0277s9536/91
Sm. Sci. Med. Vol. 33, No. 10, pp. 11X--1162, 1991
$3.00 + 0.00
Pergamon Press plc
Printed in Great Britain
SOCIAL DISINTEGRATION AND THE SPREAD OF AIDS: THRESHOLDS FOR PROPAGATION ALONG ‘SOCIOGEOGRAPHIC’ NETWORKS RODRICK WALLACE Epidemiology of Mental Disorders Research Department, New York State Psychiatric Institute, 722 W. 168 St, New York, NY 10032, U.S.A. Abstract-Previous work on the asymptotic spread of HIV infection along a low dimensional ‘sociogeographic’ network-a social network characteristically embedded within a limited geographic area-is extended to explore threshold conditions under which the infection extends widely beyond an initial set of infected individuals or communities. Results for one dimension suggest that threshold behavior is analogous to a chain reaction with criticality determined conjointly by the susceptibility of individuals within a community to a nexus of behavior conducive to rapid HIV spread and by the probability of transmission between susceptible communities. Once threshold is exceeded, a stochastic reformulation finds the asymptotic rate of transmission between communities may be markedly raised by positive correlation between susceptibility to rapid disease spread within a community and the transmissibility between communities, for example outmigration driven by social disintegration or residential instability arising from inherent structural factors associated with community susceptibility, as with male prostitution. Examination of threshold conditions for higher dimensional sociogeographic networks most likely characteristic of disease spread beyond the ‘deep ghettos’ now suffering the highest burden of infection suggests it is at least as, and likely more, effective to decrease the fraction of population susceptible to the high risk behavioral nexus as it is to lower the probability of disease transmission between susceptible individuals or communities. Thus, while purely ‘medical’ or ‘educational’ interventions may influence probabilities of transmission, socioeconomic and political structures and policies may strongly affect degrees of social disintegration or other factors determining both the fraction of a population engaging in a nexus of high risk behavior within a community and patterns of outmigration or other residential instabilities affecting the rate of spread between communities. This suggests the necessity of interventions to stabilize socially disintegrated or other affected communities, in addition to programs aimed only at decreasing the probability of disease transmission between individuals. Key words-AIDS
epidemiology, social disintegration,
sociogeographic networks, threshold conditions,
stochastic models
INTRODIJC’I’ION HIV transmission depends heavily on local customs and forms of sexual activity and the sex industry, and in some circumstances of substance use and abuse, all of which are deeply embedded in underlying socioeconomic and political structures. A long asymptomatic but infectious latent period for expression of the disease as AIDS makes almost inevitable a hidden but explosively rapid and comprehensive spread of the virus within geographicalIy concentrated populations engaging in a nexus of high risk behaviors. Potterat ef ai. [I] observed a particularly close intertwining of spatial and social patterns of gonorrhea endemicity in a minority population within a small American city, and R. Wallace [2] has adapted this view to model deterministically the asymptotic transmission of HIV as a new disease introduced on a possibly non integral dimensional ‘sociogeographic network,’ a ‘Potterat structure’ of a social network focused within a local geographic area. This picture may apply more generally to the social structures of oppressed groups within the United States, particularly when combined with rapid social
change. For example gay males, driven as ‘sexual migrants’ [3] from their communities of origin into urban ghettoes, may then experience great intensification of within-group interactions, including sexual practices effective in spreading HIV. The gay bathhouse phenomenon comes to mind as such a highly geographically concentrated interaction. Similarly, intravenous drug abusers in disintegrating urban centers, usually an oppressed minority within an oppressed minority, may respond to rapid social change by intensification of needle-sharing behavior, particularly within the context of geographically centered ‘shooting galleries’ where drugs are sold and ‘works’ rented. The recent ‘crack house’ phenomenon, a central point for the smoking of crack cocaine, and for the exchange of sex for drugs, would seem to provide yet another example. Finally, a recent study of male prostitutes in New Orleans [4] finds their activity not only to be geographically concentrated within the city, but to be highly transient, with many male prostitutes routinely traveling between ‘red light’ districts of major urban centers with high levels of HIV infection,
1155
1156
M. Fullilove follows:
RODRICK WALLACE
[5] has summarized
this pattern
as
rapid social change acting on groups marginalized by the larger society can create an intensification of risk behavior within the group. A particular and critical aspect of the changing group interaction is that previously isolated social networks become behaviorally interlinked within the group. It can be hypothesized that this subgroup interlinking occurred in the case of distinct social groups of gay men meeting in the bathhouse, of intravenous drug users meeting in the shooting gallery, and of crack users meeting in the crack house.
