INTRODUCTION
Parasites are often aggregated among their hosts such that a large proportion of the parasite population is present on only a small proportion of the host population (Crofton, Reference Crofton1971; Dobson and Merenlender, Reference Dobson, Merenlender, Toft, Aeschlimann and Bolis1991; Shaw and Dobson, Reference Shaw and Dobson1995). This type of aggregation follows a statistical trend found across a wide range of disciplines known as the ‘Pareto principle’ or ‘80/20 rule’ (80% of parasites are found on 20% of the host population) (Woolhouse et al. Reference Woolhouse, Dye, Etard, Smith, Charlwood, Garnett, Hagan, Hii, Ndhlovu and Quinnell1997; Koch, Reference Koch1999). The aggregated distribution of parasites can often be modelled using the negative binomial distribution, a discrete probability distribution with a k-exponent that defines the degree of aggregation (Crofton, Reference Crofton1971; Shaw and Dobson, Reference Shaw and Dobson1995). Heterogeneities in thse abundance of parasites among hosts arise due to variability in individuals in their rate of exposure to parasites and in their susceptibility to parasites once they have been exposed (Wilson et al. Reference Wilson, Bjørnstad, Dobson, Merler, Poglayen, Randolph, Read, Skorping, Hudson, Rizzoli, P. J., Rizzoli, Grenfell, Heesterbeek and Dobson2002). The aggregation of parasites has important implications for the epidemiology of vector-borne pathogens; if a small fraction of the host population supports the majority of the vector population then those individuals will also be the most important hosts for the transmission of pathogens (Woolhouse et al. Reference Woolhouse, Dye, Etard, Smith, Charlwood, Garnett, Hagan, Hii, Ndhlovu and Quinnell1997; Perkins et al. Reference Perkins, Cattadori, Tagliapietra, Rizzoli and Hudson2003).
Ticks are vectors of numerous bacterial, viral and protozoan pathogens of medical importance (Jongejan and Uilenberg, Reference Jongejan and Uilenberg2004) and small mammals are reservoirs of many of these pathogens (Goodman et al. Reference Goodman, Dennis and Sonenshine2005). The transmission of pathogens from one tick to another, a pre-requisite for the establishment of cycles of infection, may occur via 3 pathways (see Randolph (Reference Randolph1998) for a review). Firstly, female ticks may transmit the pathogen to eggs via transovarial transmission. Secondly, nymphs may feed on, and infect, a host leading to a systemic infection; larval ticks may then acquire an infection by ingesting the blood of the infected host, maintain the infection transstadially, and infect an alternative host during a subsequent bloodmeal as a nymph. Thirdly, ticks may become infected by co-feeding, spatially or temporally, alongside infected ticks. This ‘co-feeding transmission’ does not require the host to have a systemic infection but instead pathogens are passed from one tick to another as they feed together. Vertebrate hosts may vary in their competency to support transstadial and co-feeding transmission. For example, red grouse (Lagopus lagopus scoticus) and sheep become viraemic with louping-ill and can transmit it transstadially, hares (Lepus timidus), however, do not develop a systemic infection but play a crucial role in the persistence of the disease via non-viraemic, co-feeding transmission (Hudson et al. Reference Hudson, Norman, Laurenson, Newborn, Gaunt, Jones, Reid, Gould, Bowers and Dobson1995; Norman et al. Reference Norman, Ross, Laurenson and Hudson2004). In the case of Borrelia burgdorferi s.l., small mammals are competent for both transstadial transmission and co-feeding transmission. In contrast, sheep do not develop a systemic infection and therefore do not permit transstadial transmission, however, they do support co-feeding transmission (see Randolph et al. (Reference Randolph, Gern and Nuttall1996) for a review).
