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From Anderson and May (1991), with permission.

From Anderson and May (1991), with permission.

most bacterial and viral infections. It implies that no infection can maintain itself unless R0 is greater than unity.

In practice, we observe that the values of R0 for some common infections are in the range 318 (Table 2.1). This implies that a primary case of measles, say, can infect 11-17 other people during the course of the primary infection, and that the 11-17 secondary cases can each do the same. It is this potentially exponential increase in cases that defines the rising phase of the epidemic curve (Figure 2.1). At some point, however, the availability of susceptible people will diminish, and the achievable number of secondary infections will decline accordingly: i.e. the effective reproductive number, R, will diminish in the presence of constraints on the growth of the population of the infectious agent. This defines the falling phase of the epidemic curve (Figure 2.1).

The availability of susceptible people and the value of R0 are the crucial determinants of the dynamics of epidemics. R0 is some function of the biology of the pathogen; a measure of transmissibility and infectiousness. The availability of susceptibles (and the effective reproductive rate), however, is predominantly a function of the host population. Since birth is the primary source of new susceptibles in a stable population, this implies that infectious disease dynamics are determined by some combination of host population size and birth rate, both of which are largely independent of the pathogen itself (except in the case of a prevalent lethal infection, such as HIV).

Fig 2.1 The course of an epidemic. Infection spreads when the reproductive rate is over unity in value. As the proportion of the population infected or immune grows, the number of contacts with infection that generate new cases falls, thus the effective reproductive rate is reduced. The prevalence of infection will rise for a short time once the effective reproductive rate has fallen below 1 because of the momentum of the epidemic. When a large number of cases are present, then new cases may be generated faster than others recover, even though each case is not on average replacing itself. This carries the effective reproductive rate below unity in value at the peak prevalence, where, as the reduced reproductive rate takes effect, the prevalence of infection starts to fall. It falls until a steady endemic prevalence is reached, when the value of the effective reproductive rate is equal to unity. The endemic proportion of the population not susceptible to infection keeps the effective reproductive rate at unity (modified from Garnett and Ferguson, 1996)

Empirical evidence provides support for the importance of host demography. Studies of island and city communities have shown that a population of some 500 000 or more is required for endemic maintenance of measles (Macdonald, 1957; Bartlett, 1957). Once this population threshold is exceeded, then birth rate is an important determinant of the rate of arrival of new susceptibles, and thus the period between the end of one epidemic and the time at which sufficient susceptibles have accumulated for a new epidemic to begin. For measles, this inter-epidemic period is approximately 2 years in countries of the 'north' (e.g. the USA), with annual population growth rates <1%, but only 1 year in countries of the 'south', with rates of around 3% (e.g. Kenya) (Anderson and May, 1979; Black et al., 1966; Nokes et al., 1991).

Thus, the dynamic properties of an infection depend not only on the characteristics of the pathogen but also on those of the host populations. This is not the whole explanation, of course, but it does seem to provide a remarkably complete understanding of the dynamic behaviour of some infectious diseases. Interestingly, the rather ragged 2 year cycles for measles in cities have been analysed using current assumptions about 'chaos'. It has been suggested that the apparently random behaviour of these time-series is generated by simple and completely deterministic systems, and that measles epidemics provide one of the best examples of naturally occurring deterministic chaos (Fine et al., 1982). This offers the tantalising prospect of short-term prediction, a prospect which now seems less likely to become a practical reality (Gleick, 1987).

Estimating the Basic Reproductive Number for Microparasites

For directly transmitted (also called 'close contact') microparasitic infections, the basic reproductive number can be defined as:

Ro = pNd where p is the coefficient of transmission, N is the total population (all assumed susceptible at baseline) and d is the duration of infectiousness.

The effective reproductive number, R of an infection is unity in a steady, endemic state because each primary case exactly replaces itself. The basic reproductive number R0 is in practice discounted by the proportion, x, of the population remaining susceptible (as against contacts which have experienced infection and are now immune), where:

For a stable population, this implies that if people live an average of L years, become infected at age A, and are protected by maternal antibodies up to age D, then the fraction susceptible is given approximately by (A—D)/L. However, in developing societies, with a growing population, the relevant estimate of L is the reciprocal of the per capita birth rate, B, rather than the inverse of the death rate. Hence R0 can be crudely estimated from B/(A—D). For further discussion of this and other issues, see Anderson and May (1992).

