This peak incidence is expressed as a fraction of the total population .sx An exact expression for the peak incidence of infection in a homogeneous population ( which is equal to vy max ) is given by eqn ( 6.20 ) , which can be seen to reduce to eqn ( 11.95 ) when R 0 is significantly in excess of unity .sx The presence of carriers and of heterogeneity in sexual habits produce complications , but eqn ( 11.95 ) remains a good back-of-an-envelope estimate ( so long as R 0 is appreciably greater than unity ) , and is consistent with broad trends seen in Fig. 11.23. The result ( 11.95 ) arises essentially because a fraction f of those infected move into the class with full-blown AIDS at a rate v 1 , following the initial stages of the epidemic in which ( for R 0 large ) most of the population is infected .sx Thus observations of the incidence of clinical AIDS as the epidemic begins to peak give a very rough estimate of the combination f/D , where D=1/v 1 is the average interval between infection and diagnosed AIDS .sx For example , if a peak incidence of clinical AIDS of 2-3 per cent of the at-risk population were found each year , we could roughly estimate f approximate-sign 10 per cent if D is around 3-4 years - or f approximate-sign 50 per cent if D is around 15-20 years - and so on .sx The long-term public health issues , however , depend more on the total fraction of the at-risk population who are likely to develop clinical AIDS , that is ( for R 0 large ) upon f. As discussed above , the observed levels of seropositivity suggest R 0 is unlikely to be less than around three , and it may well have a value of five or more .sx With formula , this means the factor inside the square brackets two paragraphs above is likely to have a value of 5 years or more .sx But this value could be attained in two ways , which have different implications for estimates of f. On the one hand , if formula , we have the estimates that D is around 5 years or more , whence observations about peak incidence of AIDS lead to a crude estimate of f. On the other hand , if 1/v 2 >>1/v 1 , it could be that the value of R 0 / beta c is determined essentially by 1/v 2 , whence the above argument gives virtually no information about D , and thus is of no help in putting bounds on f. These arguments can be refined into more quantitative estimates , with the help of numerical studies of the kind reported in Figs .sx 11.22 and ) .sx An essential ambiguity remains , however , in whether diagnosed AIDS-developers and carriers do or do not have similar durations of infectiousness ; that is , whether v 1 approximate-sign v 2 or not .sx The analysis presented above suggests some ways of gathering information to resolve this issue .sx Consider first the incidence of infection ( as revealed by seropositivity ) as a function of time , among classes of individuals distinguished according to levels of sexual activity .sx As shown in Fig. 11.31(a ) , if v 1 =v 2 there is a tendency for the more sexually active individuals to acquire infection earlier , and less active individuals later , so that the ratio of more sexually active to less sexually active individuals among those infected shifts as the epidemic progresses .sx As seen in Fig. 11.31(b ) , this tendency is much more pronounced if v 1 is substantially greater than v 2 ( carriers on average remain infectious significantly longer than diagnosed AIDS-developers) :sx the more active people tend to acquire infection early on , whereas infection among the less active fraction of the population slowly builds up over time , depending primarily on the long-lived infectiousness of carriers .sx Systematically compiled serological data that discriminate among the levels of sexual activity of recently infected individuals , over time , should in this way be capable of shedding light on the ratio between v 1 and v 2 .sx graphs&captions .sx The same general considerations apply to incidence of newly diagnosed cases of full-blown AIDS , distinguished according to levels of sexual activity in the patients , as a function of time .sx Here , however , the patterns are more blurred , by virtue of the statistical distribution of times between acquiring infection and manifesting full-blown AIDS .sx Figures 11.32(a ) and ( b ) correspond to Figs .sx 11.31(a ) and ( b ) , illustrating the incidence of cases of diagnosed AIDS disease , among more sexually active individuals and among less active ones , as a function of time .sx Figure 11.32(a ) is for v 1 =v 2 and Fig. 11.32(b ) is for v 1 =0.2 yr -1 and v 2 =0.05 yr -1 .sx The trends seen in Figs .sx 11.31(a ) and ( b ) are again evident , although somewhat less pronounced .sx Again , systematic investigation of data of this kind could help determine the ratio of v 1 to v 2 .sx On another tack , it would be helpful to look at the incidence of diagnosed AIDS disease , as a function of time , among those individuals who were most sexually active .sx Most of these people are likely to have acquired infection early on , so that their distribution of incidence of disease over time gives a reasonable indication of the actual distribution of incubation times ( in a way that is not possible if all data , from more active along with less active people , are combined) .sx Such analyses could help provide an independent estimate of whether the distribution of incubation times found among those infected with HIV by blood transfusions are indeed representative of broader categories of HIV infections .sx Whether AIDS will establish itself as an endemic disease in the USA and Europe depends on whether sexual habits change sufficiently to push R 0 below unity .sx This requires the distribution in the number of sexual partners per unit time among homosexuals to decrease to a level such that .sx c' 0 .sx ( 11.96 ) .sx Here R 0 is the initial value of the basic reproductive rate for AIDS , estimates of which we have just discussed , and c and c' are the initial and later values , respectively , of the appropriate average over the distribution in sexual habits ( formula ) .sx Downward changes in the mean clearly help .sx But downward changes at the high end of the distribution help disproportionately , because they have more effect on 2 > .sx Alternatively , R 0 can be lowered by changes in the intrinsic transmission parameter beta ( 'safe sex') .sx In either event , it is clear that the magnitude