LIKE-WITH-LIKE PREFERENCE AND SEXUAL MIXING MODELS

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1 1 LIKE-WITH-LIKE PREFERENCE AND SEXUAL MIXING MODELS by Stephen P. Blythel and Calos Castillo-Chavez2,3 BU-982-M ABSTRACT August 9, 1988 We pesent two new geneal methods fo incopoating like-with-like pefeence into onesex mixing models in epidemiology. The fist is a genealization of the Sattenspiel mixing equation, while the second compises a tansfomation of a geneal pefeence function fo patnes of simila sexual activity levels. Both methods satisfy the constaints implicit in a mixing model. We then illustate how the tansfomation pefeence method behaves and compae it with the standad popotionate mixing model. INTRODUCTION In models of the dynamics of sexually tansmitted diseases (STDs) within populations with heteogeneous sexual activity, it is necessay to specify the contact pefeence (who mixes with whom). Thus, fo each level of sexual activity (numbe of new patnes pe unit time) we must know the faction of patnes coming fom all othe levels of activity. Fo pactical modeling puposes, we equie some function of activity which both makes analysis staightfowad and is a easonably accuate chaacteization of obseved mixing pattens. Until ecently the popotionate mixing model- equation (1) below-was the only desciption of the mixing pocess available in analytic fom, although abitay ules may, of couse, be applied in stochastic simulations of the inteaction of 1 Depatment of Physics and Applied Physics, Univesity of Stathclyde, John Andeson Building, 107 Rottenow, Glasgow, G4 ONG Scotland 2 Biometics Unit, Cente fo Applied Mathematics, Depatment of City and Regional Planning, 341 Waen Hall, Conell Univesity 3 To whom coespondance should be addessed

2 2 individuals. Popotionate mixing has been used extensively in a vaiety of situations by Babou (1978), Nold (1980), Andeson and May (1984), Dietz and Schenzle (1985), Andeson and Genfell (1986), Hethcote and Van Ak (1987), Castillo-Chavez et al. (1988, and in pess). While this model has also poved useful in the study of the epidemiology of STDs (see Hethcote and Yoke, 1984, fo an outstanding example), it has become clea in ecent yeas that moe ealistic mixing models ae equied fo any detailed undestanding of the tansmission dynamics of IllY -1 and fo any study of the possible value of vaious contol measues. At pesent thee ae few obust data on contact pefeence in a given community, although estimates of the distibution of numbes of sexual patnes have been deived fom vaious suveys. In the absence of detailed mixing infomation, models of HIV -1 tansmission must take into account as many mixing pattens as possible in ode that the impact of any pefeential mixing on the AIDS epidemic in a given population be bette undestood. Recently Sattenspiel (1987a and 1987b) questioned the use of popotionate mixing in dynamic models fo the spead of diseases in stuctued populations. She emphasizes in those diseases fo which the geogaphic and social stuctue plays an impotant ole. Since then he ideas have been expanded by he and he collaboatos (see Sattenspiel and Simon 1988, Sattenspiel et al. (ms.), and Jacquez et al.(in pess)). Sattenspiel et al. (ms.) and Jaquez et al. (in pess) have pesented a new mixing model which has a esticted fom of like-with-like pefeence (individuals have a bias towads othes of the same activity level) supeimposed upon a popotionate mixing backgound. Stanley and Hyman (in pess) have examined some appoximations to like-with-like mixing, and Stanley (pesonal communication) has developed a model whee the pefeence of half of the population may be specified by the modele, with the othe half being defined by the pefeences of the

3 3 fist. All of these appoaches epesent impotant and valuable contibutions to the study of STD epidemiology. In this pape we seek to extend the ange of mixing models available to the modele by intoducing two new foms which satisfy the necessay containts: (1) genealized Sattenspiel mixing, and (2) neighbohood mixing. MIXING FUNCTIONS In any one-sex model with heteogeneous sexual activity we have the mixing function p(s, ), which specifies the faction of s patnes among individuals with activity. Thee ae thee constaints which p(s, ) must satisfy fo all s and : (i) p (s, ) ;::: 0, (ii) J p(s,)d = J p(,s)ds = 1, 0 0 (iii) p (s, ) s N(s) = p (, s) N(), 00 whee N(x) is the numbe of people in the population with activity x (this is of couse a function of time. Howeve, we have suppessed time notation as (i - iii) must be tue at all times. Conditions (i) and (ii) aise because p(s,) is in effect a pobability density function, while condition (iii) expesses the equiement that the total numbe of patneships of s people with -people must equal the total numbe of patneships of -people with s-people. These constaints ae simple and obvious, but it is extaodinaily difficult to fmd functional foms fo p(s,) which satisfy them simultaneously fo all s,, and timet. We expess the standad mixing model fo popotionate mixing as