For sexually transmitted infections this relationship can be modified to give:

For vector-borne microparasites (Bolker and Grenfell, 1995) the number is given by:

and:

In these equations, a is the average biting rate per day of the vector species, ph is the likelihood of transmission to the mosquito when it takes a blood meal from an infectious human and pv is the likelihood of transmission to the human when it is fed on by an infectious mosquito. The recovery rate of humans from infectiousness, v is the inverse of the time during which a host is infectious. Likewise, the mosquito death rate, is the inverse of the mean duration of infectious-ness. V is the number of vectors and H the number of hosts in the population, so that V/H is the vector density per host. Finally, P is the proportion of vectors which when infected become infectious. There is an important asymmetry in the basic model, in that humans can be bitten by a virtually unlimited number of mosquitoes, whereas mosquitoes are limited in the number of blood meals they will take from humans. This asymmetry plays a role in determining the numbers of vectors that are, on average, likely to bite the initial infectious host and so spread the infection. This implies that the density of vectors, relative to the human host, is critical in determining the potential for an epidemic.

where m is the mean number of sexual partners and s2 the variance. The number of susceptibles is no longer included in this equation; instead, there is an estimate of the probability of encountering an infectious partner. Note also that the variance has a much greater effect than the mean on R0, which implies that the segment of the population with the highest rates of partner change makes a disproportionate contribution to persistence and spread.

Implications of Infection Dynamic Theory for Public Health Practice

Figure 2.2 shows how the values of R0 (see Table 2.1) relate to the proportion of the population that must be vaccinated to achieve 'herd immunity' and eradicate an infection (Anderson and May, 1982a). These values are approximate— they also depend on a number of genetic and

Fig 2.2 The critical proportion of the population that theoretically needs to be vaccinated to eradicate diseases with different reproductive rates. The curve shows the threshold condition between disease eradication and persistence. Typical reproductive rates for measles in developing countries and for fox rabies in Europe are shown by the dashed lines. From Anderson and May (1991), with permission

Fig 2.2 The critical proportion of the population that theoretically needs to be vaccinated to eradicate diseases with different reproductive rates. The curve shows the threshold condition between disease eradication and persistence. Typical reproductive rates for measles in developing countries and for fox rabies in Europe are shown by the dashed lines. From Anderson and May (1991), with permission social factors (see above)—but provide a surprisingly useful guide. They indicate that the higher the value of R0, the greater the coverage required to achieve eradication. This helps explain why smallpox, estimated R0 3-4, was the first disease to be eradicated from the world, and why polio (R0 5-6) may be the second. It also helps explain why the same coverage with MMR (mumps/ measles/rubella) vaccine in the USA has effectively eradicated rubella (R0 6-9), but not measles (R0 11-17), as a public health problem.

Vector-borne protozoan parasites, such as those that cause malaria, require that the vector population be taken into account in estimating R0, but the dynamics otherwise have behaviours similar to those of directly transmitted microparasites. Initial estimates of R0 for Plasmodium suggested that the value was extremely high, of the order of 50-100, with consequent discouraging prospects for control. Such high rates of transmission imply the need for 99% coverage before the age of 3 months with a vaccine that gives life-long protection (Molineaux and Gra-miccia, 1980). More recent work, however, which takes into account the antigenic diversity of Plasmodium, suggests much lower transmissibil-ity and an R0 value that is an order of magnitude less (Gupta et al., 1994). This more encouraging conclusion, for which empirical evidence is being actively sought (Dye and Targett, 1994), suggests that a practical malaria vaccine is a real possibility but argues for a vaccine that is generic rather than strain-specific.

