Ordinary Differential Equation Model in the Application of Infectious Disease Research

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1 Ordinary Differenial Equaion Model in he Applicaion of nfecious Disease Research Xiaocheng Gao Heihe Universiy Baoding Heihe China Absrac nfecious diseases (plague) are ofen popular around he world such as cholera smallpox ADS SARS HN virus. Seing up he mahemaical model of infecious disease analyzing is regulariy prevening is spread are a dauning ask. This paper which only discusses he mahemaical model of infecious diseases will spread he general laws. Firsly his paper inroduces he reader o how o use ordinary mahemaical model of differenial equaions which is esablished so ha readers are familiar wih some simple mahemaical model he modeling seps of ordinary differenial equaions. Then he paper discusses research heme applicaion of ordinary differenial equaion model in he research of infecious diseases. n he course of he sudy he firs sep will make he crowd be divided ino hree caegories including S (suscepible) (infecion) R (removed); he second sep according o specific groups is o esablish he mahemaical model of he propagaion of infecious diseases analyze he specific of all kinds of groups. The hird sep will make he model be applied o he acual analysis of propagaion of a flow of HN raionaliy validiy o infecious disease model validaion. Through wriing of his aricle propagaion process can make people beer unders maser he infecious disease so ha improve all he infecious disease awareness prevenion awareness provide a heoreical basis for he conrol of infecious diseases. Keywords - Ordinary differenial equaion; Mahemaical model; nfecious disease; Communicaion processes; A flow of HN. NTRODUCTON nfecious disease is a killer for human beings Has long been he aenion of counries all over he world. Wih he developmen of he sociey improved saniaion healh level Such as cholera smallpox so on Once raged has been effecive conrol of infecious diseases around he world. Bu New infecious diseases are sill emerge in endlessly. n many pars of he world Also puncuaed infecious disease epidemic siuaion [ ]. The SARS virus spread in our counry in nfeced more han 8 people killed more han 3 people; Repeaed oubreaks of HN flu around he world Nearly wo hous people deah. Human urgenly wan o know wha is he spread of infecious disease law? Disease ransmission behavior how o affec is populariy? How o preven he spread of i? So Clear up he spread of infecious diseases s of grea significance for human survival. On he one h Can make people more undersing masering he spread of infecious diseases in order o communicable diseases beer able o conrol is spreading. On he oher h he use of learned knowledge link heory wih realiy closely o solve pracical problems he benefi of human sociey.. BACKGROUND AND PUT FORWARD For a long ime he spread of infecious diseases are common concerns by expers from all over he world. You desperaely wan o use he exising heoreical knowledge o clear up he spread of infecious diseases. Bu o build a comprehensive sysem of he mahemaical model of infecious disease spread law is difficul. n view of his we can only in he general law of he case analysis of he spread of infecious diseases. Usually spread of infecious diseases wihin he scope of he populaion is divided ino hree caegories [3] Class: Suscepive people No infecion Bu easy o be infeced. Class: nfecive people People infeced wih he disease Will spread i o ohers. Class: Removal people solaed because of illness Or because of illness immuniy. Model ( S model) This model or immuniy due o illness only consider suscepible infeced he dynamics of wo kinds of people Model assumes ha he following: n addiion o he disease characerisics no any differences beween individuals disease infecions mixing in he crowd. Populaion quaniy is big enough only considered in he process of spreading he average effec of infecious disease process. (3)The chance of suscepible people infeced is proporional o he chance he may have come in conac wih he infeced person. (4)The disease of infecion rae was consan. (5)Regardless of people's birh deah emigraion immigraion. Assuming ha The number of suscepible infeced is abou he funcion of ime in he book of he DO.3/JSSST.a SSN: x online prin