The possibility of rapid, silent HIV transmission along sociogeographic networks seems especially significant for the traditional, and now rapidly expanding, residential zones of the socioeconomically marginalized urban and rural poor in the United States. For poor American urban communities, highly dense and geographically centered resource sharing socioeconomic networks are particularly essential to individual and family strategies for survival, as well as to community stability [6-lo]. Such networks are very likely to include fundamental rapid HIV transmission routes, in particular the local formal sex industry and the informal structure of sexual relationships that many poor women are almost required to form for individual and family survival [ 111. After initial rapid spread within male homosexual and intravenous drug-using (IVDU) populationsaffecting as well the sexual partners and children of IVDUs, HIV infection now appears poised to devastate youth of minority neighborhoods who become involved in a social-disintegration induced nexus of deviant behavior: a large and growing body of literature describes how, within disintegrated communities, the disruption of personal, domestic and community social networks and other structures, typically induced by a broad spectrum of factors, can initiate and firmly establish a nexus of intertwined and mutually reinforcing behaviors with the most severe implications for transmission of HIV infection. Behaviors induced by social disintegration include violence, criminal activity and various forms of substance abuse, along with early initiation of sexual activity, frequent sexual activity in conjunction with substance use and abuse, teen pregnancy, polyaddiction and so on [8-10, 12-181. As Osgood et al. [l9] put it, Research has firmly established that a wide range of deviant behaviors are positively correlated with one another during adolescence and early adulthood . two explanations [emerge]. The first is that engaging in one form of deviant behavior leads to engaging in others as well . The second . . . is that different behaviors are related because they have shared influences . To the degree that the same factors are major sources of all deviant behaviors, it is meaningful to speak of a general syndrome of deviance The work of Donovan and Jessor (201 . focused on covariance among deviant behaviors determin[ing] that a variety of behaviors formed a general syndrome of deviance. As regards the intertwining of substance abuse and obviously important factors in transsexuality, mission of HIV infection, Belcastro and Nicholson
[21] find
. the use of. . drugs may be a form of ‘chemical foreplay’ where they are used to enhance and culminate the coital episode. [This implies] education which segregates drugs from sexuality is inadequate. Mott and Haurin (221 in a study titled ‘The Interrelatedness of Age at First Intercourse, Early Pregnancy, Alcohol, and Drug Use Among American Adolescents’ found Independent of social class, a stable childhood environment appears to be strongly linked with a below-average tendency by a youth to engage in any of the [deviant] activities [studied].
Zabin ef al. [23], studying inner city adolescents, found a high correlation of sexual activity with substance abuse within all studied subgroups, and concluded that This relationship has potential importance in the design of intervention programs . . [and] public policy because the same comprehensive interventions may be useful in serving youths with a wide range of needs . . the cost of such programs could . . be economical if they reduce, at one and the same time, the potential effects of several components within these clusters of problem behaviors.
The converse is that public policy which enhances socioeconomic inequity, or disrupts the personal, domestic and community social networks of poor communities, may be expected to induce or exacerbate an interrelated nexus of deviant and other behaviors with the gravest implications for many problems of public health and public order, including the spread of AIDS [18,24-26). As Refs [1,2,5] suggest, such a behavioral nexus may itself be strongly geographically centered, leading to an inference that HIV infection processes may, at present, be very tightly clustered indeed within a composite ‘sociogeographic’ space. Here we will attempt to describe mathematically, first, the threshold conditions for initial rapid spread of HIV infection through dense, geographically embedded social networks in vulnerable communitiesa process we suggest is highly dependent on intertwined patterns of socially or socioeconomically determined behavior much like those described above. Second, by collapsing the dimensions of social and geographic interaction to one, we are able to reexamine asymptotic rates of transmission from a simple but powerful stochastic viewpoint. Third, we will examine complications which may be of importance if HIV infection emerges from tightlyghettoized populations-what M. interacting Fullilove has described as the ‘deep ghetto’-into a larger, more openly-structured community. The network structures described here may be ‘self-similar’ on a number of scales, that is, sociogeographically-centered interactions between individuals may be analogous to interactions between bathhouses, shooting galleries or crack houses within one city or ghettoized neighborhood, or, on a larger scale, between ghettoes in different cities. This leads to possible interpretations involving fractal percolation structures which we will not, however, pursue here. In the spirit of Pielou [27], we will try to infer from the resulting models what data-based analyses are likely to give useful scientific insight to these phenomena. We will also begin to explore certain evident
