CHAPTER 3 A REFINED MODEL FOR GONORRHEA DYNAMICS

CHAPTER 3 A REFINED MODEL FOR GONORRHEA DYNAMICS he o~.t_at _o wh_c,' eeds to be described b a T.odei for the t.rasmissio of goorrhea cosists of those _sexually active people who could he ifected by their cat acts. The model i Chapter 2 assumes that th is r u at i ~ is i ge? ius ad a i four ; however, t''1 At is too limp Le sice the populatio is really quite heterogeeous..,..ita'ble model should allow for heterogeeity b icororatir marks fgrous. The divisio ito groi.ps could be doe accord igg to A differ-,-eces i sex, sexual cotact rates, sexual behavior, age, geagriphic locat io, so ioeco m status, etc. :or exa 1~1P, so Tel '.'I duais are more active sexually tha others i the sese that they have more f.reeset chages of sex parters. Come ifected people, eseeial.l : ~ me, 3.re ess et _ ~y a :3.V'ptomat is -ed do ot seek trey met,,hi~ others have,: ymptoms wh ieh cause them to seek treatmet.? sec r, io. 7. 1 we deveio e mo? e,' for a roorulat io d `_v ided ito grow s or silbpopu -i s. We show that either the disease dies out aturally for all possible iitial levels or the disease remais edemic for all future time. " ' oreover, the umbers, o` irfectfiver ad suscept.ibles i each group approach ozero costat Level,,.-,, which are idepedet 1 iitial ieve i s. he effects of chages i the par arieter values (corres pod i sr o ep.idem _'Log ical chages) o a disease ca be determied by examiig -The result i.g chages i the edemic eau i ibriur: level,-. A method of determiig the cotact proportioate mixig assumptio is rate, amog, groups by usig a described i sectio 3.2. With this ssum t io the threshold q,,.at i7 y wh Lch determies whether The disease dies Out ur remais edemic is a average cotact umber. "od 'i 'sits differet grours are cosidered i subseuet charterss. 3.1 A Goorrhea Model with Groups t Assume that the populatio is divided ito groups ed let i be he s ize of the suhpopulu.t io iri group i. We assume that each,~rou s hoioc:e o._1 ; i the sese that ale _I iv idua, the gro'.1c, are similar. They should have the rates ofcotaa.ct with ew,sexual parters, the same mea dt.ratios; of ifect io ad the same likelihood 1). ace'.i firig Lri.-ect io Cl 1r.ig a sexual ecouter w it a Ifect, ious parter. We assume that fid ividua.f_s are either susceptible or ifer?.ios ad that, ifectious idividua -ls i a rasp have the. ame

26 sexual behaviour ad activity levels as susceptibies. Let I 1 (t) deote the prevalece i group i at time t so that the susceptible fractio i group i is 1--I i (t). We measure time t i days. Let a i j be the average umber of adequate cotacts (i.e., cotacts sufficiet for trasmissio) per uit time (oe day) of a ifective i group j with persos i group i. Sice the susceptible fractio I group i is 1--I i(t), the average umber of suscetibies i group i ifected per uit time by a ifective i group j is a ij (1- I i (t)) ad -the average umber ifected per uit time by N j t ifectives is aij.nitj(1-ti(t)). Let d i be the mea duratio of ifectio i days for a perso i group i. As i Chapter 2, we assume that each ifective i group i has a fixed chace of recoverig each day ad that the probability is 1/d i. Thus the removal rate per day from the ifectious class is rl i T i /d i. As oted i sectio 2.1, that this is equivalet to assumig that the duratios of ifectio i group i have a egative expoetial distributio (Fethcote ad Tudor, 1980). The differetial equatios for the model are d (N N. I. /d, dt i j=1 with iitial coditios I i (O) = I io for i=1,2,...,. The first term i each differetial equatio is the rate of ew ifectios or icidece i group 1 ad the secod term is the removal rate due to recovery. Figure 3.1 shows the susceptible ad ifecttive compartmets ad the trasfer rates betwee compartmets. group 1 susceptibles (~ a j N j I i )(1-I 1 ) _1 N 1 I 1 group 1 ifectives I Xj N j T j )(1 -T } group 1 susceptibles NI / [i I group ifectives Figure 3.1 Flow diagram for the model [3.1]