Larvae and nymphs of Ixodes ricinus, the vector of numerous pathogens, have been found to be highly aggregated among small mammal hosts across much of Europe, typically having a k-exponent of the negative binomial distribution between 0 and 1 (Randolph et al. Reference Randolph, Miklisova, Lysy, Rogers and Labuda1999). Often larvae and nymphs of I. ricinus are not independently aggregated but follow a coincident aggregated distribution, i.e. those individuals with large numbers of larvae also support nymphs (Randolph et al. Reference Randolph, Miklisova, Lysy, Rogers and Labuda1999; Harrison et al. Reference Harrison, Montgomery and Bown2011). For example, Harrison et al. (Reference Harrison, Montgomery and Bown2011) found that the same 20% of wood mice in Ireland fed 72% of I. ricinus larvae and all nymphs found on mice, a pattern corresponding closely to the 80/20 rule. This level of aggregation has important implications for the efficiency of co-feeding transmission as it may allow roughly twice as many larvae to feed alongside nymphs than would be possible with an independent aggregated distribution (Randolph et al. Reference Randolph, Miklisova, Lysy, Rogers and Labuda1999).
The basic reproduction number, denoted as R 0, has a number of important applications in epidemiology. It is defined as the average number of secondary cases caused by one infected individual entering a population consisting solely of susceptible individuals (Anderson and May, Reference Anderson and May1990; Diekmann et al. Reference Diekmann, Heesterbeek and Metz1990). It has a threshold value such that if R 0 > 1 a pathogen will persist when introduced, but a value of R 0 < 1 suggests that it will die out. In the case of tick-borne pathogens it can be used to identify the average number of ticks infected in the second generation after 1 infected tick enters the population, therefore, if R 0 >1, the pathogen will perpetuate within the tick-host system. Heterogeneities in the efficiency of different transmission routes of tick-borne pathogens will affect R 0 and, therefore, the likelihood of a pathogen becoming established within a system (Woolhouse et al. Reference Woolhouse, Dye, Etard, Smith, Charlwood, Garnett, Hagan, Hii, Ndhlovu and Quinnell1997). As expected, due to the applicability of the basic reproduction number, many attempts have been made to estimate R 0 for tick-borne pathogens (Randolph, Reference Randolph1998; Norman et al. Reference Norman, Bowers, Begon and Hudson1999; Randolph et al. Reference Randolph, Miklisova, Lysy, Rogers and Labuda1999; Caraco et al. Reference Caraco, Glavankov, Chen, Flaherty, Ohsumi and Szymanksi2002; Rosa et al. Reference Rosá, Pugliese, Norman and Hudson2003; Ghosh and Pugliese, Reference Ghosh and Pugliese2004; Rosa and Pugliese, Reference Rosá and Pugliese2007). Indeed, Rosa et al. (Reference Rosá, Pugliese, Norman and Hudson2003) previously investigated the role of tick aggregation on the persistence of tick-borne pathogens; however, many of these approaches were lacking in their relevance to the tick life cycle, assuming bite rates akin to insect vector models, or used derived expressions that could not be strictly interpreted as a biologically meaningful value of R 0 (Hartemink et al. Reference Hartemink, Randolph, Davis and Heesterbeek2008).
The current study uses the most up-to-date and biologically relevant next-generation model of the basic reproduction number for tick-borne pathogens (Hartemink et al. Reference Hartemink, Randolph, Davis and Heesterbeek2008) to investigate how the degree of aggregation of I. ricinus ticks on small mammal hosts (ranging from dispersed to independent aggregated to coincident aggregated distribution) affects the ability of tick-borne pathogens to become established in enzootic small mammal cycles. We use previously published tick-related and pathogen-specific data as well as simulated data to parameterize the R 0 model specific for Borrelia burgdorferi s.l., and tick-borne encephalitis virus (TBEV), 2 of the most important tick-borne pathogens of humans in Europe. The k-exponent of the negative binomial distribution was varied while other parameters were held static so that the effect of the aggregation of ticks on the persistence of tick-borne pathogens in small mammal hosts could be observed. This method also allowed the relationship between the k-exponent of the negative binomial distribution and R 0 to be defined. It has also been shown that increasing the diversity of vertebrate hosts in an ecosystem can reduce the risk of disease as only some hosts are competent for disease transmission and infective tick bites may be ‘wasted’ on non-transmission competent hosts, the so called ‘dilution effect’ (Ostfeld and Keesing, Reference Ostfeld and Keesing2000; Schmidt and Ostfeld, Reference Schmidt and Ostfeld2001). Therefore, we also investigated how the degree of aggregation of ticks on small mammal hosts affects the ability of these pathogens to become established in relation to the proportion of transmission-competent hosts on which ticks are feeding.