Social disintegration
and the spread
public policy implications. Understanding the conditions under which explosively rapid propagation can begin on geographically-embedded social networks, and the factors determining the rate of propagation, is of more than research interest, and may lead toward design of intervention programs which take us beyond TV spot ads, subway posters and videotapes which have not succeeded in stemming the spread of the disease. THRESHOLDS FOR, AND RATES OF, INFECTION ON A ONE DIMENSIONAL SOCIOCEOGRAPHIC NETWORK
We begin, in the sense of Refs [I] and [2], consideration of a number of communities embedded in a geographic context, but linked together in a one dimensional chain structure, with ‘distance’ between them defined as some possibly subtle composite of social and geographic structure which we will not characterize more precisely. Wallace [2] has inferred, from observation that the number of AIDS cases in New York City grew almost linearly in time between 1983 and 1988, that HIV transmission occurred as a traveling wave proceeding along a ‘one dimensional sociogeographic network.* This simple form, if indeed the case, probably reflects the ‘tightness’ of the urban ghettoes of male homosexuals and intravenous drug users (IVDU), whose social and geographic structures appear to be highly determined by the pressures of widespread and longstanding forms of discrimination and oppression. We assume M individuals infected with HIV enter community n = 1 and, for simplicity, seed infection according to a branching process with mean progeny p per individual. p is thus an index of the susceptibility of community n = 1 to spread of the disease. [28] Shows that the mean number of infected individuals in community 1, E(N,), wiI1 then be
E(N,)=M
the total number of infected individuals chain remains finite and is given by N lmal=
KM
1 1-p
=-
MK”-’
KM
K K [ 1-p
M
(3)
l-(P+K)
(4)
Here a, is the time version of the coupling parameter K, but now itself taken as a stochastic variable. N. is
the number of individuals infected by the outbreak in segment n at time Tn. The conditional mean time of transmission of HIV infection between segments n and n + 1, which we write as r(n), is just the mean of an exponential distribution:
(l-p)2
Z
For arbitrary n, somewhat heuristically, Wn)=o”=-
M
p,[T,+, - T,> t]=exp[-a,N,r]
Assume a fraction K of infected individuals in community n = 1 enter community n = 2, and begin spreading disease. K is thus an index of transmissibility of the disease between communities. Then the mean number of infected individuals in community 2, N2, will be
1-p
1 (1) n
texp[-a,N,t]dt
t(n) = a,,N,
=-
1 a, Nn
I0
(5)
The conditional mean rate of transmission of the process between community segment n and community segment n + 1 is taken as:
Note that if K = 0 there can be no propagation. Thus we must assume K > 0. As n +oo then E(N,) = 0 unless /L+K21
along the
Equation (2) suggests that for a ‘one dimensional sociogeographic network,’ the susceptibility of an individual community to disease spread, the term p, and the probability of disease transmission between communities, the term K, jointly determine whether disease is fully propagated along the chain or dies out with a total number of infected individuals given by equation (3), which itself has the form of a branching process with mean progeny p + K. Equation (3) will be recognized as characterizing a form of chain reaction with the condition for criticality determined equally by community susceptibility p and by the transmission probability K. This has evident implications for disease control strategies. A branching process model of epidemic spread is useful at best in the very stages of an epidemic, before ‘saturation’ of a population. If we assume the process does not die out, that it propagates, but that individual communities along the chain have finite expectation for the number of infected individuals, E(N,) < co, then we may explore the rate of disease propagation along the one dimensional chain in a standard manner which takes us beyond the deterministic approximation of Ref. [2]. We first assume the infection within a community along the chain occurs ‘instantaneously,’ compared to the rate of propagation between communities. Second, we assume the time for first transmission of the disease from one chain segment, say n infected at time T,, to an adjacent, but untouched segment, n + 1 infected at time T,, , , is determined by an exponential distribution, so that the probability that the time of transmission from segment n to n + 1 is greater than a value t is determined by
1-P
E(N,)=-..---.----
1157
of AIDS
(2)
If, on the other hand, p + K c 1, then the disease does not propagate to ‘infinity,’ and some manipulation-summing the terms of equation (I)-shows
(6) The mean rate of transmission segment n, R(n), is then just R(n) = r(l) + . . * +@I n
from segment 1 to
+Rasn-rm.
