# Contact rates density and frequencydependent transmission

For most infections, it has often been assumed that the contact rate c increases in proportion to the density of the population, N/A, where A is the area occupied by the population, i.e. the denser the population, the more hosts come into contact with one another (or vectors contact hosts). Assuming for simplicity that A remains constant, the Ns in the equation then cancel, all the other constants can be combined into a single constant P, the 'transmission coefficient', and the equation becomes:

the rate of production of new infections = P • S • I. (12.3)

This, unsurprisingly, is known as density-dependent transmission.

On the other hand, it has long been asserted that for sexually transmitted diseases, the contact rate is constant: the frequency of sexual contacts is independent of population density. This time the equation becomes:

the rate of production of new infections

where the transmission coefficient again combines all the other constants but this time acquires a 'prime', P', because the combination of constants is slightly different. This is known as frequency-dependent transmission.

Increasingly, however, it has become apparent that the assumed simple correspondence between sexual transmission and frequency dependence on the one hand, and all other types of infection and density dependence on the other, is incorrect. For example, when density and frequency dependence were compared as descriptors of the transmission dynamics of cowpox virus, which is not sexually transmitted, in natural populations of bank voles (Clethrionomys glareolus), frequency dependence appeared, if anything, to be superior (Begon et al., 1998). Frequency dependence appears to be a better descriptor than density dependence, too, for a number of (nonsexually transmitted) infections of insects (Fenton et al., 2002). One likely explanation in such cases is that sexual contact is not the only aspect of behavior for which the contact rate varies little with population density: many social contacts, territory defense for instance, may come into the same category.

Secondly, P • S • I and P' • S • I/N are themselves increasingly recognized ends of a spectrum (e.g. McCallum et al., 2001) as, at best, benchmarks against which real examples of transmission might be measured, rather than exact descriptors of the dynamics; or perhaps as ends of a spectrum along which real transmission terms could be assembled. For example, fitting the term PSTy to the transmission dynamics of granulovirus infection in larvae of the moth Plodia interpunctella revealed that the best fit was not to 'pure' density-dependent transmission, PSI, but to P'S112I014 (Figure 12.8).

density-dependent transmission frequency-dependent transmission

Figure 12.8 Estimating the transmission coefficient at various densities of (a) susceptible hosts and (b) infectious cadavers during the transmission of a granulovirus amongst moths, Plodia interpunctella, showed that the coefficient appeared to increase with the former and decrease with the latter. This is contrary to the expectations from density-dependent transmission (an apparently constant coefficient in both cases). (After Knell et al., 1998.)

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Host density

Figure 12.9 The spatial spread of damping-off disease in a population of radish plants, Raphanus sativus, caused by the fungus Rhizoctonia solani. Following initiation of the disease at isolated plants (light squares), the epidemic spreads rapidly to neighboring plants (dark squares), resulting in patches of damped-off plants (picture on the right). (Courtesy of W. Otten and C.A. Gilligan, Cambridge University.)

Figure 12.9 The spatial spread of damping-off disease in a population of radish plants, Raphanus sativus, caused by the fungus Rhizoctonia solani. Following initiation of the disease at isolated plants (light squares), the epidemic spreads rapidly to neighboring plants (dark squares), resulting in patches of damped-off plants (picture on the right). (Courtesy of W. Otten and C.A. Gilligan, Cambridge University.)

In other words, transmission was greater than expected (exponent greater than 1) at higher densities of susceptible hosts, probably because hosts at higher densities were short of food, moved more, and consumed more infectious material. But it was lower than expected (exponent less than 1) at higher densities of infectious host cadavers, probably because of strongly differential susceptibility amongst the hosts, such that the most susceptible become infected even at low cadaver densities, but the least susceptible remain uninfected even as cadaver density increases.