2 Chronicles of he as S. Assuming ha(5)the limis (N)on he oal number of people in he process of epidemic is consan can be expressed as S N. For convenience we will S as suscepible people he proporion of infeced people in he crowd so S. Assuming ha They are coninuous smooh enough. According o he number of conservaion in ( ) period of infecion is increasing he number of equals he number of infecion in period of ime. Then assuming ha (3) (4) ge N ( ) kns Make ge d ks So S + ge k ( ) d ; S This is he S model of disease of is ( ) which he proporion for he iniial feeling sick. Using variable separaion soluions of ge: k ( ) e (3) ( ) k k S e / ( ) e (4) Pu(3)ype (4)ype inoype ge k k k ( ) e / ( ) e d (5) (5) says growh rae of he disease's relaionship wih ime change. Easy o draw (5) will reach maximum a he ln( momen m k ). n addiion. You can also see m k is inversely proporional o. is no hard o unders k as he disease infecion rae infecion rae is smaller he side can reflec he local healh level is higher. See ha When have Tha is o say As ime goes on he whole human beings will all become infecious disease paiens. This conflics wih he acual siuaion. Through he analysis he following model is se up. Model ( S S model) Afer some infecious diseases such as colds diseases such as dysenery cure low immuniy. When assuming no immuniy. So o cure he paien can sill be infeced again become suscepible people. So will he following he model called S S model. Assuming ha(6): Feeling sick wih fixed raio o heal o become suscepible people. Make h On behalf of he cured rae (says he number of people per uni ime o heal he proporion of he oal). h is expressed as he average infecious disease ransmission cycle. A his imeshould be revised o k ( ) h d (6) To solve he(6)ype ge (( kh) ( ) e k h ( ) k h k (7) Among hem k / h n combinaion wih k /h is he average number of he infecive period of each paien effecive conac (conac number) simplified lim ( ). (7) Easy o ge Analysis on he ype The number of access hreshold for.nfecious diseases spread of behavior in he value of he volailiy When The number of infeced people he proporion of he oal number is on he decline will end o be consan zero namely compleely go he conrol of infecious diseases. When The number of infeced people he increase of he proporion of he oal number of depends on worh he size. n his case he infecious diseases could no be eliminaed compleely. n real life here are many diseases paiens undergoing reamen afer recovery o move ou of he ransmission sysem. For modeling convenience remember hese people ou referred o as R classificaion. Considering he siuaion We esablish a mahemaical model of he following: Model ( S R model) Model 3 model a he same assumpion R remember o remove accouns for he proporion of he oal number. This model has S R So he model was revised o DO.3/JSSST.a SSN: x online prin

3 ds ks d ks h d dr h d Remember he iniial momen suscepible people ou of proporion S S( ) ( ) R R So S + R The above equaion can be simplified o: ds ks S S d ks h. d (8) So S R Will change o (8) ge h S ( ) ds S k (9) (Among hem said relaively removal rae) soluion(9) ge: S ( ) ( S ) S ln S S Can be seen from he diagram When he disease will evenually disappear in he sysem ge S. S wih ime change he funcion of he image as shown in figure figure. Figure Figure S wih ime change he funcion wih ime change he funcion observaion daa. To solve he n o ake limi ge ge S ln( S / S ) Solving ge ( S )/(lns ln S ) S When afer he spread of infecious diseases S Subsiuion of can be given. Because in he process of he prevalence spread of infecious diseases he number of people is no a fixed value wih his in mind we will build he model. Model (Seady sae S R model was born) n he model S R will say imes he populaion of hree kinds of people. Then discussed earlier model S R can be revised o: ds ks d ks h d dr h d S n R. Make ds / d / d ge ( S ) S h/ k / h Now we discuss near balance changes Le S S ( u) ( v) among u v values are smaller And ignore iem hen ge on equaion: du uv d dv u d among h/( k) /h. The above equaions ge a abou v homogeneous differenial equaion wih consan coefficiens of he second order: dv dv v d d Using euler's mehod of undeermined exponenial funcion (also called characerisic roo mehod) [4-5] ge a homogeneous differenial equaion wih consan coefficiens of he second order of wo characerisic roo for: n S--R model parameers is very imporan. Usually he size of he esimae can be given by DO.3/JSSST.a SSN: x online prin

