Godwin Valentine UGWU, CHINEDU STEPHEN PG/MSC/14/76573 MATHEMATICAL MODEL ON GONORRHOEA TRANSMISSION FACULTY OF PHYSICAL SCIENCE

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1 UGWU, CHNEDU TEPHEN PG/MC/4/76573 MATHEMATCAL MODEL ON GONORRHOEA TRANMON FACULTY OF PHYCAL CENCE DEPARTMENT OF MATHEMATC Goin Valentine Digitally igne by: Content manager s Name DN : CN Webmaster s name O University of Nigeria, Nsukka OU nnovation Centre

2 ii MATHEMATCAL MODEL ON GONORRHOEA TRANMON BY UGWU, CHNEDU TEPHEN PG/MC/4/76573 UPERVOR: PROF. G.C.E. MBAH DEPARTMENT OF MATHEMATC UNVERTY OF NGERA, NUKKA DECEMBER, 5

3 i TTLE PAGE MATHEMATCAL MODEL ON GONORRHOEA TRANMON BY UGWU, CHNEDU TEPHEN PG/MC/4/76573 A THE UBMTTED N PARTAL FULFLLMENT OF THE REQUREMENT FOR THE AWARD OF MATER OF CENCE (M.c) N APPLED MATHEMATC N THE DEPARTMENT OF MATHEMATC, FACULTY OF PHYCAL CENCE, UNVERTY OF NGERA, NUKKA UPERVOR: PROF. G.C.E. MBAH DECEMBER, 5

4 ii CERTFCATON TH TO CERTFY THAT TH WORK WA CARRED OUT BY UGWU CHNEDU TEPHEN REG/NO PG/MC/4/76573 DEPARTMENT OF MATHEMATC UNVERTY OF NGERA, NUKKA APPROVED BY UGWU, CHNEDU.( PG/MC/4/76573) (tuent) Prof. G. C. E. MBAH (upervisor) Prof. M. O. Oyesanya (Hea of Department) External Examiner Date Date Date Date

5 iii DEDCATON This ork is eicate to Almighty Go from hom all inspiration an knolege comes from. The glory an honour be ascribe unto his mighty name Amen.

6 iv ACKNOWLEDGEMENT t is my Pleasure to express my gratitue to those ho have contribute in one ay or the other toars the success of this ork. My inestimable gratitue goes to my supervisor Prof. G.C.E Mbah, a noble, prestigious title man Omere Oha in AKU DEWA an of Mathematics Department, University of Nigeria, Nsukka, hose mathematical knolege is beyon measure. He evote his precious time to ensure that this ork is successful may Go ill bless him abunantly. am enormously inebte to the Hea of Department of Mathematics Prof. M. O. Oyesanya an other lecturers Prof. F. Ochor, Dr..E Aniaku, Dr.. Yekini, Dr. Collins Obiora an other lecturers from Maths Departments hose names are not here for their fatherly care an avice. My special thanks to my belove parents Mr. an Mrs. Ugu amuel Amu for their moral an financial support to me. praye for them to live long an reap the fruit of their labour. also ant to sho appreciation to my uncles Ugu Hyacinth an Ugu unay, My brother Tobechuku, my sisters Nneinso, Chikoi an my frien Chiimma for their prayers, love an encouragement to me. Go bless you all.

7 v ABTRACT This project presents mathematical moeling on gonorrhoea transmission. t covers the moe of transmission, causes, symptoms, complication an treatments/ controls of Neisseria Gonorrhoeae. t also inclues the schematic iagram that escribe the transmission ynamics of the isease an the moel. Finally, mathematical analysis such as equilibrium, stability an graphs ere inclue.