(7)
1158
RODRICK WALLACE
By the law of large numbers this is just given by R(n) -+ R = E(r(1)) = E((r,)E(N,)
+ Cov(a,, N,) (8a)
where Cov(a, N) is the covariance between a and N. Rewriting equation @a) slightly, leaving out the confusing subscripts, R z E(a)E(N)
f P~NQPN
(8b)
where cr2 represents variance and p correlation. Thus the rate of transmission of infection along the ‘one dimensional sociogeographic network’ is determined by complex interaction between community susceptibility to infection, described by the stochastic process N,, , and a structure of transmissibility between communities, described by the stochastic process a,. If correlation between the structure of transmission and susceptibility is zero, then the rate of transmission is just the product of the means of the two processes. If, however, community transmissibility and susceptibility are positively correlated-for example if social disintegration encourages outmigration or if susceptibility is structurally associated with transience-then the asymptotic rate of disease transmission can be much greater than the product of the means assuming large variances for those factors. See Ref. [24] for a possible case history. The ‘sociogeographic’ case histories summarized in the introduction-gay bathhouses, IVDU shooting galleries, crack houses and the configuration of male prostitution-all suggest an inevitable intertwining of a highly variable ‘risk’ behavior nexus and equally or even more variable patterns of movement in space and between or within social networks, i.e. p 2 1 and u $ 1. Equation (8b) suggests such consonance may result in very rapid spread of HIV infection. In Africa, migration in search of work, ‘contract’ labor at a distance from home without family, along with long distance truck transport patterns, have perhaps combined with the sex industry to create similar conditions causing similarly rapid disease spread in a more general population. Such matters are not so directly treated from a purely deterministic viewpoint, suggesting that stochastic methods may be required for proper understanding of disease transmission between communities which are embedded in sociogeographic networks. THRESHOLD
CONDITIONS
FOR HIGHER
spread patterns, both in conventional time and space as well as ‘sociogeographic’ space. In two or more dimensions the process of transmission of infection between communities is much complicated by the possibility of more than one route connecting them within the sociogeographic net. We begin by supposing a very large (asymptotically an infinite) number of communities in a geographically embedded network of social links, and that a fraction 0 < P, ,C1 of these communities participate in a nexus of deviant or other behaviors leading to rapid spread of HIV infection. These we term susceptible communities. We assume that the probability of transmission of HIV infection between linked communities participating in the nexus of high risk behaviors is 0 < PE< 1. We term the set of communities and connecting links the graph G, and ask the question, given an initial set of HIV-infected communities S, what is the probability, P = P(Py,PE,S,G) that HIV infection spreads beyond any finite set of communities. In other words, we ask for the probability that HIV infection becomes ubiquitous among susceptible communities. This is a version of the fundamental question of a difficult and subtle branch of applied mathematics called, for obvious reaons, percolation theory. See, for example Refs [29-331. Following Hammersely’s development closely [29], for fixed S and G the function P(P,,PE)is defined on the square 0 < P, < 1, 0 < PE d 1 and is clearly a nondecreasing function of its two arguments P, and PE, taking the values P(0,0)= 0 and P(1,1) = 1. A fundamental theorem of percolation theory states that P > 0 only on the upper right hand corner of the square, the shaded region of Fig. 1 (taken from Ref. [29]). P = 0 elsewhere in the unshaded region, provided that S is a finite set. That is, there is a threshold region for P > 0 determined as a function of P, and PE:if these probabilities are below certain values, HIV infection will be contained within a finite population of susceptible communities and will
DIIMENSIONS
We have suggested, in Refs [2] and [5] and above, that the initial stages of HIV spread in the United States have been along ‘one dimensional sociogeographic networks,’ largely defined by oppressive social constraints acting on already spatially ghettoized populations. The next phases of the AIDS epidemic may involve ‘looser’ sociogeographic networks within the general population, possibly of higher dimension, that is, establishment of HIV infection as a sexually transmitted disease outside of the ghettoes. Higher dimensionality introduces complications of some note, and may be expected to result in different
-
P”
Fig. 1. From Ref. [29], structure of threshold behavior for percolation of HIV infection throughout a social network. P, is the fraction of behaviorally-susceptible individuals in the network, and Pt; is the probability of HIV transmission between susceptible individuals.