4 4 N() p (s, ) = J u N(u) du 0 ( 1 ) Hee p(s,) is actually independent of s, and may be intepeted as saying that the faction of patnes taken by any individual in the population fom individuals with activity is popotionate to the total numbe of patneships fomed by all -people, and clealy satisfies (i)-(iii). The Sattenspiel mixing function is an extension of equation (1) to include a pefeence of individuals fo patnes with exactly the same activity level. In the continuous vaiables and s used hee, he function is N() p (s, ) = (1 - a) ---- J u N(u) du 0 + a o(s- ) ' (2) whee o(s-) is a Diac delta function and the constant a epesents the bias towads patnes of exactly the same activity. Although vey useful fo modeling puposes, and sufficient to demonstate that even a small bias towads like-with-like can have a pofound effect on epidemiological pattens, equation (2) is athe esticted as a geneal model of pefeence. A moe geneal altenative to popotionate mixing has been deived by Stanley (pesonal communication), and takes the fom

5 5 N() p (, s) s N(s), <s p (s, ) = _s_f_(_s,...;) N_(;...)_ (1 - J p (s, u) du)' s >s ( 3) J f(s, u) u N(u) du 0 0 whee p(,s) fo <sis abitaily specified by the modele to suit available data, and the est of the values ae deived fom this constaint. The function f(s,) appeas to be abitay, and may be used to fine-tune the behaviou of p(s,) to the modele's needs. It may be shown that equation (3) satisfies (i)- (iii). This geneal mixing function is potentially of geat value in modeling studies. We now intoduce two new mixing functions which satisfy the constaints (i)- (iii). GENERALIZED SATTENSPIEL MIXING The fist mixing model is a diect genealization of Sattenspiel's additive equation (2) to allow pefeences fo patnes with activities which ae abitay multiples of one's own. This takes the fom m N() - La i ai N(a i ) j u N(u) du ; ~I ' a ; i=l ~ ~ s p(s, ) = i a. u (--) 0 m (4) Hee thee ae m delta-functions with weights {a i}, descibing the pefeence of individuals with activity s fo individuals with activity s/al' s/~,..., s/am.

6 6 Fo like-with-like pefeence we might have a 1 = 1, and the othe ai aanged as multiples and factions of unity, with the weights {a) at a maximum foi= 1 and deceasing as the ai get lage o smalle than unity. It is elatively easy to show that equation ( 4) satisfies (i) - (iii) povided that the sum of the weights, I ai m, is lage enough (geate than MAX {a ia ix N(x)} is sufficient). It may in pactice be a seious deficiency that thee ae "gaps" in the pefeence function between the abitaily chosen positions of the delta-functions. Nonetheless equation (4) may be useful fo peliminay investigations of a like-with-like pefeence distibuted aound s =. A NEIGHBORHOOD MIXING FUNCTION Instead of the delta-function model of equation (4), we should like to be fee to specify like-with-like pefeence by some abitay function with well-undestood popeties. In paticula we wish to use "neighbohood" functions which expess pefeence as a continuous function with a single peak at = s, falling off to eithe side. We know of no such functions which may be used diectly, satisfying (i)- (iii). Even an isolated delta-function equies some tansfomation, the simplest example of which is ( ) N() ~( _ s) p s, = s N(s) This example povides a clue as to how one might make use of an abitay function, say <j>(s,), as ou pefeence function. We must ask: "What tansfomation of the function

7 7 <l>(s,) satisfies (i)- (iii)?" If we estict ou choice of <1> to functions with the popety <l>(s-) = <l>(-s), and state that +co J <I> (y) dy = 1 ' then we fmd that the tansfomation p(s, ) = : N() P() P(s) + ~() <l>(s _ ), J u N(u) P(u) du 0 ( 5) satisfies (i)-(iii). In (5), P(x) = 1 - ~ J u N(u) <l>(x - u) du, ( 6) 0 and A is a constant. We conside this constant futhe below. It is tivial to show that equation (5) satisfies constaints (ii) and (iii); the value of A must be lage enough to give P(x) > 0 fo all x, which in tun is sufficient to satisfy (i). When <l>(s-) is not a deltafunction at s =, then the choice 00 A = J u N(u) du ( 7) 0 is sufficient; fo a delta-function P(x) involves point values athe than integals, and A > MAX{xN(x)} is necessay and sufficient.