8 vi TABLE OF CONTENT Title Page Certification Deication Acknolegment Abstract Table of Contents CHAPTER ONE i ii iii iv v vi. ntrouction. Aims an objective of tuy 4. cope of tuy 4.3 Limitations of tuy 5.4 Motivation of tuy 5.5 Present tuy 6 CHAPTER TWO Literature Revie 7 CHAPTER THREE 3. What is Gonorrhoea? 3. Classification of Neisseria 3.3 Moe of Transmission 3

9 vii 3.4 ymptoms of Gonorhoea in Men ymptoms of Genorrhoea in Women Complications of Gonorrhoea Treatments/Controls of Gonorrhoea 5 CHAPTER FOUR 4. Notations an Assumption of the Moel 8 4. The Flo Diagram 4.3 The Moel for Gonorrhoea Transmission 4.4 The Control Flo Diagram 4.5 The Moel for Control Transmission 4.6 Equilibrium Analysis 4.7 Disease Free Equilibrium 4.8 Enemic Equilibrium Basic Reprouctive Number (R o ) 4 4. tability Analysis 3 4. conitions Necessary for table or Unstable ystem Graph 36 CHAPTER FVE 5. Discussion ummary / Conclusion Recommenation 4 REFERENCE

10 CHAPTER ONE NTRODUCTON Many people are in chaos about the negative influence of Gonorrhea isease. t has renere a lot of men an omen infertile in protracte case an complications. Gonorrhea is a sexual transmitte isease cause by pathogen neisseria gonorrhoeae. The bacteria can live insie the cells of the cervix (entrance to the uterus), the urethra (tube here urine comes out), the rectum, the throat an occasionally the eyes. The bacteria are foun mainly in the semen an vaginal fluis of omen ho have the infection. Gonorrhea is mainly passe from one person to another through sexual contact. Anyone ho is sexually active can get it an pass it on. You on t nee to have lots of sexual partners. Gonococcus infection oes not seem to confer immunity more precisely none of the immunologic efense mechanism has been shon to consistently prevent re-infection. This may be ue the great avancement of gonococcus surface antigens in orer for an iniviual to become infecte; gonococci must attach themselves to mucosal surface so that they ill not be ashe aay by urine an mucosal flos. The gonococci have hair like appenages calle Pili hich enhance their attachment lymphote activation

11 an prouction of migration inhibition factor to various gonococcus antigens has to be shon, hoever the presence of a cellular immune response oes not protect against re infection. niviual ith pelvic inflammatory isease the presence of bactericial antiboy oes not protect iniviuals from either subsequent reinfection or PD. Nevertheless, no vaccine hich prevents gonorrhea is no available one vaccine containing gonococcus pili elicite the prouction of antiboies both in bloo an in the secretion of the urinogenital tract, here they may prevent bacterial attachment (Marx 98). Another gonococcus Pili vaccine stimulate the prouction of antiboies in the bloo, but ha only a slightly temporary effect in the secretary antiboies of the urinogenital tract since there are about ientifiable strains of gonococci ith ifferent pili, a gonorrhea vaccine oul have to contain a variety of pili. Both gonorrhea an meningococcal meningitis are infections of mucosal surfaces an because the recent evelope vaccine for meningitis only provies immunity for several months, it is possible that a gonorrhea vaccine oul only provie temporary immunity vaccine that prouce permanent immunity are generally irecte at viral iseases.

12 3 Gonorrhea has three striking epiemiology characteristics hich must be incorporate into a moel. The first is that gonococcus infection oes not confer protective immunity. t says that iniviuals are susceptible again as soon as they recover from infection inee, this lack of protective immunity makes gonorrhea very ifferent from other iseases such as measles, mulps, chicken pox e.t.c. econly, iniviuals ho acquire gonorrhea become infectious ithin a ay or to. Therefore, the latent perio is very short compare other viral iseases such as ays for measles, 5ays for chicken pox e.t.c. Thirly, the seasonal oscillations in gonorrhea incience are very small (Less than %). n contrast, the inciences of isease such as influenza, measles, mulps an chickenpox often vary seasonally by factor of 5 to 5 or more. Thus moels hich use constant values for parameter provie goo approximation for gonorrhea. Because of the numerous complex interactions in a population, it is ifficult to comprehen the large scales ynamics of isease sprea ithout the formal structure of a mathematical moel. An epiemic moel uses a microscopic escription (the role of an infections iniviual to preict the