MATERIALS AND METHODS
Data generation
Theoretical datasets for larvae, nymphs and adults were generated from the negative binomial distribution using the ‘rnbinom’ function in package ‘base’ within the R statistical software environment, available from www.r-project.org. Three parameters were required to simulate the data; the ‘number of observations’, n, (the host population size, set at 500 for all simulations), ‘size’ (also known as the dispersion parameter or the k-exponent of the negative binomial distribution, hereafter referred to as the ‘k-exponent’ or ‘k’) and ‘mu’ (the mean, in this case the mean number of each tick life stage present on the small mammal population). For each life stage, datasets were created assuming 3 distributions using a range of values of the k-exponent. The first dataset had a starting value of k = 0·1 and increased in increments of 0·1 until k = 1·0, thus representing natural levels of aggregation of ticks on small mammals in Europe (Randolph et al. Reference Randolph, Miklisova, Lysy, Rogers and Labuda1999). The second dataset represented a dispersed independent distribution of ticks on hosts beginning at k = 1·5 and increasing in increments of 0·5 until k = 30·5 at which point the variance was approximately equal to the mean and the negative binomial distribution became Poisson distributed (mean = 6, variance = 6·67, goodness of fit to the Poisson distribution χ 2 = 18·5, d.f. = 14, P = 0·184). The third dataset represented a coincident aggregated distribution and utilized data generated for the independent aggregation of ticks in dataset 1 (0·1⩽k ⩽ 1·0), but in this case 80% of nymphs that were previously independently distributed among the host population were re-distributed randomly among the 20% of individuals most parasitized by larval ticks, thus conforming to the 80/20 rule and achieving a coincident aggregated distribution with values of k ranging between 0·1 and 1·0. As adult ticks were so rare among the datasets (mu = 0·001), all adults were re-distributed randomly among the top 20% of individuals most parasitized by larval ticks for the coincident aggregated dataset. Therefore, 86 individual datasets were created each for larval, nymphal and adult ticks, 10 for the independent aggregated distribution (0·1⩽k ⩽ 1·0), 66 for dispersed distribution (1·5⩽k ⩽ 30·5) and 10 for the coincident aggregated distribution (0·1⩽k ⩽ 1·0). Increments of 0·1 were used from 0·1 to 1·0 to increase the sample size for this range of values of the k-exponent, as they represented natural levels of aggregation as previously stated.
Values of ‘mu’, the average number of each tick life stage were set as 6, 0·2 and 0·001 for larvae, nymphs and adults respectively (cited by Hartemink et al. Reference Hartemink, Randolph, Davis and Heesterbeek2008). Although the presence of adult ticks is rare among small mammals the model is parameterized to indicate whether the pathogen is likely to become established or persist in small mammal cycles and we therefore assume that the tick population is supported by larger mammals within the system e.g. sheep, hares or deer. The data generated for larvae, nymphs and adults were used to calculate parameters within the R 0 model of Hartemink et al. (Reference Hartemink, Randolph, Davis and Heesterbeek2008) that are affected by the degree of aggregation of ticks on hosts i.e. those involved with co-feeding: C LL, C NL, C AL, C LN, C NN, C AN, C LA, C NA, C AA (for example, C LL is ‘the mean number of larvae co-feeding with a larva’, see Table 1 for a full list of parameter descriptions). A set of these parameters was produced for each of the 86 individual datasets across the range of the dispersed distribution (1·5⩽k ⩽ 30·5), the independent aggregated distribution (0·1⩽k ⩽ 1·0) and the coincident aggregated distribution (0·1⩽ k ⩽ 1).
