadis koncentracia
.
amgvarad, naxSirbadis (II) oqsidis daJangvis siCqare maqsimaluria, roca daJangvis Semcveloba airTa narevis 33,3%-s Seadgens.
5.6 maqsimaluri ganaTebulobis dadgena fotoqimiuri procesebisTvis
mTeli rigi qimiuri reaqciebis warmarTvisTvis iyeneben sinaTles (fotoqimia). xSirad dRis wesrigSi dgeba amocana _ ra simaRleze unda ganlagdes sareaqcio sistemidan (WurWlidan) sinaTlis wyaro, rom misi ganaTebuloba maqsimaluri iyos.
nax. 5.6.1-ze moyvanilia amocanis pirobis Sesabamisi sqema (igulisxmeba, rom ganaTebis are sxivebis perpendikularuli araa). cnobilia formula, romlis mixedviTac ganaTebuloba
, (5.6.4)
sadac k proporciulobis koeficientia, r manZili sinaTlis wyarosa da obieqts Soris, _ sinaTlis sxivis dacemis kuTxe.
nax. 5.6.1-dan cxadia, rom
da
.
Tu SevitanT parametrebis am mniSvnelobas (5.6.4) formulaSi, gveqneba:
. (5.6.5)
x
nax. 5.6.2
davafiqsiroT a da x vcvaloT 0-dan -mde. x-is mniSvnelobaTa Soris unda vipovoT iseTi, romlisTvisac udidesia. maqsimumis wertilis sapovnelad, Teorema 5.6.2-is Tanaxmad, unda ganvixiloT
,
e. i.,
gantoleba. radgan da , amitom -ze gamravlebis Semdeg miviRebT
, e. i.,
kvadratul gantolebas, romlis dadebiTi fesvia:
. (5.6.6)
SevniSnoT, rom, rogorc es (5.6.5)-dan gamomdinareobs,
da ,
xolo intervalze dadebiTia.
vinaidan (5.6.6)-iT gansazRvruli x0 x-is erTaderTi mniSvnelobaa, sadac warmoebuli 0-is tolia, cxadia, rom maqsimumis wertilia (ix. nax. 5.6.2).
rodesac
,
maSin
.
maSasadame, ganaTebuloba udidesia, rodesac dacemis kuTxis tangensi -is tolia. es ki maSin xdeba, roca
.
|