13 4 macroscopic behavior of isease sprea through a community. The purpose of mathematical moels is to achieve a better unerstaning of ho the biological an sociological mechanism influence isease sprea. The negative influence of gonorrhea in our society cannot be overemphasize. Therefore, all hans shoul be on eck to alleviate an eraicate gonorrhea isease in our system.. AM AND OBJECTVE OF THE TUDY We stuy the gonorrhea isease hich impaire reprouctive level principally to:. Unerstan hat the isease is. Unerstan ho it is contracte, controlle using chemotherapy an other means. 3. llustrate schematic interaction or flo iagram involving agents that contribute in the mechanism of transmission.. COPE OF TUDY This research ork is on ho gonorrhea is transmitte an moel. t also illustrate: (i) chematically the agents involve in transmission of gonorrhea (ii) The sensitivity of the system

14 5 Hoever this ork oes not cover the quantity of neisseria gonorrhea that can cause gonorrhea, level of immunity of an iniviual, cultural, religious belief an geographical area into consieration..3 LMTATON OF TUDY This ork is mainly limite to practical but e epen on the existing ata from biologist an researchers on Gonorrhoea isease, fit in, in orer to valiates the moels that ill be built..4 MOTVATON OF TUDY t makes me to be in state of insomnia on my vie to both men an omen that are infertile as a result of gonorrhea isease. Therefore, it gives me the enthusiasm to explore:. Biological control of the isease. Mechanism an meium of transmission 3. The mathematical moel that govern the transmission of gonorrhea in a given system. 4. Control measurement through mathematical moel rather than chemotherapy means.

15 6 nee, if all these strategies ill be aopte an put into practice, it ill eraicate gonorrhea..5 PREENT WORK/TUDY This ork eals ith gonorrhea isease in these perspectives:. The causes, symptoms, moe of transmission, complication, requirement for the test an treatments / controls of gonorrhea.. t encompasses the flo iagram hich schematically illustrate the reaction of agents involve the ynamics of the isease. 3. The mathematical moel that preicts quantity or behavior of the compartment involve in the transmission of the isease. 4. The veritable analysis that preict an control the ynamics of the isease.

16 7 CHAPTER TWO LTERATURE REVEW An aequate contact is a irect or inirect contact that is sufficient for transmission of infection if the iniviual contacte is susceptible. The concept of an aequate contact is necessary since transmission of infection sometimes oes not uring sexual intercourse beteen an infective an a susceptible, moreover, the probability of transmission from an infections man to a susceptible omen is greater than the probability hen the roles are reverse. f all people contacte ere susceptible, then the contact number oul be the average number of people infecte by one infective uring the infectious perio. The contact number for gonorrhea epens on the sexual behaviour of the population being consiere. The contact number has also been calle the reprouctive rate (DELZ, 975). ome omen are more likely to get PD than others. Women ho use intrauterine evices (UDs) are estimate to be about 3 times as likely to get PD from gonococal infection as those ho use no contraceptive measures. Women ho have multiple sex partners an omen ho have ha PD before are more likely to get PD. Women ho use oral contraceptives are less likely to get PD. Women ho use barrier methos of contraception

17 8 such as iaphragms, conoms, spermicial foams, jellies or creams, are also somehat protecte against PD (WHO, 978). Penicillase- proucing Nesseria gonorhoeae (PPNG) consists of strains of gonococci hich are resistant to all forms of pencillin. penicillinase is an enzyme that estroys penicillin. Although the stanar treatment for gonococcal infection is the aministration of penicillin. More expensive antibiotics are neee to cure PPNG, the treatment of gonorrhoea coul rapily become more ifficult an generally less effective so that incience rates coul increase. Control of PPNG transmission inclue testing some gonococcal isolate for PPNG, rapi ientification an tracing of sexual partners, of PPNG infection; an of the folloing ith spectinomycin (CDC, 98). Gonococcal infection oes not seem to confer immunity; more precisely, none of the immuniologic efense mechanisms has been shon to consistently prevent reinfection (NAD, 98). This may be ue to the great antigenic iversity of gonococcal surface antigens. n orer for an iniviual to become infecte the gonococci must attach themselves to mucosal surfaces so that they ill not be ashe aay by urine an mucosal flos. The gonococci have hair-like appenages calle Pili on their surfaces hich facilitate their attachment. Local antiboies to Neisseria gonorrhoeae have