Ecological parameters for Ixodes ricinus
† Numbers in superscript refer to the following sources cited by Hartemink et al. (Reference Hartemink, Randolph, Davis and Heesterbeek2008): 1Randolph and Craine (Reference Randolph and Craine1995); 2Randolph (Reference Randolph2004); 3Randolph et al. (Reference Randolph, Miklisova, Lysy, Rogers and Labuda1999); 4Gray (Reference Gray2002): 5Humair et al. (Reference Humair, Rais and Gern1999): 6S.E. Randolph, unpublished manuscript.
* Parameters calculated in the current study across a range of values of the k-exponent. Please refer to the Supplementary data (online version only) for equations used to calculate each element within the next generation matrix and for the structure of the matrix.
The next generation matrix
When diseases are transmitted directly, estimating R 0 is straightforward but vector-borne diseases are more complex as they utilize different ‘infected types’ i.e. hosts and vectors. This is made increasingly complex when additional transmission routes and host and vector types are present; however, the next-generation matrix provides a method to incorporate these complexities. The matrix is based on the concept of types at birth i.e. the different states at which individuals become infected. Hartemink et al. (Reference Hartemink, Randolph, Davis and Heesterbeek2008) identified 5 types at birth for tick-borne infections, (1) eggs (via transovarial transmission), (2) larvae, (3) nymphs, and (4) adults (via bloodmeals) and (5) vertebrate hosts. For a system with 5 types at birth the next generation matrix will be a 5 × 5 matrix of elements with every type at birth being infected by another (see Supplementary material for elements and the structure of the next generation matrix, online version only). If we consider the element k 31: the number of nymphs (type at birth 3) infected by a tick infected as an egg (type at birth 1), which has the formula k 31 = (s Lø LNCNL+s Ls Nø NNCNN+s Ls Ns Aø ANCNA) h c, it has 3 terms as ticks infected as eggs can infect nymphs after their first, second and third bloodmeals while the survival to each stage (s), the transmission efficiency from tick to tick (ø) and the aggregation of ticks (C) is taken into account. Of course, some of the elements of the matrix are not possible and therefore equal 0, for example, k 15, as tick eggs cannot be infected by a vertebrate host.
Calculation of R0
Values of R 0 were calculated for B. burgdorferi s.l. and tick-borne encephalitis virus (TBEV) using the next generation matrix of Hartemink et al. (Reference Hartemink, Randolph, Davis and Heesterbeek2008). Tick-related parameters involved with co-feeding transmission derived from datasets described above (C LL-C AA, Table 1) and previously published pathogen-specific parameters (Tables 1 and 2) were entered into equations to calculate each element of the matrix (please see Supplementary material for equations used to calculate each element, online version only). Previously published parameters were held constant while each of the 86 parameter sets (C LL-C AA) were substituted into the model in turn, thus allowing values of R 0 for each pathogen to be calculated as a function of the degree of aggregation. R 0 was calculated via the spectral decomposition of the parameterized next-generation matrix using the eigen(matrix) function in package ‘base’ in R. Values of R 0 were plotted as a function of the k-exponent and non-linear regression performed (Sigmaplot 10, Systat Software Inc.) for 2 datasets, one as the dispersed distribution approached the independent aggregated distribution and the second as dispersed distribution approached the coincident aggregated distribution.
Ecological parameters for Borrelia burgdorferi s.l. and tick-borne encephalitis virus
† Numbers in superscript refer to the following sources cited by Hartemink et al. (Reference Hartemink, Randolph, Davis and Heesterbeek2008): 1Randolph et al. (Reference Randolph, Gern and Nuttall1996); 2Gern and Rais (Reference Gern and Rais1996); 3Labuda et al. (Reference Labuda, Jones, Williams, Danielova and Nuttall1993); 4Labuda et al. (Reference Labuda, Kozuch, Lys, Suss and Kahl1997); 5Randolph and Craine (Reference Randolph and Craine1995); 6S.E. Randolph, unpublished manuscript; 7Kurtenbach et al. (Reference Kurtenbach, Dizij, Seitz, Margos, Moter, Kramer, Wallich, Schaible and Simon1994); 8Hubálek and Halouzka (Reference Hubálek and Halouzka1998). Please refer to the Supplementary data (online version only) for equations used to calculate each element within the next generation matrix and for the structure of the matrix.