of the changes needed to drive R 0 below unity depend on how large R 0 originally was ; as we have seen , this number is uncertain , but is unlikely to be less than five or so .sx All the above discussion has treated AIDS as a disease of homosexuals .sx Needle-sharing among intravenous drug users involves the same ideas , with different values for the transmission factors beta and c , but the same parameters characterizing the course of infection .sx For such transmission , beta c can be estimated from initial doubling times , and the above discussion repeated , mutatis mutandis .sx The same is true for heterosexual transmission , which is currently low in developed countries but high in many sub-Saharan African countries .sx As we have seen , simple models for purely heterosexual transmission suggest that in the earlier stages of the epidemic - before any subgroup becomes saturated with infection - the incidence of HIV infection and of AIDS cases among males and among females grows roughly exponentially at the same rate in both populations .sx The doubling rate ( which previously was beta c - 1/ D ) is now approximately formula , where D is the average duration of infectiousness .sx In developing countries , we would expect to find the female-to-male transmission probability , beta ' , and probably also the male-to-female transmission probability , beta , less than the transmission probability among homosexuals .sx We would also expect the appropriately averaged rate of acquiring new sexual partners by women and by men , c' and c , to be less than the corresponding quantity for homosexuals in large cities in the developed world in the early 1980s .sx As was discussed in the section on epidemiological parameters , data from recent studies in the United Kingdom suggest a factor 10 to 20 difference in the average rate at which male homosexuals and male heterosexuals acquire new partners ( see Fig. ) .sx Overall , therefore , we would expect the doubling rate to be smaller - and the doubling time to be longer - for HIV infections transmitted along hetero - sexual chains than along homosexual ones in developed countries .sx Whether or not a self-sustaining epidemic can be generated by purely heterosexual transmission in developed countries depends on the magnitude of the basic reproductive rate for such transmission ; this quantity is given approximately by formula .sx Such an epidemic is likely to take off if R 0 exceeds unity , and not otherwise .sx At present , uncertainty surrounds each one of the five quantities entering into this expression for R 0 ( especially the female-to-male transmission probability , beta ' ) , nor are there enough data to attempt the kind of indirect inference of R 0 from population-level considerations , as we did for homosexual transmission in developed countries .sx We believe that reasonable estimates span a range of R 0 values from below unity to above unity .sx Bearing all this in mind , our guess is that , in developed countries , HIV could well spread by purely heterosexual transmission within relatively promiscuous subgroups .sx The long-term possibility of a much more generally disseminated epidemic seems to us to depend crucially on whether a significant fraction of HIV infectees remain asymptomatic carriers ( never developing AIDS ) , transmitting infection with some low probability essentially for the remainder of their sexually active lives .sx If this is so , then quite modest levels of promiscuity could result in HIV infections spreading slowly , over many decades , among a population who would not think of themselves as promiscuous .sx In our earlier language , such a long duration of infectiousness among a significant fraction of infectees corresponds to the basic reproductive rate , R 0 , for heterosexually transmitted HIV being appreciably greater than unity .sx This brings us to a final and important point to do with predicting AIDS cases .sx It might at first seem that the larger the fraction of HIV infectees who go on to develop AIDS and die , the larger the eventual number of deaths from the epidemic .sx This is not necessarily so .sx Contrast the following two possibilites , neither of which can be completely ruled out by currently available data :sx in Case A , 30 per cent of those infected develop AIDS , remaining infectious typically for 5 years and then dying , while the remaining 70 per cent of those infected remain asymptomatic yet infectious for 30 years ; in Case B , 100 per cent of those infected develop AIDS , remaining infectious typically for 8 years and then dying .sx Notice immediately that the characteristic duration of infectiousness is larger in Case A ( 0.3x5+0.7x30=22.5 years ) than in Case B ( 8 years ) ; this means that the basic reproductive rate for HIV infection is almost three times greater under assumption A than under B , if the transmission probability is assumed to be the same in both cases .sx The total number of deaths caused by AIDS in either case depends on the fraction ever infected with HIV and on the fraction of those infected who develop AIDS .sx For Case A , with its substantially larger basic reproductive rate , a larger fraction of the population will acquire HIV infection , but a smaller proportion ( 30 per cent ) of these will die .sx In contrast , in Case B a smaller fraction will be infected , but all who are will die .sx We simply cannot say whether Case A or Case B will result in a larger total number of deaths , until we have made a detailed analysis of the non-linear way in which the differences in the basic reproductive rates are likely to affect the numbers ever infected .sx The results of such a calculation are sensitive to the amount of variability in degrees of sexual activity ; the greater this variability , the more likely that Case A will lead to more deaths than Case B , in defiance of simple intuition .sx For homosexually transmitted HIV infection in developed countries , our preliminary calculations suggest that the total number of AIDS deaths increases as f ( the fraction of HIV infectees going on to develop full-blown AIDS ) increases up to around 50 per cent or so , but that total deaths remain roughly constant , or can even decrease , as f increases toward 100 per cent .sx As emphasized at the outset , current evidence cannot rule out f values as low as 20-30 per cent nor as high as 100 per cent .sx