18 9 been foun on the mucosal surfaces hich inhibit attachment of gonococci to epithelial cells; hoever there antiboies isappear after the gonococcal infection ens. Gonorrhea is an exception; most other irectly- transmitte iseases confer significant levels of immunity. ince gonococcal infection oes not appear to confer protective immunity of substantial resistance, acquire immunity cannot be a saturation factor for gonorrhea. An aequate contact of an infective ill not result in transmission of gonococcal infection only if the contacte iniviual is also an infective. This is calle preemption effect since an alreay infections iniviuals cannot be infecte by another infective strictly speaking the infecte iniviual coul acquire an aitional strain. Preemption seems to be the only possible saturation factor for gonorrhea since infection oes not confer immunity (Herbert. 984). (Ramakishore an N. Pattabhiramacharyulu, ) establishe the effect of cure rate an infective rate on the sprea an control of the iseases. For this they have solve a pair of non linear equations hich represents the groth rates of promiscuous population an infective by employing the rung-kutta metho of fourth orer. When F> an Po/k > here f is the specific cure rate, Po is the initial population an k is the carrying capacity. Then the cure rate is greater than infective rate an initial

19 population is greater than infective rate an initial population is greater than carrying capacity. n this case, e observe that the infective are ecrease for increasing values of cure rates. Here only healthy population oul survive. Also hen f > an P o /K < cure rate is grater than infective rate an initial population is les than carrying capacity. n this case e observe that infective are graually increasing to some extent for small cure rates an these are falling on hen cure rate is increasing. Gonococcus germ is sprea from person to person by sexual contact. A fe ays after the infection there is usually itching an burring of the genital area, particularly hile urinating. About the same time, ischarges evelops hich males ill notice, but hich females may not notice. nfecte omen may have no easily recognizable symptoms, even hile the isease oes substantial internal amage. Gonorrhea can only be cure by antibiotics (usually penicillin). Hoever, treatment must be given early if the isease is to be stoppe from oing serious amage to the boy. f untreate, gonorrhea can result in blinness, sterility, arthritis, heart failure an ultimately, eath (G. Prabhakararao 3). (Liana ana Meia- Rios et al 5) establishe that eucation affects behaviours that may fall into to categories- prevention an isease status aareness. t is sho that isease eucation has to be multifacete an

20 inclue reuctions in the average number of contacts an aareness that leas to testing. t also shon, in magnitue P(increase uration of asymptomatic infective) an (the portion of time the population spens in goo behaviour) are the next most sensitive eucation parameters. The analysis suggest that public health effort shoul be concentrate on encouraging people to practice safe sex more often an to be teste more frequently. These initiative oul be better aime at a reucing (contact rate).

21 CHAPTER THREE 3. WHAT GONORRHOEA? Gonorrhoea is the clinical isease resulting from infection ith the Gram-negative Diplococcus Neisseria gonorrheae. Gonorrhea is cause by bacteria hich are foun mainly in the semen an vaginal fluis of men an omen ho have the infection. The primary sites of the nfection are the mucous membrane of the urethra, enocervix, rectum, pharynx an conjunctiva. t can be painful an can cause serious health problems such as infertility in both men an omen. Neisseria gonorrhoeae may co-exisit ith other genital mucosal pathogens notably Tricholomas vaginalis, cania albicans an Chlamyia trachomatis. f symptoms are present, they may be attributable to the coexisting pathogen. 3. CLAFCATON OF NEERA A family of bacteria of hich three varieties causes isease: Neisseria meningitiis causes meningococcal meningitis an septicaemia. t is ivie into three groups: A, B an C; Group B account for most meningitis cases in the UK, mostly affecting chilren. The secon variety is Neisseria gonorrhoeae cause Gonorrhoea. The bacteria are gram-negative cocci usually occurring in pairs.