To give an indication of the mean effect each degree of aggregation had on the basic reproduction number in relation to the proportion of transmission-competent hosts on which I. ricinus is feeding, values of R 0 were averaged across the datasets for each of the 3 levels of aggregation and the difference between the means of each distribution (ΔR 0) calculated for a range of values of h c, (10–100% at 10% intervals). In the current study we define small mammals as ‘competent hosts’ as they facilitate both transstadial and co-feeding transmission for both pathogens and are considered to be the most important reservoirs of these pathogens in Europe. Although sheep support co-feeding transmission of B. burgdorferi s.l., they are not thought to be important for the epidemiology of this pathogen in most of its range (Hartemink et al. Reference Hartemink, Randolph, Davis and Heesterbeek2008).
RESULTS
Increasing levels of aggregation of ticks on hosts led to an increase in values of R 0 for both pathogens. When values of R 0 were plotted as a function of k the effect of aggregation on R 0 values could be clearly seen. The independent aggregated distribution formed at values of the k-exponent between 0·1 and 1·0 caused values of R 0 to increase in comparison to values observed when ticks were dispersed, leading to an inverse first-order polynomial relationship between k and R 0 when the dispersed distribution converged on the independent aggregated distribution (non-linear regression, R2adj = 0·87, F = 478·66, d.f. = 68, P < 0·0001, f = 3·87 + (0·02/x), (B. burgdorferi s.l.) and R2adj = 0·47, F = 115·94, d.f. = 68, P < 0·0001, f = 0·73 + (0·001/x) (TBEV), adjusted R2, F-value, degrees of freedom, P-value and relationship of the line of the first order polynomial, Fig. 1a and b). The coincident aggregated distribution also caused an increase in R 0 relative to the dispersed distribution but to a greater degree than the independent aggregated distribution. This relationship also followed an inverse first-order polynomial (R2adj = 0·77, F = 231·66, d.f. = 68, P < 0·0001, f = 3·88 + (0·05/x), (B. burgdorferi s.l.) and R2adj = 0·62, F = 60·74, d.f. = 68, P < 0·0001, f = 0·72 + (0·001/x) (TBEV).
Values of R 0 calculated for (a) Borrelia burgdorferi s.l. and (b) tick-borne encephalitis virus, plotted as a function of the degree of aggregation of ticks on small mammal hosts (k-exponent of the negative binomial distribution) ranging from dispersed (1·5⩽k ⩽ 30·5) to independent aggregated (0·1⩽k ⩽ 1) and from the dispersed (1·5⩽k ⩽ 30·5) to the coincident aggregated distributions (0·1⩽k ⩽ 1). Small values of the k-exponent indicate high levels of aggregation, larger values indicate low levels of aggregation. Plots of first-order polynomial relationships between k and R 0 for the dispersed distribution converging on the independent aggregation and for the dispersed distribution converging on the coincident aggregated distribution for each pathogen are also presented.