4 4 So equaion can solve: v Ce Ce u Ce C e clearlywhen u v are ending o zero visible when he balance is sable.. NFECTOUS DSEASE MODEL ANALYSS OF NFLUENZA A (HN) Through he analysis of he a/hn flu HN spread of pahology known he a/hn flu infecion period is shorerso in he sudy does no consider he birh deah raes of infecion period herefore on he basis of he above model S R we will crowd ino 4 ypes: S class class ang R class E class( E class represens he populaion of he incubaion period).make: N Says infecious diseases o he oal populaion: E ncubaion period said momen in a percenage of he populaionpopulaion S said Momen of suscepible populaion says momen of infeced populaion R said ime ou of proporion Said he average daily conac rae Said in he incubaion period average daily raes Said he average daily rae of exinfecious disease model S R is esablished based on he above hn infecion rule of differenial equaion model is esablished: ds S d de S E d d E d dr d S E R S S E E R R (3) To solve (3) should firs find ou he average daily funcion relaion beween he conac rae imenow he inroducion of new variablesmake A said d influence infecious disease degree said he daily new increase he number of deahs b said new increase in he number of infeced v said he new increase in he number of suspeced cases.based on he weigh number in he mahemaical modeling combining he relaionship beween hese variables can ge he following formula: d b v A q q q max ( ) max ( ) 3 max ( ) q Among hem: q q 3 d b v A d b v (4) is he relaive weigh of respecively. n order of prevenion conrol of a/hnthe healh secor in conrolling epidemic prevenion effors x ( x ) vigilan srengh play y ( y ( ) ) an imporan role in he whole process heir common influences people's prevenive measuresake m.5 x.5 y. Said a he same imebecause paiens a day on average he number of effecive conac accouns for he proporion of healhy populaion So is a funcion of prevenive measures m.when When m m ake he maximum value increasing would be reducedwhen m ge. Through he analysis of he m by using he saisical daa relaionship of of swine flu (3-6-6) daa for he iniial daa for fiing ge parameers q.53 q.34 q [6] The above parameers obained daa ino (3) by fiing he daa comparing wih acual saisics daa specific resuls such as able : DO.3/JSSST.a SSN: x online prin

5 TABLE PREDCTED VALUE AND ACTUAL VALUE STATSTCS ABOUT THE NUMBER OF NFECTED PEOPLE NUMBER OF LKELY NFECTED PERSON AND NUMBER OF PEOPLE OUT OF DSEASE Number of infeced The number of likely The number of people Dae people infeced person ou of disease Predicive The acual Predicive The acual Prediciv The acual value value value value e value value Can be seen from he given in able he number of he sick person suspeced disease qui hose prediced value acual value Based on S R infecious disease model o build a model of influenza a (HN) (3) can be prediced in Sepember 9 influenza a HN enered he peak in May he infecious diseases will be fully eliminaed. V. CONCLUSON This paper aimed a he differen assumpions four ypes of infecious disease model is esablished followed by model S he model S S he model S R he model S R of he seady sae was born he four models are expounded in he general case he spread of infecious diseases. And hen hrough he analysis of he a/hn flu HN ransmission mechanism is propagaion law is in line wih he infecious disease model S R on his basis furher esablish he swine flu HN spread rule of he differenial equaion model. Verified in his paper he mahemaical models of infecious diseases has cerain use value in real life can predic he propagaion law of cerain infecious diseases. Bu because my abiliy is limied no considering he acual condiion of he comprehensive sysem of various facors only in he general law of he case analysis of he spread of infecious diseases so in he model are made of differen level assumpions almos did no consider communiy oubreaks of infecious disease so if here is a communiy oubreak he disorion degree of model predicion would have been improved a ha ime should be adjused. REFERENCES [] Q.Y.Jiange al: Mahemaical Model Higher Educaion Press China vol. No.3 pp [] G.Y.Leie al: Mahemaical model of he hou Cambridge Universiy Press China vol. 5 No.3 pp [3] S.J.Yang H.Rene al: Epidemiology People's Medical Publishing House China vol. No.3 pp [4] G.X.Wange al: Ordinary Differenial Equaion Higher Educaion Press China vol.8 No.8 pp [5] The healh minisry's press office. The a/hn flu sympoms prevenion vol.5 No.8pp [6] The healh minisry's press office. The minisry of healh of he a/hn flu prevenion conrol informaion vol. No. pp.769. DO.3/JSSST.a SSN: x online prin