22 3 A thir variety is morax-ella catarrhalis. This occurs in the nose an throat an sometimes causes ear infection an lo-grae infection of the respiratory tract. 3.3 MODE OF TRANMON Gonorrhoea is usually passe from one person to another uring sex. The bacteria can live insie the cells of the cervix (entrance to the uterius), the urethra (tube here urine comes out), the rectum, the throat an occasionally the eyes. One can become infecte ith Gonorrhoea if one comes into contact ith ischarge from the vagina, throat, urine or rectum of an infecte person. The infection is most commonly sprea through: Unprotecte vaginal, anal or oral sex. haring sex toys if you on t ash them. Gonorrhoea can also be passe from a pregnant oman to her baby uring chil birth. This can cause inflammation an ischarge in the baby s eyes (cojuctivities). Gonorrhoea can be treate ith antibiotics hen one is pregnant or hen one is breast feeing.

23 4 n omen it is possible for the bacteria to sprea in the vaginal secretions to the rectum. one oesn t nee to have anal sex for this to happen. Gonorrhoea, bacteria that come into contact ith the eye can cause conjuctivities. This is uncommon in aults. One cannot catch gonorrhoea from kissing, hugging, sharing baths or toels, simming pool or from sharing cups, plates or cutlery. 3.4 YMPTOM OF GONORHOEA N MEN An unusual ischarge from the tips of the penis-the ischarge may be hite, yello or green. Pain hen passing urine Rarely pain or tenerness in the testicles Rectal infection in homosexual men is asymptomatic but may cause anal pain (7%) Pharyngeal infection is usually asymptomatic (>9%) 3.5 YMPTOM OF GENORRHOEA N WOMEN An unusual vaginal ischarge hich may be thin or atery, yello or green Loer abominal pain or tenerness Gonorrhoea is a rare cause of intermenstrual bleeing or menorrhagia.

24 5 Urethral infection may cause ysuria (%) but not frequently. nfection at the enocervix is frequently asymptomatic. Pharyngeal infection is usually asymptomatic (>9%) 3.6 COMPLCATON OF GONORRHOEA n men, untreate infections may ascen to involve prostate, seminal vesicles, epiiymis an testis. Neglecte cases may be complicate by chronic urethral stricture an in more avance cases, by permanent sterility. n omen, ascening infection, leas to acute inflammation of the fallopian tubes (alpingitis) an ovaries. t can also cause pelvic inflammatory isease (PD) a conition in hich the fallopian tube become scarre blocking the path of the seconary oocyte into the uterus. This ill lea to resultant infertility an an increase risk of ectopic pregnancy. Less common Gonorrhoea can cause inflammation of the joints an tenons an skin lesions. 3.7 TREATMENT/CONTROL OF GONORRHOEA Eucational programs either in clinics or in the meia might make the sexually active population more aare of the symptoms an seriousness of gonorrhoea so that people ho suspect that they might be infecte oul seek examination an treatment sooner.

25 6 CREENNG: The primary part of this program involves culture testing large numbers of omen for gonorrhoea. Although some of the culture tests are for patients ith symptoms ho have come to clinics, many of the tests are for omen ithout symptoms ho are having a gynecologic examination. Contact Tracing an ntervieing Contact investigation or tracing attempts to ientify contacts of knon infectives an to encourage contacts to be checke as soon as possible contact investigation for gonorrhoea sometimes consists of eucating knon infectives about the seriousness of gonorrhoea an asking them to encourage their contacts to be examine. Early ientification of infectives by contact tracing reuces their infectious perio an, consequently, can reuce the chance of transmitting the infection. Use conoms (male or female) every time you have vaginal or anal sex. f you have oral sex, use a conom to cover the penis, or a latex of polyurethane (soft plastic) square to cover the female genitals or male or female anus. Avoi sharing sex toys. f you o share them, ash them or cover them ith a ne conom before anyone else use them.

26 7 Gonorrhoea can be treate ith antibiotics such as ciprofloxacin, ofloxacin, ampilicilin, azithromycin, cefixime, ceftriaxone, spectinomycin, ampicilin. Antibiotic treatment is an essential control measure for gonorrhoea. The stanar treatment regimen shoul cure more than 95% of cases.

