Neutral model.

The key quantity in this problem is the number of parasites in (or associated with) a host, which we denote by N. Since the number of parasites varies from host to host, N is a random variable and our model describes the statistics of N. Assuming there is no vertical transmission of parasites, the essence of the model is that a host is born parasite free and then has a random number of encounters with a parasite/source of parasites. An encounter is characterised by its success, as measured by the number of parasites that are ultimately carried by the host, due to the encounter.

We denote the number of encounters of a host with a parasite/source of parasites by ℰ, and the success of the j’th encounter by S_j where j = 1, 2, …, ℰ. The number of parasites in a host, N, is then given by a sum of successes of the different encounters of the host with a parasite/source of parasites: (1) We take the S_j and ℰ to be independent random variables whose possible values are 0, 1, 2, … and the value of the sum is taken to zero if ℰ equals zero. The S_j are all taken to have the same probability distribution, and hence have identical expected values and identical variances, which we write as E[S] and Var(S), respectively.

Equation (1) constitutes a fairly general viewpoint for the infection of hosts by a single parasite species. While Eq. (1) looks simple, this appearance is deceptive; N is a quantity which is composed of a random number (ℰ) of random variables (the S_j) and hence constitutes a compound random variable [22], which requires two probability distributions to characterise it. A number of different biological scenarios may be considered for Eq. (1). The variable, ℰ, can represent the number of independent encounters of a given host with individual parasites. Alternatively, ℰ can represent the number of encounters of a host with sources of parasites such as: infected sites (for air, soil or water-borne diseases), infected vectors (for vector-borne diseases) or infected hosts for directly transmitted parasites. Similarly, the variable S_j could represent the viability of a single parasite, and hence would only take two values, namely 0 or 1, corresponding to death or survival of the parasite. However, the variable S_j could also account for multiple infections/contacts and/or within host parasite viability and reproduction, in which case S_j is capable of taking values larger than 1, and the discrete distribution it follows would reflect this feature.

It is worth mentioning that the causes of the random variability in success can be associated with the parasite or with the host. Such variability can be due to a finite random sampling of parasite diversity in each host, just as it can reflect random variability in host resistance or condition. But, whatever the definition of ℰ and S_j and their causes, the key assumption of this neutral model is that the levels of random variability in encounters and success are the same for all hosts. In other words, encounters and success are realizations of random variables whose distribution is common to all individuals, and hosts are thus ‘statistically equivalent’. Accordingly, there is no intrinsic difference between hosts and these variations can be seen as a form of demographic stochasticity, whose effects on parasite distribution can be evaluated from the neutral model.

Heterogeneous model.

The above model can now be expanded to incorporate host heterogeneity, by allowing different types of hosts, labelled t = 1, 2 … h. Consider a randomly captured host of type t. The number of parasites in the host is then related to the following: (i) the number of encounters, specific to a host of type t, that the host has with a parasite/source of parasites, and which we write as ℰ_t, and (ii) the success that is associated with each encounter of a host of type t, which we write as $S_j^{(t)}$. The number of parasites in a randomly captured host can be written as (2) In this equation the H_t indicate the type of host captured. Only one of the H_t’s takes the value of unity, while the remainder are zero, and the probability with which H_t = 1 is p_t. Thus explicitly, (H₁, H₂, …, H_h) constitutes a multinomial random variable with parameters 1 and (p₁, p₂, …, p_h).

As in the neutral model, there are various alternative biological scenarios that can lead to heterogeneity in the rate at which different hosts encounter parasites. The rate of encounters can vary with the level of host foraging activities [23, 24], the characteristics of the host habitat [25–27], the spatial and temporal co-distributions of hosts and vectors [28–31]. Similarly, the rate of parasite success in the host can depend on the individual level of immunity [32–34] or physiological status [35–38].

Neutral model.

All results given below, for the neutral model, are established in Methods A.