Another perspective on the importance of R0 is provided by sexually transmitted infections. In this case R0 is largely dependent on the rate of sexual partner change (see above). The rates of partner change required for HIV to persist and spread are much lower than for most other STDs despite a low transmission probability, largely because of the long duration of infectiousness. It appears that where there is a high probability of transmission (perhaps because of concurrent predisposing STDs) (Laga et al., 1994) or a high rate of partner change, HIV can spread rapidly, while elsewhere the virus may require its full infectious period of some 10 years to spread (Anderson et al., 1991). In the latter case, the epidemic may develop over a period of decades rather than months, which may help explain the marked global variation in the rate of development of the HIV/AIDS pandemic.

STDs also provide a good example of how population-mixing behaviours affect R0 and the prospects for control. As with all other types of infections, some (often few) individuals are more likely to acquire and transmit infection, perhaps because of behavioural or genetic characteristics. If these individuals mix randomly in the population, then the effective reproductive number is greater than the simple average. If those with high risk tend to mix with others with high risk (assortative mixing), then the reproductive number is likely to be high within this group and low outside it (Garnett and Anderson, 1993). This may imply that some infections can only persist 1) because of the existence of the high risk group, with obvious implications for the targeting of control (Garnett and Anderson, 1995).

Controlling Disease in Low-income Countries

Population dynamic theory has particular relevance for the design of control programs for developing societies, where populations tend to have high intrinsic rates of increase, low average age and high density. One effect that we have already considered is how the high rate of population increase may result in short inter-epidemic periods and, therefore, much more frequent epidemics.

Age effects may be even more important, since Ro is related to the average age at first infection. This is illustrated by Figure 2.3, which shows the age profiles for measles, mumps and rubella seroprevalence in the same Caribbean community. It is apparent that the age at first infection scales in direct proportion to incidence and in inverse proportion to Ro. This indicates that the opportunity to vaccinate—after the decline of maternal antibody but before the occurrence of natural infection—is shortest for measles and is inversely proportional to Ro. The greater the value of Ro, the narrower is the vaccination 'window'. At extreme values of Ro, which may occur in some developing societies, vaccination at a single age may be insufficient to eradicate infection, even with ioo% coverage (McLean and Anderson, 1986).

Population density may also be important in designing an intervention strategy. Populations are typically distributed heterogeneously in space, with some people living in dense urban aggregations (high R) and others isolated in villages (low R). These rural-urban differences

Fig 2.3

Age-specific seroprevalence of mumps, measles and rubella in a St Lucian community (modified from Cox et al, 1989)

Fig 2.3

Age (years)

Age-specific seroprevalence of mumps, measles and rubella in a St Lucian community (modified from Cox et al, 1989)

tend to be much greater in developing societies, and can result in values of R0 which are, on average, greater than suggested by estimates based on the assumption of spatial homogeneity. This implies that eradication will be more difficult to achieve in such societies. It may also imply a need to target vaccination coverage in relation to group size, with denser urban populations receiving higher coverage (Anderson and May, 1990).

The high transmission rates found in many developing societies also tend to increase the risks associated with incomplete or partial coverage. Population dynamic theory predicts that, in general, incomplete vaccination coverage will reduce the rate of transmission and increase the age at first infection. In the case of rubella, this age-shift has particularly important consequences, since the primary public health concern is to prevent women contracting rubella once they have reached child-bearing age and so avoid congenital rubella syndrome. The carefully documented experiences of rubella vaccination in the north (Ukkonen and von Bronsdorff, 1988), indicate a remarkable coincidence between theoretical prediction and observation (Anderson, 1994). This could have important implications for developing societies, where R0 is typically higher, such that first infection with rubella is at 5 years of age rather than the 8-9 years seen in the north. The same robust theory predicts that vaccinating 50% of all 2 year-olds in developing societies could double the incidence of congenital rubella syndrome. This argues strongly against the introduction of MMR vaccination, particularly in developing societies, without first ensuring the protection of women of child-bearing age.

The Special Case of Helminth Infections

Helminth infections (called 'macroparasites' in this context) have fundamentally different population dynamic characteristics from the microparasites, since the dynamics are primarily determined by the number of worms present (the intensity of infection), rather than the number of hosts infected (Anderson and May, 1979; Bundy, 1988). The basic reproductive number for worm infection is defined in terms of the number of female offspring produced by a female worm in her lifetime, so that the basic unit of transmission is the individual worm, not the individual case infection. Intensity is also believed to be the major determinant of morbidity. Thus, the major aim in controlling helminth infection should be to reduce the overall worm burden in a population, rather than to reduce the number of cases of infection.