Average values of R 0 were increased for both pathogens with increasing levels of aggregation relative to those obtained when ticks followed a dispersed distribution, these values also increased in relation h c (Table 3.). For B. burgdorferi s.l., the average value of R 0 increased by a minimum of 0·012(0·75%) from 1·594 to 1·606 at h c = 10% and by a maximum of 0·066(1·49%) from 4·444 to 4·510 at h c = 100% when ticks followed the independent aggregated distribution relative to the dispersed distribution. When ticks followed a coincident aggregated distribution the average value of R 0 increased by 0·033(2·07%) from 1·594 to 1·627 at h c = 10% and by 0·297(6·68%) from 4·444 to 4·741 at h c = 100% relative to the dispersed distribution. Therefore, the threshold value of R 0 > 1 was achieved for B. burgdorferi s.l. regardless of the level of aggregation of ticks on hosts and even when the proportion of competent hosts on which I. ricinus had fed was very low (10%). In the case of TBEV, the average value of R 0 increased by 0·003(1·32%) from 0·227 to 0·230 at h c = 10% and by 0·028(3·26%) from 0·858 to 0·886 at h c = 100% when ticks followed an independent aggregated distribution relative to a dispersed distribution. When ticks followed a coincident aggregated distribution, the effect was much greater, the average R 0 value increased by 0·029 (12·78%) from 0·227 to 0·256 at h c = 10% and by 0·342(39·86%) from 0·858 to 1·200 at h c = 100%. In contrast to B. burgdorferi s.l., values of R 0 for TBEV only achieved the threshold value when h c reached 80% and only when ticks followed a coincident aggregated distribution. In this case the coincident aggregated distribution of ticks on hosts as well as ticks feeding on a high proportion of transmission competent hosts is essential for the threshold value of R 0 to be achieved and the pathogen to persist.
Mean values of R 0 for Borrelia burgdorferi s.l. and tick-borne encephalitis virus for different distributions of Ixodes ricinus ticks on small mammal hosts in relation to the proportion of transmission-competent hosts on which I. ricinus is feeding (h c)
DISCUSSION
Using different levels of aggregation from the dispersed to the independent aggregated to the coincident aggregated we showed that values of R 0 became elevated with progressive levels of aggregation, making it increasingly likely for these pathogens to become established and to persist. In the case of Borrelia burgdorferi s.l., values of R 0 were always above the threshold value for the establishment of the disease and therefore the level of aggregation of ticks on hosts and the proportion of competent hosts on which ticks are feeding has little effect on whether this pathogen becomes established or not. In contrast, the threshold value of R 0 for TBEV was only achieved when the proportion of transmission-competent hosts on which ticks were feeding reached 80% and only when the ticks followed a coincident aggregated distribution. This suggests that any decrease in the proportion of competent hosts on which ticks are feeding, for example through vaccination or increasing the diversity of potential vertebrate hosts (assuming they are less competent transmission hosts), will cause a reduction in R 0 and the pathogen will become less likely to become established or to persist. For TBEV, the distribution of ticks on hosts has a much more definitive role in whether the pathogen becomes established or not. Without the increased transmission efficiency afforded to the pathogen via this enhanced potential for the co-feeding transmission route this pathogen would not become established in small mammal hosts at all. Results of the current study are consistent with those of Hartemink et al. (Reference Hartemink, Randolph, Davis and Heesterbeek2008) and Randolph et al. (Reference Randolph, Gern and Nuttall1996) who also found that the threshold value of R 0 was easily achieved for B. burgdorferi s.l. with or without co-feeding transmission but the threshold value for TBEV was only achieved when co-feeding transmission was present.