We begin with Eq. (1), which relates the number of parasites of an individual host, N, to the number of encounters that they have with a parasite/source of parasites, ℰ, and the ultimate success of these parasites on different encounters, i.e., the S_j. A direct consequence of Eq. (1) combined with the assumptions made above is that the expected value of the number of parasites of a host, E[N], is related to the expected number of encounters, E[ℰ], and the expected level of success, E[S], as (3) Additionally, the relationship between the variance of N and the various statistics of ℰ and S_j can be shown to be (4) Equations (3) and (4) are standard results for compound random variables (see e.g., [22]), and can be found in Methods A, as part of the analysis.

It is important to point out that Eqs. (3) and (4) apply for any distributions of ℰ and S_j, and hence for any distribution of N that results from Eq. (1). In particular, Eqs. (3) and (4) apply whether or not N has a negative binomial distribution. This is important since, using our framework, it can be shown that conditions need to be satisfied, by the distributions of encounters and success, in order to obtain a negative binomial distribution (see Methods A). These conditions appear to be rather specific for the distribution of successes, as we show in the second example in Methods A.

As expected, the mean number of contacts and successes combine in a symmetric way to produce the mean number of parasites in a host [12]. However, the asymmetric way that the summary statistics of encounters and success enter the result for the variance in the number of parasites in a host, Eq. (4), means that these two processes cannot be regarded as having equivalent effects on the variance.

We note that the variance of N depends on the variance of the number of contacts and the variance of success, however, these variances are weighted differently in Eq. (4), namely by the squared mean of success and the mean of contact, respectively. Such an asymmetry of the weightings allows the distribution of parasites in hosts to take a variety of forms. We thus used a measure of aggregation that is applicable to a general distribution, namely the variance-to-mean-ratio of the number of parasites in hosts. Values of the variance-to-mean-ratio that are greater than 1 are associated with aggregation [39]. Using Eqs. (3) and (4) this ratio can be expressed as the sum of two terms, involving the variance-to-mean-ratio in encounters and success: (5) The two contributions in this equation, of the variance-to-mean-ratios of encounters and success, are weighted differently; the average success multiplies the contribution of the variance-to-mean-ratio in encounters. The simplicity of this quantitative outcome of our neutral model leads to two general insights into the emergence of aggregation from random variation associated with encounters and success. First, aggregation decreases with the mean number of contacts, E[ℰ], until the level of aggregation reaches an asymptotic level of Var(S)/E[S], and only reflects randomness in success (Fig. 1A). Second, the mean level of success, E[S], can have a more complex effect on aggregation than the mean number of contacts because of its presence in two places in Eq. (5) (Fig. 1B).

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Figure 1. This figure illustrates how aggregation varies with either host-parasite encounters or parasite success (in infecting hosts).

In Panel A we have adopted the values Var(S)/E[S] = 1 and Var(ℰ)E[S] = 1. We show how parasite aggregation varies with the mean number of encounters. The level of aggregation decreases with the number of encounters, and asymptotically approaches a value that depends only on the variance-to-mean ratio of parasite success, i.e., Var(S)/E[S]. In Panel B we have adopted the values Var(ℰ)/E[ℰ] = 1 and Var(S) = 0.5. We show how parasite aggregation varies with the mean parasite success in infecting hosts. The level of aggregation initially decreases with the average success of parasites in infecting their hosts until a minimum is reached at a value of E[S] of $M=\sqrt{\mathrm{\text{Var}}(S)E[ℰ]/\mathrm{\text{Var}}(ℰ)}$, as indicated on the abscissa. The level of aggregation then starts to increase, with an asymptotically achieved slope that is directly proportional to the variance-to-mean-ratio of encounters, i.e., Var(ℰ)/E[ℰ].

https://doi.org/10.1371/journal.pone.0116893.g001

Any increase in E[S] (at a fixed value of Var(S)) decreases the variance-to-mean-ratio of success, and thus lowers the level of aggregation. However, increasing E[S] simultaneously increases the contribution of randomness in encounters by magnifying the differences between hosts that had low and high levels of contacts with parasites. The balance between these two antagonistic effects leads to an intermediate level of average success of $E[S]=\sqrt{\mathrm{\text{Var}}(S)E[ℰ]/\mathrm{\text{Var}}(ℰ)}$ that corresponds to a lowest level of aggregation of $2\sqrt{\mathrm{\text{Var}}(S)\mathrm{\text{Var}}(ℰ)/E[ℰ]}$, while at larger E[S] the level of aggregation asymptotically changes as E[S]Var(ℰ)/E[ℰ].