27 8 CHAPTER FOUR 4. NOTATON AND AUMPTON OF THE MODEL The population uner consieration is ivie into five compartments hich change ith time t. () is the susceptible class hich consists of those iniviual ho can contract the isease. () is the infective class hich consists of those ho are transmitting the isease to others. (E) is the expose class hich consists of those ho have contacte the isease but not yet infectious. n this class the isease is uner incubation an the iniviual cannot transmit the isease to others. ( T ) is the treate class hich consists of those ho have contacte the isease an been treate from the isease. t consists of the iniviual from the infective class an Expose class that is properly treate. (R) is the remove class hich consists of those ho are remove from the susceptible- infective interaction by recovery ith immunity, isolation or treatment. n this class, iniviuals recover ith temporary immunity so that they eventually become susceptible again. Π is the recruitment rate. t consists of the births an immigration is the natural eath rate is the contact rate is the rate hich expose become infectious

28 9 ŋ is the treatment rate of expose is the treatment rate of the infective is the eath rate inuce by the isease. ω is the rate at hich treate class become recovere δ is the rate hich the recovere become susceptible is the interaction of the susceptible an the infective E E E E Assume those recruite into system are not infective or contacte the isease. an is the control measure But < >, R >, n >, T >, >, E>.

29 4. THE FLOW DAGRAM Π δ R E ŋ ω T 4.3 THE MODEL FOR GONRROHOEA TRANMON E R T Π δ R ( ) ( )E ( δ )R T E ( ) n E ( ) T

30 4.4 THE CONTROL FLOW DAGRAM Π δ R (-) E ŋ() T () 4.5 THE MODEL FOR CONTROL TRANMON E Π ) δr ( ) E ( ) E E ( R WT δr R T ( ) E ( ) ( ) T ( ). E ( ) (

31 4.6 EQULBRUM ANALY Consiering the ifferential equation belo s π δr...() E E E E...() T E ( ) T...(3) R T ( δ ) R...(4) E ( )...(5) Here e oul like to kno hat oul happen to the isease, gonorrhoea in a long run, ill it ie out or ill it establish itself in the population an become enemic. Then to fin the equilibrium point to the equations. t suffices to equate each E R T equation to zero i.e 4.7 DEAE FREE EQULBRUM Here e consier a case hen there is no isease in the system. ince no isease in the system no boy is recovere. Therefore E,. T, n, R

32 3 o π δr o o o o o π π o o o o o π (, E, R ), O, O, O, O T n ENDEMC EQULBRUM Then to fin the equilibrium point to the equations, it suffices to equate each equation to zero from equation () π δr π δr ( ) π δr π δr * ( ) from equation () E E E E ( )E E ( )

33 4 From equation (3) T E ( ω) T O ( ω) E T E T ( ω) From equation (4) R T ( δ ) R O R * ( Tδ ) from equation 5 E ( ) O ( ) E E * ( ) BAC REPRODUCTVE NUMBER (RO) The basic reprouctive number, enote by R o is a parameter use to etermine ho long a isease ill prevail in a particular population an this etermine by the value of R o. f the basic reprouctive number Ro < it means that an iniviual prouces an average less than one infecte person, an by that calculation it means that ith time the isease ill ie out from

34 5 the population. But if R o >, it means that an infecte person prouces more than one infecte person in the population. o the isease ill surely remain in the population. o for the isease to ie out of the population ith time R o must be less than one. R o (FV - ) here (A) enotes the spectral raius of a matrix A an the spectral raius (FV - ), is the biggest nonnegative eigenvalue of the next generation matrix the square matrices F an V of orer (M M) here m is the number of infecte classes, efine by F i vi F ( x ) an ( x ) ith / j, j m xj xj such that f is nonnegative, V is a nonsingular m- matrix an X is the isease free equilibrium point (DFE). consiering E ( FE)E E ( ) T E ( ) T E,. T

35 6 F F i [ ] E E Y E V T i Y V The inverse of V Y V FV Y

36 7 To fin FV v or Consiering the estimate value of parameter from R. Rama kishore an N. ch. pattabhirama charulu an Herbert Hethcote 984