Turning from the contact rate, c, to the UN term, there has usually been a simplifying assumption that this can be based on numbers from the whole of a population. In reality, however, transmission typically occurs locally, between nearby individuals. In other words, use of such a term assumes either that all individuals in a population are intermingling freely with one another, or, slightly more realistically, that individuals are distributed approximately evenly across the population, so that all susceptibles are subject to roughly the same probability of a contact being with an infectious individual, IN. The reality, however, is that there are likely to be hot spots of infection in a population, where I/N is high, and corresponding cool zones. Transmission, therefore, often gives rise to spatial waves of infection passing through a population (e.g. Figure 12.9), rather than simply the local hot spots

Figure 12.10 The effect of resistant forms in slowing down the spread of damping-off epidemics caused by the fungus Rhizoctonia solani. (a) Progress of epidemics in populations following the introduction of R. solani into a susceptible population (radish, Raphanus sativus: o), a partially resistant population (mustard, Sinapsis alba: •) or a 50 : 50 mixture of the two (a). (b) A simulation showing that when 40% of the plants in a population are of a resistant variety, the spread of a damping-off epidemic following its introduction can be prevented. White squares are resistant plants, black squares are infected, and gray squares susceptible. Infection can only be transmitted to an adjacent plant (sharing a 'side'). Here, the epidemic can spread no further. (Courtesy of W. Otten, J. Ludlam and C.A. Gilligan, Cambridge University.)

overall rise in infection implied by a global transmission term like PSL This illustrates a very general point in modeling: that is, the price paid in diminished realism when a complex process is boiled down into a simple term (such as PSJ). None the less, as we shall see (and have seen previously in other contexts) without such simple terms to help us, progress in understanding complex processes would be impossible.

12.4.4 Host diversity and the spatial spread of disease

The further that hosts are isolated from one another, the more remote are the chances that a parasite will spread between them. It is perhaps no surprise, then, that the major disease epidemics known amongst plants have occurred in crops that are not islands in a sea of other vegetation, but 'continents' - large areas of land occupied by one single species (and often by one single variety of that species). Conversely, the spatial spread of an infection can be slowed down or even stopped by mixtures of susceptible and resistant species or varieties (Figure 12.10). A rather similar effect is described in Section 22.3.1.1, for Lyme disease in the United States, where a variety of host species that are incompetent in transmitting the spirochete pathogen 'dilute' transmission between members of the most competent species.

In agricultural practice, resistant cultivars offer a challenge to evolving parasites: mutants that can attack the resistant strain have an immediate gain in fitness. New, disease-resistant crop varieties therefore tend to be widely adopted into commercial practice; but they then often succumb, rather suddenly, to a different race of the pathogen. A new resistant strain of crop is then used, and in due course a new race of pathogen emerges. This 'boom and bust' cycle is repeated endlessly and keeps the pathogen in a continually evolving condition, and plant breeders in continual employment. An escape from the cycle can be gained by the deliberate mixing of varieties so that the crop is dominated neither by one virulent race of the pathogen nor by one susceptible form of the crop itself.

In nature there may be a particular risk of disease spreading from perennial plants to seedlings of the same species growing close to them. If this were commonly the case, it could contribute to the species richness of communities by preventing the development of monocultures. This has been called the Janzen-Connell effect. In an especially complete test for the effect, Packer and Clay (2000) showed for black cherry, Prunus serótina, trees in a woodland in Indiana, first, that seedlings were indeed less likely to survive close to their parents (Figure 12.11a). Second, they showed that it was something in the soil close to the parents that reduced survival (Figure 12.11b), though this was only apparent at high seedling density, and the effect could be removed by sterilizing the soil. This suggests a pathogen, which high densities of seedlings, close to the parent, amplify and transmit to other seedlings. In fact, dying

Figure 12.10 The effect of resistant forms in slowing down the spread of damping-off epidemics caused by the fungus Rhizoctonia solani. (a) Progress of epidemics in populations following the introduction of R. solani into a susceptible population (radish, Raphanus sativus: o), a partially resistant population (mustard, Sinapsis alba: •) or a 50 : 50 mixture of the two (a). (b) A simulation showing that when 40% of the plants in a population are of a resistant variety, the spread of a damping-off epidemic following its introduction can be prevented. White squares are resistant plants, black squares are infected, and gray squares susceptible. Infection can only be transmitted to an adjacent plant (sharing a 'side'). Here, the epidemic can spread no further. (Courtesy of W. Otten, J. Ludlam and C.A. Gilligan, Cambridge University.)