Although we aim at providing a mechanistic alternative to the negative binomial distribution to describe parasite distributions, it is worth mentioning that the standard measure of aggregation, the parameter k of the negative binomial distribution distribution, can be expressed in terms of the summary statistics of ℰ and S_j (see Methods A). This could be used when the specific conditions for the distribution to be a negative binomial distribution are fulfilled (see Methods A).

Heterogeneous model.

All results given below, for the heterogeneous model, are established in Methods B.

The above results can be expanded to multiple types of hosts by using Eq. (2) that relates the number of parasites of an individual host, N, to random variables associated with encounters and success, for different host types. The mean and variance of the number of encounters are then conditional on the host type, and we write these as E[ℰ∣t] and Var(ℰ∣t), respectively. Similarly, we denote the mean and variance of the success of a parasite to infect hosts of type t by E[S∣t] and Var(S∣t), respectively. We find that the expected number of parasites in a host is a weighted average over contributions from different possible host types: (6) (see Methods B for details) where p_t denotes the frequency of host type t. Thus, the mean number of parasites in a host depends on the mean number of encounters and success in each host type and on the way these are correlated across host types. Intuitively, we expect the total number of parasites to be larger when the hosts with the highest rate of encounters with parasites are the more susceptible ones. This appears more obviously in an alternative expression for E[N] that is completely equivalent to Eq. (6), namely (7) Here $\mu ={\sum }_{t=1}^h{p}_{t}E[ℰ\mid t]$ and $m={\sum }_{t=1}^h{p}_{t}E[S\mid t]$ represent the means of encounters and successes, across different host types, while $\mathrm{\text{Cov}}(ℰ,S)={\sum }_{t=1}^h{p}_t(E[ℰ\mid t]-\mu )(E[S\mid t]-m)$ measures how encounter and success are correlated with each other across different host types. Equation (7) clearly shows that the mean number of parasites is linearly related to the covariance between encounters and success.

A negative correlation between encounters and infection success can emerge as a result of processes related to host group size. The rate of encounter with directly transmitted parasites is positively correlated with the size of the group in many host species [40]. But at the same time, infection success of ectoparasites may decrease with the number of individuals through enhanced allo-grooming behaviour [31, 41]. Alternatively, a positive correlation can result from variation associated with home range. A small home range size in mammals is usually associated with increased host densities that favours parasite encounters and simultaneously affects host condition, which has a positive effect on infection success [16].

The variance of the distribution of parasites in hosts, Var(N), that incorporates random individual variation and actual heterogeneity in contacts and success can also be derived (see Methods B) and is given by (8) This result indicates that there are contributions to Var(N) from individual variation that is within host types (first sum), and also from effects of heterogeneity between host types (second sum).

The expression in Eq. (8) can be approximated, to provide a more explicit relationship between the variance in parasite numbers in hosts and various correlations involving encounters and success. We achieve this by determining the leading corrections to the results of the homogeneous model (see Methods B for details). This leads to an equation for the variance in the number of parasites in a host of the form (9) where ${\sigma }^2={\displaystyle \sum _{t=1}^h{p}_t}\text{Var}\left(ℰ|t\right)$ and ${\nu }^2={\sum }_{t=1}^h{p}_t\mathrm{\text{Var}}(S\mid t)$ stand for means, across all host types, of the variances of encounter and success. The leading two terms in Eq. (9) are equivalent to the terms in the homogeneous model, Eq. (4), so that conclusions on the asymmetric contributions of the levels of contact and success remain, although they now apply to averages across host types. Obviously, the quadratic correlation terms can potentially make the whole expression of Eq. (9) substantially more complicated, as one should expect from the level of complexity incorporated into the heterogeneous model. However, these additional terms can be fully identified from Eqs. (B3) and (B9) of Methods B, and they are related to the variances and covariances of the basic summary statistics (mean and variance) describing the distribution of encounters and success conditional on host types (see Eq. (B10) in Methods B).