Lajmaovich ad Yorke (1976) proved that the model [3.1] is we -,-' osed. That is, uique solutios of [3.1] exist for all time, deped cotiuously o the iitial data, ad are always betwee -0 ad 1. The.x coefficiet matrix A i the liearizatio of [3-11 is give by A = L-D where L = [ a i jid j ] ad D is a diagoal matrix with Ni,/d, as the etry i the ith row ad colum. Let s(a) be the stability modulu.s of A, i.e., the maximum real part of the eigevalues of A. They proved the followig theorem. THE01RHM 3.1. Assume that the model is irreducible, that is, the populatio caot be split ito two subpopulat ios that do ot cotact each other. The solutios of [3.1 ] approach the eauii ibr ium poit at the origi if s(a)40 ad they approach a uique positive equilibrium poit if s(a)>o, provided there is some ifectio i some group iitially. Thus goorrhea will die out if the parameter values are such that s(a)<0 ad will approach a edemic steady state if s(a)>o. Oe practical implicatio of the theorem above is that it allows us to focus o the positive equilibrium poit ad to see how it chages whe parameter values chage or whe cotrol procedures are added. Let P i >0 be the equilibrium prevalece (the fractio of group i that is ifectious at equilibrium). Thus the E i are the solutios of the simultaeous quadratic equatios obtaied whe the right sides of [3.1] are set equal to zero. From the quadratic equatios, the equilibrium icidece i group i is equal to the equilibrium prevalece E i times the group size ii i divided by the mea duratio d i. Figure 3.2 shows the typical behavior of solutio paths as they approach a edemic equilibrium poit. Oe of the strikig features of Theorem 3.1 is the qualitative dyamical coclusio that equatios [3.11 have a uique equilibriu- poit, either strictly positive or zero, which is the limit of every solutio startig out from a state where ifectio is preset. Hirsch (1984) has show that this coclusio also holds for a geeralizatio of equatios [3.11. I his differetial equatios, the icidece ad removal terms are give by fuctios which satisfy certai coditios. His model is so geeral that it is ot possible to give a procedure for decidig whether the equilibrium poit correspods to a edemic steady state or to die out of the disease. However, the geerality of his model strogly suggests that ay observed fluctuatios i the icidece are ot due to the itrisic dyamics of the disease so that they must be due to fluctuatios i 27

28 1.0 4.6 fective Fractio I Figure 3.2. Solutio paths approachig the edemic equilibrium whe s(a) > 0. 1.0 poit

29 epidemiolog.ica or evirometal factors or i reportig. 3.2 Proportioate Mixig Amog Groups The cotact rates A i j i the cotact matrix ca be determied methodically by usig some assumptios regardig the iteractios of the groups. The "proport;ioate mixig" approach explaied i Nold (1980) assumes that the umber of adequate cotacts betwee two groups is proportioal to the relative sexual activities of the two groups. A ecouter will refer to oe or more episodes of sexual itercourse with a ew parter. For example, if group 1 has 10% of all ecouters ad group 2 has 40% of all ecouters, the i a proportioate mixig model, the fractio of all ecouters which are betwee groups 1 ad 2 is.10 x.40. The frequecy of ecouters is a better measure of sexual activity that is likely to trasmit ifectio tha the frequecy of sexual itercourse, sice ecouters are ew to become ifected or to trasmit the ifectio. umber is the Let aj be the activity level of group j, of ecouters of a perso i group j per average time betwee ecouters for a perso be the probability that a ifective i group q j j ifectio durig a ecouter with a susceptible, i.e., opportuities which is the average uit time. Thus 1/al i group j. Let trasmits the that there is a adequate cotact. Let m ij be the fractio of ecouters made by a average ifective of group j with persos i group i. Notice that the sum of each colum i the mixig matrix M is 1. From these defiitios it follows that the average umber of adequate cotacts per uit time of a ifective i group j with differet parters i group i is a = am j q ij j i j The average umber of ecouters per uit time is A = a.n The fractioal activity level of group i defied by i=1 b i = a i N i /A is a measure of the relative sexual activity of group i. Notice that b i = 1. The proportioate mixig assumptio is i=l that the ecouters of a perso are distributed i proportio to the fractioal activity levels, i.e., m ij = b i. The cotact umber k j for group j, which is the umber of adequate cotacts made by a typical ifective i group j durig the duratio of ifectio, satisfies k j - g j a j d j. If tij is the umber of adequate cotacts with group i of a group j ifect ive durig a average case, the t ij a ij d j - aim. g j d j -- m ij k j. The x matrix T W [t ij ] is called the trasmissio matrix. I the proportioate mixig model, = bikj