37 8 Parameters Values π.4... δ E 5 T R 5 R (...)(..) < /

38 9 consiering the control moel E T R FOR CONTROL MODEL ( ) ( ) π δr ( ) ( ( ) )E ( ( )) E ( ) ( ) E ( ) T ( δ )R T Consiering the infectious class V F i E,, T ( ) ( ) V i F ( ( ) ) E ( Y ( ) E ( ) [ ( ) ( ) T E ( ( ) ) ( ( )

39 3 V then ( ) ` Y Y to compute A FV here A the eigentalue an is ientit (3 x3) matrix

40 3 B A C A A A ( B A) A or A B therefore Ro ( ) ( ( ) )( ( ) )) π ( ) ( ( ) )( ( ) )).4..5(.8).(. (.8 )..) (.(.5 ).. ).4.4x < Therefore, equations 4.7,4.8,4.9,4.,4. are the respective equilibrium points of the moel. From equation 4.7 an increase in the recruitment rate (π) hich may be by birth or immigration into the system ill lea to increase of susceptible class an ecrease in eath rate an contact rate increases the population of susceptible class.

41 3 From equation 4.8 An increase in the interaction of susceptible an infective (contact rate) creates room for increase in the expose class an increase in rate at hich Expose become infective treate an infective non-create reuces the number of expose class. From equation 4.9 an increase in the rate at hich expose become infective treate increases the infective treate class an ecrease in the eath rate ill also increase ith population of this class. From equation 4. increase in recovery rate increases the remove class hich may be as a result of immunity or treatment an increase in epletion rate (δ) ill efinitely reuces the remove population. From equation 4. an increases in the rate at hich expose become infective associate ith ecrease in natural eath rate an isease inuce eath rate ill nee to increase in the class of infective. 4. TABLTY ANALY Consiering π δr E ( )E R ( δ R T )

42 33 E ( ) T T E ) ( etting R P δ π E Q R U T ) ( δ E V ) ( T E Z ) ( the Jacobian matrix K of the moel becomes T T T T T T Z Z R Z E Z Z V V R V E V V U U R U E U U Q Q R Q E Q Q P P R P E P P R E K ),,,, ( P 6 Where

43 34 Z Z R Z E Z Z V V R V E V V U U R U E U U Q Q R Q E Q Q P P R P E P P T T T T T δ δ,,,,,,,,,,,,,,,,,,,, The P 6 becomes R E K T n δ δ,,,, Then to fin the eigenvalues of the matrix above e have δ δ Then e have δ δ K

44 35 Therefore ) ( ) ( ) ( ) ( ) ) ) ( )( ) ( ( )( ) ( )( ) ( ( ) ( δ δ here,,,, an are the eigenvalues But >, >, n >, T >, R> 4. CONDTON NECEARY FOR TABLE OR UNTABLE YTEM f any of the, > j then the system is unstable f all the, < j then the system is stable for the fact that natural eath, isease inuce eath (), interaction of susceptible an infective (), rate at hich Recovere become susceptible (δ), Recovery rate () an treatment rate cannot be negative an >, > O, n > T >, R>. Therefore the system is stable. ince all the j are all negative ith respect to the conitions above.

45 36 Figure : This is a graph shoing Gonorrhea transmission moel ithout controls/treatments.

46 37 Figure : This shos the graph for Gonorrhea transmission moel ith controls/treatments.

47 38 CHAPTER FVE DCUON OF THE REULT The population moele is ith five compartmental states available: susceptible(s), expose (E), infective () infective treate ( T ) an Remove class(r) Equation () escribes the susceptible class hich preicts that increase in recruitment rate increases the population of susceptible class an ecrease in eath rate ith increase in Remove class is proportional to the susceptible class. Equation () says that Expose class is proportional to the susceptible an infective. When there is increase in the interaction of susceptible an infective ill generate more tenency for people to be in the expose class. Equation (3) inicates that Recovery class is proportional to infective treate class an inversely proportional to eath rate. From equation (4) states that infective increases ith increase in the expose class an ecreases ith increase eath rate (natural an isease inuce). From equation (5) says that infective is proportional to expose. ince, hen people are expose an they become infectious. From equation4. inicates the reprouctive number for gonorrhoea trainission. t