The variance-to-mean-ratio of the number of parasites amongst hosts can, using Eqs. (7) and (9), be approximated as (10) where, as above, the quadratic correlation terms take into account leading effects of between-type correlations. Thus again, the leading terms in Eq. (10) on the right hand side are equivalent to the results of the homogeneous model (Eq. (5)), although they now apply to means across host types. Accordingly, the highest levels of aggregation are expected when strong levels of random individual variation in encounters (σ²) are associated with high average rates of success (m) as the latter magnifies the former. Similarly, aggregation increases when there is large random individual variation in success (ν²) are associated with high average rates of encounters (μ). It is not obvious how to identify host-parasite systems that could serve as examples of such co-variations since random individual variation in encounters and success are typically not measured in the field. However, there are a few published experimental results that suggest that rodents exposed to water-borne, air-borne or even soil-transmitted parasites could be good candidates for measurement of the level of such demographic stochasticity (see discussion). As expected, parasites will also be strongly aggregated when hosts with a high average rate of encounters are those on which parasites are successful, since this would lead to high mean parasite load and a lower variance to the mean ratio.

As explained above, such a positive correlation could be associated with a small host home range, while a negative correlation is expected in host groups of large size.

The quadratic correlation terms of Eq. (10) are directly related to those appearing in Eq. (B12) of Methods B, and thus also correspond to variances and covariances of basic summary statistics of the distributions of encounters and success. These expressions make up an explicit link between the patterns of aggregation and basic statistics on the level of host random individual variation and actual heterogeneity in the processes of encounters and success. Although no general results can be derived from these quadratic correlations because of their remaining complexity, Eq. (9) and (10) provide the theoretical background necessary to gain a better knowledge of the parasite distribution, if combined with a good empirical knowledge of the undoubtedly system-specific correlations between encounters and success.

General results.

We begin with a discrete random variable N, which takes the values 0, 1, 2, …. The probability generating function for N is defined by $G_N(\lambda )=E[{\lambda }^N]={\sum }_{n=0}^{\infty }\mathrm{\text{Prob}}(N=n){\lambda }^n$ where E[…] denotes an expected (or average value) and λ is a real variable. All information about the distribution of N is contained in G_N(λ) [22]. Henceforth we shall refer to probability generating functions simply as generating functions.

We take N to represent the number of parasites in a randomly picked host, as given by Eq. (1) of the main text, namely $N={\sum }_{j=1}^{ℰ}{S}_j$ where ℰ is the number of exposures of a host to a parasite/source of parasites and S_j is the ultimate success of a parasite on the j’th exposure. The value of N is taken to be 0 if ℰ = 0. We assume ℰ and the S_j are all independent random variables that can take the values 0, 1, 2, …. We take all S_j to have identical distributions, with expected value E[S] and variance Var(S). The generating function of N, that follows from Eq. (1), is [22] (A1) Differentiating Eq. (A1) once or twice with respect to λ and then setting λ = 1 leads to results for E[N] and E[N(N − 1)] that correspond to the following relations between the means and variances of N, ℰ and S: (A2)

Particular results on the negative binomial distribution.

To establish some particular but informative results, we take N to have a negative binomial distribution. The form of this distribution that we use follows Anderson and May [6]. It has parameters m and k and is defined by (A3) where Γ(x) denotes Euler’s Gamma function. This distribution has a mean of E[N] = m and a variance of Var(N) = m²/k + m. The parameter k is often used as a measure of aggregation of the distribution and it can be approximated as (A4)

Constraints on the distribution of ℰ and S that lead to a negative binomial distribution.