brium if W>1, provided there iitially. i s 30 The average cotact umber for this model with proportioate mixig is K = b i k i, which is the weighted average of the cotact i=1 umbers of the groups with the fractioal activity levels used as weights. It is the average umber of persos cotacted by a average ifective durig the ifectious period. We ow prove that this average cotact umber is a threshold parameter which determies whether goorrhea dies out (741) or remais edemic (K>1). The characteristic equatio for the trasmissio matrix T is det(t-ai) _ (-1) a -1 (a-k) = 0. We assume below that T is irredu- cble, i which cible, agai meas that the whole populatio caot be split ito two subpopulatios which do ot iteract with each other. The lemmas below are from Nold (1980). LEMMA 3.2. If T is a square matrix with oegative elemets, the T has a real, simple eigevalue p(t), called the Perro eigevalue, which is equall to its spectral radius. LEMMA 3.3. The outbreak eigevalue m o = s(a) for [3.1] has the same sig as r(t)-1 where r(t) is the spectral radius of T. THEOREM 3.4. I the proportioate mixig model the solutios of [3.1] approach the origi if 7<1 ad they approach a uique positive PROOF. some equili-- From the characteristic equatio ad Temma 3.2, the Perro eigevalue p(t) = K is equal to the spectral radius r(t). By Lemma 3.3, r(t) = K<1 is equivalet to the outbreak eigevalue satisfyig m V = s(a)<0. The theorem ow follows from Theorem 3.1. 1 ifectio i some group We ow develop some relatioships that will be useful i -Later chapters. Usig several defiitios above, a algebraic maipulatio leads to a ij N./N i = (k i /g i d i )b j g j so that [3.11 becomes di. k.(1-i) i dt t - ( I1 b j g j l j ) l q. d. j= I [3.21 for i = 1,2,...,. This is a coveiet values appearig are ofte available. The edemic equilibrium trevaleces I i right sides of [3.2] equal to solutios of form sice the parameter are foud by settig the zero so they are the otrivial

31 h j g j E j)kq i = E :i [3.3] for i = 1,2,...,. Defie the _average_ equilibrium ifectivity_ h by h = j ~ 1 b j g j h j. [3.4] The fractioal ifectivity of group j defied by = bjgjfj/h [3.5] measures the relative ability of group j to trasmit the if.ect :io. From [3.3] ad [3-41 we fid that the edemic equilibrium prevaleces E i must satisfy I? i = hk i /(g 1 +hk i ) [3.61 The equatios [3.4] ad [3.61 yield g t b i k i /(g i +hk.) = 1 [3.71 which is equivalet to a th degree polyomial for h. For example if =2, the the quadratic equatio is k 1 k2 h 2 [ q k + q 1 k 2 - (q 1 b 1 +q 2 b 2 ) k1 k 2 ] h - qi q 2 R-1) = 0. [3.81 The edemic prevaleces are foud of [3.81. Sice the icidece i group from i is N i [3.6] usig the positive root h times the summatio terms i [3.21, the total icidece of the populatio per year divided by the populatio size (i.e., the umber of cases per perso per year) is Y = 365[ Ni(j 1~ ~ i )k i 0-)k0 -F i )/g i d i ] i =1 = [3.9] Usig [3.31 the umber of cases per perso per year satisfies Y = 365 ~ N i F i Id. -1 i x3.10]