The differential response of these two pathogens to co-feeding transmission lies in differences in their systemic infection duration (Randolph et al. Reference Randolph, Gern and Nuttall1996). The systemic infection of B. burgdorferi s.l. in a rodent host is 120 days but only 2 days for TBEV (Randolph et al. Reference Randolph, Gern and Nuttall1996). When considering transstadial transmission, rodents infect ticks with B. burgdorferi s.l. over a long period of time, furnishing the system with large numbers of infected ticks that in turn infect large numbers of hosts. This leads to a greater overall level of transmission between hosts and vectors and allows values of R 0 to be maintained above the threshold value for establishment. In contrast, rodents can only infect ticks with TBEV over a short period. Thus many fewer ticks become infected with TBEV than with B. burgdorferi s.l. and values of R 0 for TBEV are reduced so that they do not reach the threshold value for establishment of the pathogen. However, the extra boost in overall transmission efficiency supplied by co-feeding transmission, driven by the coincident aggregated distribution of ticks, is enough in some cases to increase values of R 0 above the threshold value. This explains why natural cycles of B. burgdorferi s.l. are widespread while those of TBEV are highly focal and patchy in distribution; only certain locations provide the correct climatic condition for the seasonal synchrony of larvae and nymphs on small mammal hosts that contribute to the coincident aggregated distribution that drives co-feeding transmission (Randolph et al. Reference Randolph, Gern and Nuttall1996; Randolph and Rogers, Reference Randolph and Rogers2000). Warm summers allow the rapid development of ticks while rapid autumn cooling sends larvae into winter diapause causing them to emerge in synchrony with nymphs in spring and facilitates the coincident aggregation of ticks on hosts and, therefore, co-feeding transmission (Randolph and Rogers, Reference Randolph and Rogers2000; Gatewood et al. Reference Gatewood, Liebman, Vourc'h, Bunikis, Hamer, Cortinas, Melton, Cislo, Kitron and Tsao2009). In contrast to TBEV, locations with climatic conditions that promote the activity of nymphs prior to larvae often have the strongest transmission cycles for B. burgdorferi s.l (Kurtenbach et al. Reference Kurtenbach, Hanincová, Tsao, Margos, Fish and Ogden2006). As co-feeding transmission is less important for B. burgdorferi s.l, this activity profile promotes the persistence of this pathogen as it allows time for vertebrate hosts to become infected by nymphs, develop a systemic infection, and to subsequently infect larvae. In addition, genospecies of Borrelia that persist for longer in vertebrate hosts may be favoured by climate-driven asynchrony of immature ticks while those genospecies that have a shorter systemic may be driven by seasonal synchrony (Kurtenbach et al. Reference Kurtenbach, Hanincová, Tsao, Margos, Fish and Ogden2006; Gatewood et al. Reference Gatewood, Liebman, Vourc'h, Bunikis, Hamer, Cortinas, Melton, Cislo, Kitron and Tsao2009). This can lead to geographical variations in the distribution of Borrelia genospecies driven by climate (Gatewood et al. Reference Gatewood, Liebman, Vourc'h, Bunikis, Hamer, Cortinas, Melton, Cislo, Kitron and Tsao2009). While the degree of aggregation of ticks on hosts at a given point in time has implications for pathogen transmission, there is also a temporal aspect due to the seasonality of immature ticks in response to climate.
While the coincident aggregated distribution caused an increase in R 0 so did the independent aggregated distribution, albeit to a lesser extent. In this case, it is not that individuals have an increased chance of hosting both larvae and nymphs but that the independent aggregation of larvae and nymphs is such that occasionally some individuals will host nymphs and, by chance, large numbers of larvae at the same time. This will lead to an ‘accidental’ increase in R 0 but to a lesser extent than when individuals consistently host nymphs and larvae together as under the coincident aggregated distribution.