1159
Social disintegration and the spread of AIDS not spread throughout the network, under these approximations. A simplified version of this threshold behaviorqualitatively different from the threshold behavior described earlier in one dimension-is not difficult to understand. Assume, first, a strict two dimensional integer network of social connections between only susceptible communities, and a probability of infective transmission along one step of the network of O
Sm. Sci. Med. Vol. 33, No. 10, pp. 11X--1162, 1991
$3.00 + 0.00
Pergamon Press plc
Printed in Great Britain
SOCIAL DISINTEGRATION AND THE SPREAD OF AIDS: THRESHOLDS FOR PROPAGATION ALONG ‘SOCIOGEOGRAPHIC’ NETWORKS RODRICK WALLACE Epidemiology of Mental Disorders Research Department, New York State Psychiatric Institute, 722 W. 168 St, New York, NY 10032, U.S.A. Abstract-Previous work on the asymptotic spread of HIV infection along a low dimensional ‘sociogeographic’ network-a social network characteristically embedded within a limited geographic area-is extended to explore threshold conditions under which the infection extends widely beyond an initial set of infected individuals or communities. Results for one dimension suggest that threshold behavior is analogous to a chain reaction with criticality determined conjointly by the susceptibility of individuals within a community to a nexus of behavior conducive to rapid HIV spread and by the probability of transmission between susceptible communities. Once threshold is exceeded, a stochastic reformulation finds the asymptotic rate of transmission between communities may be markedly raised by positive correlation between susceptibility to rapid disease spread within a community and the transmissibility between communities, for example outmigration driven by social disintegration or residential instability arising from inherent structural factors associated with community susceptibility, as with male prostitution. Examination of threshold conditions for higher dimensional sociogeographic networks most likely characteristic of disease spread beyond the ‘deep ghettos’ now suffering the highest burden of infection suggests it is at least as, and likely more, effective to decrease the fraction of population susceptible to the high risk behavioral nexus as it is to lower the probability of disease transmission between susceptible individuals or communities. Thus, while purely ‘medical’ or ‘educational’ interventions may influence probabilities of transmission, socioeconomic and political structures and policies may strongly affect degrees of social disintegration or other factors determining both the fraction of a population engaging in a nexus of high risk behavior within a community and patterns of outmigration or other residential instabilities affecting the rate of spread between communities. This suggests the necessity of interventions to stabilize socially disintegrated or other affected communities, in addition to programs aimed only at decreasing the probability of disease transmission between individuals. Key words-AIDS
epidemiology, social disintegration,
sociogeographic networks, threshold conditions,
stochastic models
INTRODIJC’I’ION HIV transmission depends heavily on local customs and forms of sexual activity and the sex industry, and in some circumstances of substance use and abuse, all of which are deeply embedded in underlying socioeconomic and political structures. A long asymptomatic but infectious latent period for expression of the disease as AIDS makes almost inevitable a hidden but explosively rapid and comprehensive spread of the virus within geographicalIy concentrated populations engaging in a nexus of high risk behaviors. Potterat ef ai. [I] observed a particularly close intertwining of spatial and social patterns of gonorrhea endemicity in a minority population within a small American city, and R. Wallace [2] has adapted this view to model deterministically the asymptotic transmission of HIV as a new disease introduced on a possibly non integral dimensional ‘sociogeographic network,’ a ‘Potterat structure’ of a social network focused within a local geographic area. This picture may apply more generally to the social structures of oppressed groups within the United States, particularly when combined with rapid social
change. For example gay males, driven as ‘sexual migrants’ [3] from their communities of origin into urban ghettoes, may then experience great intensification of within-group interactions, including sexual practices effective in spreading HIV. The gay bathhouse phenomenon comes to mind as such a highly geographically concentrated interaction. Similarly, intravenous drug abusers in disintegrating urban centers, usually an oppressed minority within an oppressed minority, may respond to rapid social change by intensification of needle-sharing behavior, particularly within the context of geographically centered ‘shooting galleries’ where drugs are sold and ‘works’ rented. The recent ‘crack house’ phenomenon, a central point for the smoking of crack cocaine, and for the exchange of sex for drugs, would seem to provide yet another example. Finally, a recent study of male prostitutes in New Orleans [4] finds their activity not only to be geographically concentrated within the city, but to be highly transient, with many male prostitutes routinely traveling between ‘red light’ districts of major urban centers with high levels of HIV infection,
1155
1156
M. Fullilove follows:
RODRICK WALLACE
[5] has summarized
this pattern
as
rapid social change acting on groups marginalized by the larger society can create an intensification of risk behavior within the group. A particular and critical aspect of the changing group interaction is that previously isolated social networks become behaviorally interlinked within the group. It can be hypothesized that this subgroup interlinking occurred in the case of distinct social groups of gay men meeting in the bathhouse, of intravenous drug users meeting in the shooting gallery, and of crack users meeting in the crack house.