8 8 AN EXAMPLE In this section we conside a simple example fo which p(s, ) can be calculated. We ae not hee concened with a time-vaying activity distibution (which would be the case in a eal application o a dynamic model), and choose the convenient exponential fom -ks N(s) = Nke, ( 8 ) whee N(s) is the distibution of sexual activity in the population, N is the total population size, and 1/k: is the mean sexual activity. Fo the neighbohood pefeence functions <j>(s, ), we choose C - c Is- l $:s, ) = I e ( 9) which becomes moe shaply peaked as c inceases. Using equation (7) we have A= N/k: fo this case. It is tivial to calculate the expession fo P(x) and p(s,) given equations (8) and (9), and in Figues (1) to (10) we pesent some illustative examples. In the figues, we have gaphed p(s,) as a function of fo diffeent values of s and fo a vaiety of values of c and k, with A = N/k:. In Figs (1) to (3) we illustate p(s,) fok= 0.1 and c = 0.5, and s = 1.0, 5.0, and10.0 espectively. In this case p(s,) etains the shaply peaked fom of <j>(s,) except when s is small, in which case p(s,) is much smoothe. This case coesponds to a vey naow neighbohood function, with 50% of the aea unde <j>(s,) lying in the inteval = s ± 2ln2, and a lage aveage activity: 1/k: = 10.0 patnes pe unit time. In Figs (4) to (6) we illustate p(s,) fok= 0.5 and c = 0.1, with the same ange of s values. In this case the neighbohood function is vey boad, and contibutes vey little to the shape of p(s,), which always behaves as N() (that is, like popotionate mixing).

9 9 In Figs (7) to (10) we illustate the case k = 0.25 and c = 1.0 fo s = 1.0, , and 20.0, espectively. Although hee the neighbohood function is naow, the mean sexual activity is small and the inteplay between N() and p(s,) is complicated. The essential fom of p(s,) is a mixtue of popotionate and like-with-like mixing. At a small s (less than 1/k, Fig (7)), p(s,) is vey much like <j>(s,), but with a moe ponounced tail. Ass is inceases (Figs (8) to (10)), the component due to <j>(s,) deceases, until by the time s = 20.0 popotionate mixing is pedominant. We emak that the fidelity of the tansfomation p(s,) to the undelying neighbohood function <j>(s,), given equations (8) and (9), depends upon the width of <j>, the mean activty 1/k, and the value of s in elation to 1/k. CONCLUSION We have pesented two new like-with-like mixing functions, one based on popotionate mixing biased at m values of the atio s/, and the othe based on a tansfomation of a geneal neighbohood function <j>(s,). A simple example fo a static population indicates that the second mixing function behaves like the neighbohood function, povided that the latte is shaply peaked and the mean activity in the population is elatively high. In othe cases popotionate mixing may be egained, with o without a level of bias towads like-with-like pefeence. These esults suppot some of the numeical expeiments of Hyman and Stanley (1988, and in pess) egading the ole of the width (vaiance) of the neighbohood pefeence function and its elationship to popotionate mixing. Much wok emains to be pefomed befoe we have a complete undestanding of the tansfomation method fo an abitay neighbohood function, and the behavio of this p(s,) in a fully dynamic epidemiological model must be investigated. This wok is in

10 10 pogess and we hope to epot on it in futue publications. Finally, we speculate that if estimates fo N(s) (the activity distibution in the population) and <j>(s,) (tendency fo likewith-like mixing) can be obtained fom suvey esults, then examination of the tansfomation p(s,) of equation (5) may be able to tell us whethe o not the like-with-like pefeence is impotant in a given population, and thus whethe a popotionate mixing desciption is adequate, o a moe complicated model is equied. Acknowledgments The authos expess thei appeciation to J. M. Hyman, E. A. Stanley, and S. A. Colgate fo thei stimulating convesations and L. Sattenspiel fo he povocative ideas.