It is convenient to express the generating function of N in terms of the parameter (A5) and then find that (A6)

When N has the distribution of Eq. (A3), the equation of the generating function, Eq. (A1), imposes a relationship between the generating functions G_ℰ(λ) and G_S(λ), namely (A7) Equivalently, this equation expresses a relationship between the probability distributions of ℰ and S_j. We note that if we specify either G_ℰ(λ) or G_S(λ), we can exploit Eq. (A7) to determine the other generating function. This generating function can then be used to determine the corresponding probability distribution. Note however, that this procedure does not always work: it sometimes gives rise to an invalid probability distribution, because some of the ‘probabilities’ obtained are negative. We give two illustrative examples of this usage of Eq. (A7). In the first example we determine the distribution of ℰ, after assuming the distribution of S. In the second example, the distribution of S is determined after assuming the distribution of ℰ. In this case we find that a probability distribution for S follows only for restricted values of some of the parameters.

Example 1

If S can only take the values 0 and 1, and these occur with probabilities 1 − α and α respectively, then G_S(λ) = 1 − α + αλ. It follows from Eq. (A7) that (1 + r − λr)^−k = G_ℰ(1 − α + αλ). Solving this equation for G_ℰ(λ) yields G_ℰ(λ) = (1 + r/α − λr/α)^−k which, on comparison with Eq. (A6), corresponds to ℰ having a negative binomial distribution with parameters m/α and k: (A8)

This example leads to E[ℰ] = m/α, E[S] = α, Var(ℰ) = m/α + m²/(α² k) and Var(S) = α(1 − α).

Example 2

If ℰ has a Poisson distribution, with Prob(ℰ = n) = βⁿ e^−β/n! for n = 0, 1, 2, … then G_ℰ(λ) = exp(βλ − β). Using this result in Eq. (A7) yields (1 + r − λr)^−k = exp(βG_S(λ) − β), and this can be directly solved for G_S(λ) with the result G_S(λ) = 1 − (k/β) ln (1 + r − λr). Expanding G_S(λ) in powers of λ leads to (A9) This distribution can be viewed as a mixture of two distributions: a unit probability mass located at s = 0 with weight 1 − (k/β) ln (1 + m/k), and a logarithmic distribution with overall weight (k/β) ln (1 + m/k). The values of the parameters m, k and β cannot take any values, but must be chosen so the probabilities of different values of S (in particular S = 0) are all non-negative. For example, at fixed values of k and m, the parameter β must have some restrictions placed upon it; in this case we can only consider values of β which yield 1 − (k/β) ln (1 + m/k) ≥ 0 so that Prob(S = 0) ≥ 0. Defining (A10) we thus require β ≥ β_min for the probabilities of all values of S to be non-negative.

This example leads to E[ℰ] = β, E[S] = m/β, Var(ℰ) = β and Var(S) = (m/β)(1 + m/k − m/β).

Mean of N.

We determine the mean of N by differentiating G_N(λ) once with respect to λ and then setting λ = 1. This yields (B4) Equation (B4) can be rewritten in the alternative form (B5) where C_{a, b, c, d} is defined in Eq. (B3) and explicitly, C_{1, 0, 1, 0} is the covariance (B6)

Variance of N.

We first determine the expected value E[N(N − 1)] by differentiating G_N(λ) (Eq. (B2)) twice with respect to λ and then setting λ = 1. This yields (B7) From this result and Eq. (B4) we obtain (B8) Hence having multiple types leads to an extra level of averaging (as indicated by the sums in Eq. (B8)) plus an additional positive term in the variance reflecting an aspect of variation between different host types (the final sum in Eq. (B8).

The expression in Eq. (B8) is somewhat complicated, but we can approximate it, under the assumption that deviations between different host types that are higher than quadratic order can be neglected. Using the delta technique (see e.g., [60]), we obtain the expression (B9) which contains a leading term that is derived from average over all types, and various quadratic correlation terms that arise from between type variation.

We can write the C_{a, b, c, d} that appear in Eq. (B9) in a more suggestive form, using T to denote a random variable whose value corresponds to the type of host captured. We then have (B10) Using the above forms for the C_{a, b, c, d} we can write Eqs. (B5) and (B9) as (B11) (B12)