The possibility of rapid, silent HIV transmission along sociogeographic networks seems especially significant for the traditional, and now rapidly expanding, residential zones of the socioeconomically marginalized urban and rural poor in the United States. For poor American urban communities, highly dense and geographically centered resource sharing socioeconomic networks are particularly essential to individual and family strategies for survival, as well as to community stability [6-lo]. Such networks are very likely to include fundamental rapid HIV transmission routes, in particular the local formal sex industry and the informal structure of sexual relationships that many poor women are almost required to form for individual and family survival [ 111. After initial rapid spread within male homosexual and intravenous drug-using (IVDU) populationsaffecting as well the sexual partners and children of IVDUs, HIV infection now appears poised to devastate youth of minority neighborhoods who become involved in a social-disintegration induced nexus of deviant behavior: a large and growing body of literature describes how, within disintegrated communities, the disruption of personal, domestic and community social networks and other structures, typically induced by a broad spectrum of factors, can initiate and firmly establish a nexus of intertwined and mutually reinforcing behaviors with the most severe implications for transmission of HIV infection. Behaviors induced by social disintegration include violence, criminal activity and various forms of substance abuse, along with early initiation of sexual activity, frequent sexual activity in conjunction with substance use and abuse, teen pregnancy, polyaddiction and so on [8-10, 12-181. As Osgood et al. [l9] put it, Research has firmly established that a wide range of deviant behaviors are positively correlated with one another during adolescence and early adulthood . two explanations [emerge]. The first is that engaging in one form of deviant behavior leads to engaging in others as well . The second . . . is that different behaviors are related because they have shared influences . To the degree that the same factors are major sources of all deviant behaviors, it is meaningful to speak of a general syndrome of deviance The work of Donovan and Jessor (201 . focused on covariance among deviant behaviors determin[ing] that a variety of behaviors formed a general syndrome of deviance. As regards the intertwining of substance abuse and obviously important factors in transsexuality, mission of HIV infection, Belcastro and Nicholson
[21] find
. the use of. . drugs may be a form of ‘chemical foreplay’ where they are used to enhance and culminate the coital episode. [This implies] education which segregates drugs from sexuality is inadequate. Mott and Haurin (221 in a study titled ‘The Interrelatedness of Age at First Intercourse, Early Pregnancy, Alcohol, and Drug Use Among American Adolescents’ found Independent of social class, a stable childhood environment appears to be strongly linked with a below-average tendency by a youth to engage in any of the [deviant] activities [studied].
Zabin ef al. [23], studying inner city adolescents, found a high correlation of sexual activity with substance abuse within all studied subgroups, and concluded that This relationship has potential importance in the design of intervention programs . . [and] public policy because the same comprehensive interventions may be useful in serving youths with a wide range of needs . . the cost of such programs could . . be economical if they reduce, at one and the same time, the potential effects of several components within these clusters of problem behaviors.
The converse is that public policy which enhances socioeconomic inequity, or disrupts the personal, domestic and community social networks of poor communities, may be expected to induce or exacerbate an interrelated nexus of deviant and other behaviors with the gravest implications for many problems of public health and public order, including the spread of AIDS [18,24-26). As Refs [1,2,5] suggest, such a behavioral nexus may itself be strongly geographically centered, leading to an inference that HIV infection processes may, at present, be very tightly clustered indeed within a composite ‘sociogeographic’ space. Here we will attempt to describe mathematically, first, the threshold conditions for initial rapid spread of HIV infection through dense, geographically embedded social networks in vulnerable communitiesa process we suggest is highly dependent on intertwined patterns of socially or socioeconomically determined behavior much like those described above. Second, by collapsing the dimensions of social and geographic interaction to one, we are able to reexamine asymptotic rates of transmission from a simple but powerful stochastic viewpoint. Third, we will examine complications which may be of importance if HIV infection emerges from tightlyghettoized populations-what M. interacting Fullilove has described as the ‘deep ghetto’-into a larger, more openly-structured community. The network structures described here may be ‘self-similar’ on a number of scales, that is, sociogeographically-centered interactions between individuals may be analogous to interactions between bathhouses, shooting galleries or crack houses within one city or ghettoized neighborhood, or, on a larger scale, between ghettoes in different cities. This leads to possible interpretations involving fractal percolation structures which we will not, however, pursue here. In the spirit of Pielou [27], we will try to infer from the resulting models what data-based analyses are likely to give useful scientific insight to these phenomena. We will also begin to explore certain evident