11 11 Refeences Andeson, R.M. and May, R.M.: Spatial, tempoal, and genetic heteogeneity in host populations and the design of immunization pogammes. IMA J. of Math. Applied in Med. & Biol. 1, (1984). Andeson, R. M. and Genfell, B. T.: Quantitative investigations of diffeent vaccination policies fo the contol of congenital ubella syndome (CRS) in the United Kindom. J. Hyg., Camb. 96: (1986). Babou, A.D.: MacDonald's model and the tansmission of bilhazia. Tans. Roy. Soc. Top. Med. Hyg. 72, 6-15 (1978). Castille-Chavez, C., Hethcote, H., Andeasen, V., Levin, S. A., and Liu, W-m. Cossimmunity in the dynamics of homogeneous and heteogeneous populations. In: (T. G. Hallam, L. G. Goss, and S. A. Levin, eds.)mathematical Ecology. pp Wold Scientific Publishing Co., Singapoe (1988). Castille-Chavez, C., Hethcote, H.W., Andeasen, V., Levin, S.A., and Liu, W-m.: Epidemiological models with age stuctue, popotionate mixing, and cossimmunity. J. Math Biology (in pess). Dietz, K. and Schenzle, D.: Popotionate mixing models fo age-dependent infection tansmission. J. Math. Biol. 22, (1985). Hethcote, H.W. and Yoke, J.A.: Gonohea, tansmission dynamics, and contol. Lectue Notes in Biomathematics 56, Spinge-Velag, Belin, Heidelbeg, New Yok, Tokyo (1984). Hethcote, H.W., and Van Ak, J.W.: Epidemiological models fo heteogeneous populations: popotionate mixing, paamete estimation and immunization pogams. Math. Biosci. 84: (1987). Hyman J. M. and Stanley E. A.: Using mathematical models to undestand the AIDS epidemic. Math. Biosci. (1988). Hyman J. M. and Stanley E. A.: The effects of social mixing pattens on the spead of AIDS. In:(C. Castille-Chavez, S. A. Levin, and C. Shoemake, eds.).mathematical Appoaches to Ecological and Envionmental Poblem Solving Lectue Notes in Biomathematics, Spinge-Velag (in pess). Jaquez, J. A., Simon, C. P., Koopman, J., Sattenspiel, L., and Pey T.: Modeling and analizing IllV tansmission: the effect of contact pattens. Math. Biosci. (in pess). McLean, A. R. and R. M. Andeson.: Measles in developing counties. Pat I. Epidemiological paametes and pattens. Epidem. Inf. 100: (1988). Nold, A.: Heteogeneity in diseases-tansmission modeling. Math. Biosci. 52, (1980). Sattenspiel, L.: Population stuctue and the spead of disease. Human Biology. 59: (1987).

12 12 Sattenspiel, L.: Epidemics in nonandomly mixing populations: a simulation. Ameican Jounal of Physical Anthopology. 73: (1987). Sattenspiel, L.and Simon, C.P.: The spead and pesistence of infectious diseases in stuctued populations. Math. Biosci. (1988, in pess). Sattenspiel, L., Koopman, J., Simon, C. P., and Jacquez, J.: The effects of population stuctue on the spead of IllY infection (ms.).

13 13 FIGURE CAPTIONS Fig 1. Behavio of (a) <J>(s,) and (b) p(s,) fok= 0.1, c = 0.5, and s = 1.0. Fig 2. Behavio of (a) <J>(s,) and (b) p(s,) fok= 0.1, c = 0.5, and s = 5.0. Fig 3. Behavio of (a) <j>(s,) and (b) p(s,) fok= 0.1, c = 0.5, and s = Fig 4. Behavio of (a) <J>(s,) and (b) p(s,) fok= 0.5, c = 0.1, and s = 1.0. Fig 5. Behavio of (a) <j>(s,) and (b) p(s,) fok= 0.5, c = 0.1, and s = 5.0. Fig 6. Behavio of (a) <j>(s,) and (b) p(s,) fok= 0.5, c = 0.1, and s = Fig 7. Behavio of (a) <J>(s,) and (b) p(s,) fok= 0.25, c = 1.0 and s = 1.0. Fig 8. Behavio of (a) <J>(s,) and (b) p(s,) fok= 0.25, c = 1.0, and s = 5.0. Fig 9. Behavio of (a) <j>(s,) and (b) p(s,) fok= 0.25, c = 1.0, and s = Fig 10. Behavio of (a) <J>(s,) and (b) p(s,) fok= 0.25, c = 1.0, and s = 10.0.

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15 I " L '~ ~.J <1> p

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17 <l> p \.\

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20 ( -, ) <I> p

21 p \

22 <\> 0.5 ( ) p I \ ' : -~

23 p (!:)

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