Reducing the average intensity of infection may not, however, be enough to achieve significant control. Worm burdens exhibit heterogeneity amongst individuals and so reducing the average burden may still leave some individuals with sufficient burden to cause morbidity and sustain transmission. Furthermore, individuals appear to be predisposed to high (or low) infection intensity (Schad and Anderson, 1985; Bundy et al., 1985; Elkins et al., 1986), such that those with heavy worm burdens tend to reacquire above-average intensity infection, even after successful treatment (Keymer and Pagel, 1990; Hall et al., 1992). Such 'wormy' individuals are at greater risk of morbidity and make a disproportionate contribution to the transmission of infection; they would appear, therefore, to be the most important targets for treatment or vaccination. Yet they may be heavily infected precisely because of some failure of immunocompetence, which suggests that a helminth vaccine must be able to convert low responders into high responders if vaccination on a community scale is to be successful (Anderson and May, 1985). Fortunately there is some evidence to suggest that predisposition relates more to exposure than immunological resistance (Chan et al., 1994), but resolution of the causes of predisposition is clearly important to determining the likely success of control measures. Here we will consider how an understanding of population dynamics might help guide the development of putative helminth vaccines, and has helped the evolution of more effective approaches to control by chemotherapy (Chan et al., 1995).

Helminth population dynamics are important in terms of determining the optimal age for delivery of any future helminth vaccine. Immunisation in the pre-school years may be inappropriate for the helminthiases, since for the schistosomes and some of the major nematode species the peak of intensity, and of potential risk of disease, is attained several years later (Bundy, 1988; Anderson and Medley, 1985). This has potential consequences for the required duration of protection induced by vaccination.

Figure 2.4 shows the results of a preliminary model of the impact of vaccination on helminth dynamics, and compares the effects of vaccine delivery at different ages. The results indicate that if the protection induced has a half-life of some 12 years or less, there is benefit in vaccination at 5 years of age rather than 1 year of age. With increasing duration of protection, however, delivery of a vaccine as part of the EPI package becomes the preferred option. This implies that only a vaccine giving very long-lived protection would qualify for delivery as part of the EPI package. The results also indicate that the benefits of a vaccine giving short-lived protection, with either strategy, are likely to be small (much less than 50% reduction in mean worm burden) and that all strategies tend to shift the peak of intensity and potential morbidity into the older age classes, with unknown consequences.

Similar, although rather more dramatic, consequences are predicted (Crombie and Anderson, 1985; Woolhouse, 1991) if it is assumed that the low parasite infection intensity in adults is a consequence of acquired immunity. In this case, vaccine-induced protection may prevent the natural acquisition of immunity with age, and the acquired immunity model predicts that the eventual loss of protection will result in more rapid acquisition of infection in the older age classes. Furthermore, the peak intensity may exceed that observed in the absence of vaccination unless the protection persists beyond an age (5-15 years) at which infection exposure is negligible.

The markedly convex age profiles of infection intensity, with mean worm burdens for schisto-somes and more nematodes showing a peak in the 'school-age' population, have consequences for the population dynamics and hence control of helminths (Anderson and May, 1982b). Field studies have shown that these intensely infected age groups may harbour 70-90% of the total worm population (Bundy and Medley, 1992) and theory predicts that these age groups would therefore make a disproportionate contribution to transmission (and morbidity). Control strategies which specifically target this high-risk group of school-age children have shown that this approach reduces transmission of schistosomes to a rate similar to that achieved when the whole population is treated (Butterworth, 1991) and reduces transmission of nematodes to the untreated adult population (Bundy et al., 1990). Furthermore, models of helminth dynamics accurately predict the outcome of chemotherapy control measures (Chan and Bundy, 1997). Thus, theoretical prediction is again supported by observation, and has led to the practical implementation of school-age targeted control programs (Savioli et al., 1992; Partnership for Child Development, 1997).

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