Social disintegration
and the spread
public policy implications. Understanding the conditions under which explosively rapid propagation can begin on geographically-embedded social networks, and the factors determining the rate of propagation, is of more than research interest, and may lead toward design of intervention programs which take us beyond TV spot ads, subway posters and videotapes which have not succeeded in stemming the spread of the disease. THRESHOLDS FOR, AND RATES OF, INFECTION ON A ONE DIMENSIONAL SOCIOCEOGRAPHIC NETWORK
We begin, in the sense of Refs [I] and [2], consideration of a number of communities embedded in a geographic context, but linked together in a one dimensional chain structure, with ‘distance’ between them defined as some possibly subtle composite of social and geographic structure which we will not characterize more precisely. Wallace [2] has inferred, from observation that the number of AIDS cases in New York City grew almost linearly in time between 1983 and 1988, that HIV transmission occurred as a traveling wave proceeding along a ‘one dimensional sociogeographic network.* This simple form, if indeed the case, probably reflects the ‘tightness’ of the urban ghettoes of male homosexuals and intravenous drug users (IVDU), whose social and geographic structures appear to be highly determined by the pressures of widespread and longstanding forms of discrimination and oppression. We assume M individuals infected with HIV enter community n = 1 and, for simplicity, seed infection according to a branching process with mean progeny p per individual. p is thus an index of the susceptibility of community n = 1 to spread of the disease. [28] Shows that the mean number of infected individuals in community 1, E(N,), wiI1 then be
E(N,)=M
the total number of infected individuals chain remains finite and is given by N lmal=
KM
1 1-p
=-
MK”-’
KM
K K [ 1-p
M
(3)
l-(P+K)
(4)
Here a, is the time version of the coupling parameter K, but now itself taken as a stochastic variable. N. is
the number of individuals infected by the outbreak in segment n at time Tn. The conditional mean time of transmission of HIV infection between segments n and n + 1, which we write as r(n), is just the mean of an exponential distribution:
(l-p)2
Z
For arbitrary n, somewhat heuristically, Wn)=o”=-
M
p,[T,+, - T,> t]=exp[-a,N,r]
Assume a fraction K of infected individuals in community n = 1 enter community n = 2, and begin spreading disease. K is thus an index of transmissibility of the disease between communities. Then the mean number of infected individuals in community 2, N2, will be
1-p
1 (1) n
texp[-a,N,t]dt
t(n) = a,,N,
=-
1 a, Nn
I0
(5)
The conditional mean rate of transmission of the process between community segment n and community segment n + 1 is taken as:
Note that if K = 0 there can be no propagation. Thus we must assume K > 0. As n +oo then E(N,) = 0 unless /L+K21
along the
Equation (2) suggests that for a ‘one dimensional sociogeographic network,’ the susceptibility of an individual community to disease spread, the term p, and the probability of disease transmission between communities, the term K, jointly determine whether disease is fully propagated along the chain or dies out with a total number of infected individuals given by equation (3), which itself has the form of a branching process with mean progeny p + K. Equation (3) will be recognized as characterizing a form of chain reaction with the condition for criticality determined equally by community susceptibility p and by the transmission probability K. This has evident implications for disease control strategies. A branching process model of epidemic spread is useful at best in the very stages of an epidemic, before ‘saturation’ of a population. If we assume the process does not die out, that it propagates, but that individual communities along the chain have finite expectation for the number of infected individuals, E(N,) < co, then we may explore the rate of disease propagation along the one dimensional chain in a standard manner which takes us beyond the deterministic approximation of Ref. [2]. We first assume the infection within a community along the chain occurs ‘instantaneously,’ compared to the rate of propagation between communities. Second, we assume the time for first transmission of the disease from one chain segment, say n infected at time T,, to an adjacent, but untouched segment, n + 1 infected at time T,, , , is determined by an exponential distribution, so that the probability that the time of transmission from segment n to n + 1 is greater than a value t is determined by
1-P
E(N,)=-..---.----
1157
of AIDS
(2)
If, on the other hand, p + K c 1, then the disease does not propagate to ‘infinity,’ and some manipulation-summing the terms of equation (I)-shows
(6) The mean rate of transmission segment n, R(n), is then just R(n) = r(l) + . . * +@I n
from segment 1 to
+Rasn-rm.