the Janzen-Connell effect

Figure 12.11 (a) The relationship between distance to parent, initial seedling germination (a) and probability of seedling survival over time (dashed lines: o, after 4 months; •, after 16 months); n = 974 seedlings from beneath six trees. (b) The effect of distance from parent, seedling density and soil sterilization on seedling survival when seedlings were grown in pots containing soil collected close to or far from their parents. In high-density treatments, survival was significantly greater after the soil collected close to the tree was sterilized. (P < 0.0001). (c) Seedling survival in control and pathogen inoculation treatments (n = 40 per treatment). Control 1, potting mix only; control 2, 5 ml of sterile nutrient-rich fungal growth medium plus potting mix; P1, P2 and P3, three 5 ml replicates of pathogen inoculum plus potting mix. Survival was significantly lower in pathogen treatments compared with controls after 19 days (X2 = 13.8, d.f. = 4, P < 0.05). (After Packer & Clay, 2000.)

Figure 12.11 (a) The relationship between distance to parent, initial seedling germination (a) and probability of seedling survival over time (dashed lines: o, after 4 months; •, after 16 months); n = 974 seedlings from beneath six trees. (b) The effect of distance from parent, seedling density and soil sterilization on seedling survival when seedlings were grown in pots containing soil collected close to or far from their parents. In high-density treatments, survival was significantly greater after the soil collected close to the tree was sterilized. (P < 0.0001). (c) Seedling survival in control and pathogen inoculation treatments (n = 40 per treatment). Control 1, potting mix only; control 2, 5 ml of sterile nutrient-rich fungal growth medium plus potting mix; P1, P2 and P3, three 5 ml replicates of pathogen inoculum plus potting mix. Survival was significantly lower in pathogen treatments compared with controls after 19 days (X2 = 13.8, d.f. = 4, P < 0.05). (After Packer & Clay, 2000.)

seedlings were observed with the symptoms of 'damping off', and the damping-off fungus, Pythium sp., was isolated from dying seedlings and itself caused a significant reduction in seedling survival (Figure 12.11c).

12.4.5 The distribution of parasites within host populations: aggregation

Transmission naturally gives rise to an ever-changing dispersion of parasites within a population of hosts. But if we freeze the frame (or more correctly, carry out a cross-sectional survey of a population at one point in time), then we generate a distribution of parasites within the host population. Such distributions are rarely random. For any particular species of parasite it is usual for many hosts to harbor few or no parasites, and a few hosts to harbor many, i.e. the distributions are usually aggregated or clumped (Figure 12.12).

In such populations, the mean density of parasites (mean number per host) may have little meaning. In a human population in which only one person is infected with anthrax, the mean density of Bacillus anthracis is a particularly useless piece of information. A more useful statistic, especially for microparasites, is the prevalence of infection: the proportion or percentage of a host population that is infected. On the other hand, infection may often vary in severity between individuals and is often clearly related to the number of parasites that they harbor. The number of parasites in or on a particular host is referred to as the intensity of infection. The mean intensity of infection is then the mean number of parasites per host in a population (including those hosts that are not infected).

Aggregations of parasites within hosts may arise because individual hosts vary in their susceptibility to infection (whether due to genetic, behavioral or environmental factors), or because individuals vary in their exposure to parasites (Wilson et al., 2002). The latter is especially likely to arise because of the local nature of transmission, and especially when hosts are relatively immobile. Infection then tends to be concentrated, at least initially, close to an original source of infection, and to be absent in individuals in areas that infection has yet to reach, or where it was previously but the hosts have recovered. It is clear, for example, even without explicit data on the distribution of parasites amongst hosts, that the parasites in Figure 12.9, at any one point in time, were aggregated at high intensities around the wave front - but absent ahead of and after it.