In addition to the climatic conditions that allow synchronous feeding of larval and nymphal ticks, a number of other biotic factors are involved. Questing larvae are themselves clumped in their environment as they emerge from egg masses laid down by females (Randolph and Steele, Reference Randolph and Steele1985; Daniels and Fish, Reference Daniels and Fish1990). Sexually active adult male hosts often cover wider ranges than other individuals within the population and so may encounter more ticks in the environment (Randolph, Reference Randolph1977; Boyer et al. Reference Boyer, Réale, Marmet, Pisanu and Chapuis2010). These individuals are often the largest within a population (due to maturity and sexual size dimorphism) and there is evidence to suggest that larger individuals have greater tick loads than smaller individuals (Gallivan and Horak, Reference Gallivan and Horak1997; Harrison et al. Reference Harrison, Scantlebury and Montgomery2010). Sexually active males also have higher levels of testosterone that may depress innate and acquired resistance to ticks in small mammal hosts and lead to increased attachment rates (Hughes and Randolph, Reference Hughes and Randolph2001). As a result of these factors affecting the exposure and susceptibility of individuals to parasites, adult males are often the ‘20%’ in the ‘80/20 rule’ i.e. they are the fraction of the host population that contributes most to the transmission of parasites and vector-borne diseases (Perkins et al. Reference Perkins, Cattadori, Tagliapietra, Rizzoli and Hudson2003; Ferrari et al. Reference Ferrari, Cattadori, Nespereira, Rizzoli and Hudson2004; Bouchard et al. Reference Bouchard, Beauchamp, Trudel, Milford, Lindsay, Belanger and Ogden2011). Perkins et al. (Reference Perkins, Cattadori, Tagliapietra, Rizzoli and Hudson2003) demonstrated that within a population of Apodemus flavicollis, 20% of the hosts were responsible for 74% to 94% of transmission of TBEV to ticks and that adult males of high body mass were the most likely members of this group. Similarly, Ferarri et al. (Reference Ferrari, Cattadori, Nespereira, Rizzoli and Hudson2004) found that the 20% of A. flavicollis most infected with the nematode, Heligmosomoides polygyrus, (consisting of 62% males) accounted for 73% of the eggs expelled in faeces.
Although the aggregation of ticks on certain individuals, such as adult males, is common there may be exceptions and this will have consequences for the persistence of tick-borne pathogens. If tick life stages aggregate on different hosts temporally, for example, nymphs on adult males early in the season as a result of elevated testosterone or increased movement, and larvae on pregnant females or juveniles later in the season due to reduced immunity, then pathogens such as B. burgdorferi s.l would be less likely to persist (see Mannelli et al. Reference Mannelli, Bertolotti, Gern and Gray2012).
Identifying the most important demographic for the transmission of pathogens within a population has important implications for their control. By targeting the fraction of the population most accountable for the transmission of parasites, transmission potential can be reduced. Conversely, targeting individuals in the population outside the important ‘20%’ of key hosts may have little or no effect on the overall epidemiology pathogens (Perkins et al. Reference Perkins, Cattadori, Tagliapietra, Rizzoli and Hudson2003; Ferrari et al. Reference Ferrari, Cattadori, Nespereira, Rizzoli and Hudson2004). The development of vaccines for B. burgdorferi s.l. in natural rodent hosts, such as white-footed mice, Peromyscus leucopus, has focussed on outer surface protein A (Osp A) which is involved with the attachment of spirochetes in the midgut of ticks (Pal et al. Reference Pal, De Silva, Montgomery, Fish, Anguita, Anderson, Lobet and Fikrig2000). Recent field studies have indicated that vaccination of small mammal reservoirs may be a viable option for controlling Lyme borreliosis. Tsao et al. (Reference Tsao, Wootton, Bunikis, Luna, Fish and Barbour2004) demonstrated that the prevalence of B. burgdorferi s.l. infection in I. scapularis could be reduced by between 19% and 25% via vaccination of white-footed mice while Telford et al. (Reference Telford, Cunningham, Waltari and Hu2011) suggested that over half the population of white-footed mice could be targeted with nest-box deployed bait that could be used for vaccination. Although these measures are likely to reduce the number of nymphs infected with B. burgdorferi s.l. in the environment and, therefore, the risk to public health, data from the current study suggest that they are unlikely to eliminate the pathogen from the system altogether.
The current study uses the most up-to-date and biologically relevant model of the basic reproduction number, R 0, for tick-borne pathogens to investigate how the degree of aggregation of ticks on small mammal hosts, from the dispersed to the independent aggregated to the coincident aggregated distribution, affects the persistence of tick-borne pathogens in enzootic small mammal cycles. In addition, the relationship between the k-exponent of the negative binomial distribution and R 0 is defined for the first time.
ACKNOWLEDGEMENTS
This work was supported by a University of Pretoria Post-doctoral Fellowship awarded to A. Harrison and the DST-NRF South African Research Chair of Behavioural Ecology and Physiology awarded to N.C. Bennett.