(7)
1158
RODRICK WALLACE
By the law of large numbers this is just given by R(n) -+ R = E(r(1)) = E((r,)E(N,)
+ Cov(a,, N,) (8a)
where Cov(a, N) is the covariance between a and N. Rewriting equation @a) slightly, leaving out the confusing subscripts, R z E(a)E(N)
f P~NQPN
(8b)
where cr2 represents variance and p correlation. Thus the rate of transmission of infection along the ‘one dimensional sociogeographic network’ is determined by complex interaction between community susceptibility to infection, described by the stochastic process N,, , and a structure of transmissibility between communities, described by the stochastic process a,. If correlation between the structure of transmission and susceptibility is zero, then the rate of transmission is just the product of the means of the two processes. If, however, community transmissibility and susceptibility are positively correlated-for example if social disintegration encourages outmigration or if susceptibility is structurally associated with transience-then the asymptotic rate of disease transmission can be much greater than the product of the means assuming large variances for those factors. See Ref. [24] for a possible case history. The ‘sociogeographic’ case histories summarized in the introduction-gay bathhouses, IVDU shooting galleries, crack houses and the configuration of male prostitution-all suggest an inevitable intertwining of a highly variable ‘risk’ behavior nexus and equally or even more variable patterns of movement in space and between or within social networks, i.e. p 2 1 and u $ 1. Equation (8b) suggests such consonance may result in very rapid spread of HIV infection. In Africa, migration in search of work, ‘contract’ labor at a distance from home without family, along with long distance truck transport patterns, have perhaps combined with the sex industry to create similar conditions causing similarly rapid disease spread in a more general population. Such matters are not so directly treated from a purely deterministic viewpoint, suggesting that stochastic methods may be required for proper understanding of disease transmission between communities which are embedded in sociogeographic networks. THRESHOLD
CONDITIONS
FOR HIGHER
spread patterns, both in conventional time and space as well as ‘sociogeographic’ space. In two or more dimensions the process of transmission of infection between communities is much complicated by the possibility of more than one route connecting them within the sociogeographic net. We begin by supposing a very large (asymptotically an infinite) number of communities in a geographically embedded network of social links, and that a fraction 0 < P, ,C1 of these communities participate in a nexus of deviant or other behaviors leading to rapid spread of HIV infection. These we term susceptible communities. We assume that the probability of transmission of HIV infection between linked communities participating in the nexus of high risk behaviors is 0 < PE< 1. We term the set of communities and connecting links the graph G, and ask the question, given an initial set of HIV-infected communities S, what is the probability, P = P(Py,PE,S,G) that HIV infection spreads beyond any finite set of communities. In other words, we ask for the probability that HIV infection becomes ubiquitous among susceptible communities. This is a version of the fundamental question of a difficult and subtle branch of applied mathematics called, for obvious reaons, percolation theory. See, for example Refs [29-331. Following Hammersely’s development closely [29], for fixed S and G the function P(P,,PE)is defined on the square 0 < P, < 1, 0 < PE d 1 and is clearly a nondecreasing function of its two arguments P, and PE, taking the values P(0,0)= 0 and P(1,1) = 1. A fundamental theorem of percolation theory states that P > 0 only on the upper right hand corner of the square, the shaded region of Fig. 1 (taken from Ref. [29]). P = 0 elsewhere in the unshaded region, provided that S is a finite set. That is, there is a threshold region for P > 0 determined as a function of P, and PE:if these probabilities are below certain values, HIV infection will be contained within a finite population of susceptible communities and will
DIIMENSIONS
We have suggested, in Refs [2] and [5] and above, that the initial stages of HIV spread in the United States have been along ‘one dimensional sociogeographic networks,’ largely defined by oppressive social constraints acting on already spatially ghettoized populations. The next phases of the AIDS epidemic may involve ‘looser’ sociogeographic networks within the general population, possibly of higher dimension, that is, establishment of HIV infection as a sexually transmitted disease outside of the ghettoes. Higher dimensionality introduces complications of some note, and may be expected to result in different
-
P”
Fig. 1. From Ref. [29], structure of threshold behavior for percolation of HIV infection throughout a social network. P, is the fraction of behaviorally-susceptible individuals in the network, and Pt; is the probability of HIV transmission between susceptible individuals.
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Social disintegration and the spread of AIDS not spread throughout the network, under these approximations. A simplified version of this threshold behaviorqualitatively different from the threshold behavior described earlier in one dimension-is not difficult to understand. Assume, first, a strict two dimensional integer network of social connections between only susceptible communities, and a probability of infective transmission along one step of the network of O