48 39 shos that on a long run iniviual ill not be able to transmit the isease to others since Ro <. From equation 4.3 is the reprouctive number for control moel hich is efinitely less than. By comparison, Ro from equation 4.3 is far less than Ro in equation 4.. This must be obtainable since application of control measure goes along reucing the transmission faster than hen there is control measure on the system. Equation 4.6 inicates the isease free equilibrium state. At this point susceptible increases ith increase in recruitment rate an ecreases ith increase in eath rate. The stability analysis states categorically that the system is stable. Then applying any control measures such as Eucation programme an use of conom one can preict vivily hat ill happen in the system. nee, applying any of the control measures can regulate the transmission of gonorrhoea. From Figure. The infective treate is very lo as result of no control measure is applie in the system hile in figure shos greater increase in infective treate as a result control measure introuce in the system. Therefore more people from expose class an infective class ill resort for treatments.

49 4 From figure the infective an expose class shos greater increase because there is no control measure but in figure. t reuces rastically as a result of control measure. Because, use of control measure such conom an eucational programme can enlighten on the oes of the isease an fe people are infecte.

50 4 UMMARY/ CONCLUON Application of mathematical moel on isease ynamics has become inispensable tools in preiction an analyzing of some iseases especially gonorrhea. This enables us to euce the orientation, Biology of the isease an mathematical relationship among the agents involve in the transmission.

51 4 RECOMMENDATON The oes of gonorrhoea in our society cannot be over emphasize. This has attribute to millions of people renere infertile an eath in some cases yearly. Therefore, to arrest this malay, calls for iniviuals, communities an Government intervention tremenously. Eucation programme an the use of conom shoul be consiere extensively in the control measures. Even other auxillary treatment can also be applie. Applying the mathematical moel an other measures above ill combat the transmission of gonorrhoea in our society toay.

52 43 REFERENCE Bailey N.T. J. (975). The mathematical theory of infectious iseases. Griffin, Lonon. Braun M. (973). Applie mathematical sciences, Vol. 5: Differential Equations an their applications. pringer, Ne York. Centers for Disease Control, 98a. Peicillinase-preucing neisseria gonorrhoeae-los Angeles, California, Morbility an mortality eekly Report 9, Centers for Disease Control, 7: CDC fact sheet: Gonorrhoea Cook K. L. an York J. A. (978). ome equations moeling groth processes an gonorrhea epiemics. Biomathematics 6: 75-. Cornelius, (97). easonality of gonorrhea in the limite state. H.. M. H.A health Rep. 86, Dietz, K., (975). Transmission an control of arbovirus iseases, in Epiemiology, M 974 Utah Conference proceeings. AM, Philaelphia, 4-. HethcooteH W. York. J.A (984). Lecturer notes in biomathematics, Vol. 56: Gonorrhea transmission ynamics an controls. pringer- Verlag, He ielberg. Hethcote HW. Yorke J.A., (979). Gonorrhea Transmission Dynamics control lectures notes in Biomathematics J.N. Kapur (). Mathematical moels in Biology, East-West press private limite. Jing FU. et al (5) Asymptotic behaviour of a multigroup epiemic moel ith stochastic perturbation:

53 44 National nstitute of Allergy an nfectious Disease (98). exually transmitte iseases: 98 status report, NH Publication No Paul ine van en Driessche an James aemough, (): reprouctive numbers an sub- threshol enemic equilibria for compartmental moels of isease transmission, mathematical biosciences 8, Prahbakararao, (3). Mathematical moeling of gonorrhea isease a case stuy ith reference to Anantapur District- Anhrapraesh- nia. R. Rama Kishore an N. Ch. Pattabhiramacharylulu (). A numerical approach for the sprea of gonorrhea in homosexuals. Vol. 6. rinivas N.C. (99). ome Mathematical aspects of Moeling of Bio- Meical ciences. PhD Thesis. Kakateya University. rinivas N.C., Pattabhiramacharyulu N.Ch. 99. ome mathematical aspects of sprea an stability of time elay gonorrhea. Difference science journal. 4: Worl health organization, (978). Neisseria gonorrhea an gonococcal nfections, technical Report series 66, Geneva.