Using the basic reproduction number to assess the risk of transmission of lumpy skin disease virus by biting insects

Abstract In recent years, lumpy skin disease virus (LSDV) has emerged as a major threat to cattle outside Africa, where it is endemic. Although evidence suggests that LSDV is transmitted by the bites of blood sucking arthropods, few studies have assessed the risk of transmission posed by particular vector species. Here this risk is assessed by calculating the basic reproduction number (R 0) for transmission of LSDV by five species of biting insect: the stable fly, Stomoxys calcitrans, the biting midge, Culicoides nubeculosus, and three mosquito species, Aedes aegypti, Anopheles stephensi, and Culex quinquefasciatus. Parameters relating to mechanical transmission of LSDV were estimated using new analyses of previously published data from transmission experiments, while vector life history parameters were derived from the published literature. Uncertainty and sensitivity analyses were used to compute R 0 for each species and to identify those parameters which influence its magnitude. Results suggest that S. calcitrans is likely to be the most efficient at transmitting LSDV, with Ae. aegypti also an efficient vector. By contrast, C. nubeculosus, An. stephensi, and Cx. quinquefasciatus are likely to be inefficient vectors of LSDV. However, there is considerable uncertainty associated with the estimates of R 0, reflecting uncertainty in most of the constituent parameters. Sensitivity analysis suggests that future experimental work should focus on estimating the probability of transmission from insect to bovine and on the virus inactivation rate in insects.

The stable fly, Stomoxys calcitrans, has been incriminated as a potential vector in Israel by comparing its seasonal abundance with the seasonality of LSD cases (Kahana-Sutin, Klement, Lensky, & Gottlieb, 2017) and because of its ability to transmit the closely related capripox virus between sheep and goats (Kitching & Mellor, 1986;Mellor, Kitching, & Wilkinson, 1987). The potential for the biting midge, Culicoides nubeculosus, and two other mosquito species, Culex quinquefasciatus and Anopheles stephensi, to transmit LSDV has also been assessed, though for these three species attempts at transmission were unsuccessful (Chihota, Rennie, Kitching, & Mellor, 2003).
In this study, the risk of transmission of LSDV posed by different insect species is explored by estimating the basic reproduction number for five species of biting insect (S. calcitrans, C. nubeculosus, Ae. aegypti, An. stephensi, and Cx. quinquefasciatus). These species were selected as their potential role in LSDV transmission has been investigated previously (Chihota, Rennie, Kitching, & Mellor, 2001 and they represent insect species (or at least genera) that are relevant to Europe. The basic reproduction number, denoted by R 0 , is the 'average number of secondary cases arising from the introduction of a single infected individual into an otherwise susceptible population' (Diekmann & Heesterbeek, 2000). An outbreak can occur only if R 0 > 1 and, consequently, R 0 provides a means to assess the risk posed by each vector species. Using a transmission model, an expression for R 0 is derived that shows how it relates to the underlying transmission processes in insect vectors and cattle. These constituent parameters are then estimated using data from the published literature. In particular, data from transmission experiments involving the five putative vector species (Chihota, Rennie, Kitching, & Mellor, 2001 are re-analysed using Bayesian methods to quantify the uncertainty in parameters relating to mechanical transmission. In addition, the latent and infectious periods for LSDV are estimated from the outcome of challenge experiments (Babiuk et al., 2008;Tuppurainen, Venter, & Coetzer, 2005), again using Bayesian methods. Finally, uncertainty and sensitivity analyses are used to calculate R 0 and to determine which of the constituent parameters has the greatest influence on its magnitude.

| Basic reproduction number for LSDV
For an infection transmitted mechanically by biting insects the basic reproduction number can be written as, A full mathematical derivation of this expression for R 0 , including the transmission model on which it is based, is presented in the Text S1. However, the expression (1), can be understood heuristically as follows. After it feeds on an infected animal, an insect remains infected (and infectious) until the virus becomes inactivated or it dies, a period which lasts on average 1/(γ + μ) days, where γ is the virus inactivation rate and μ is the vector mortality rate. During this time it will bite susceptible cattle a times per day (where a is the reciprocal of the time interval between blood meals) and a proportion, b, of these bites (i.e. the probability of transmission from insect to bovine) will result in a newly infected host. Once infected, a bovine will remain infectious for the duration of its infectious period, which lasts 1/r I days on average. During this time the host will be bitten by susceptible insects on average m × a times per day (here m = N/H is the vector to host ratio and N and H are the number of vectors and hosts, respectively), a proportion, β, of which will result in a newly infected vector (i.e. the probability of transmission from bovine to insect).

| Mechanical transmission of LSDV by insects
Data on mechanical transmission of lumpy skin disease virus to cattle by five species of biting insect (S. calcitrans, C. nubeculosus, Ae. aegypti, An. stephensi, and Cx. quinquefasciatus) were extracted from the published literature (Chihota, Rennie, Kitching, & Mellor, 2001. This provided the number of positive insects (i.e. those for which viral DNA was detected; virus isolation was only carried out for a small number of pooled samples) and the number of insects tested after feeding on a LSDV-infected bovine (S. calcitrans, C. nubeculosus, or Ae. aegypti) or a bloodvirus mix via a membrane (An. stephensi and Cx. quinquefasciatus) at each day post feeding (Table S1). It also provided information on whether or not transmission occurred when batches of insects that had previously fed on an infected animal or received an infected blood meal were allowed to refeed on a naïve bovine (Table S1).
These data were used to estimate the virus inactivation rate (γ), the probability of transmission from bovine to insect (β), and the probability of transmission from insect to bovine (b) for each species.
Parameters were estimated in a Bayesian framework to facilitate the incorporation of uncertainty in estimates of R 0 . The likelihood for the data is given by, where Y i and N i are the number of positive insects and number of insects tested at t i days post feeding, respectively, and I j is a variable indicating whether (I j = 1) or not (I j = 0) transmission occurred when insects were allowed to refeed on naïve animal j at t j days post initial feed. In Eq (2), is the probability that an insect is positive at t i days post feeding and, is the probability that transmission occurred from infected insects to bovine j at t j days post initial feed. The probability (3), is the (2) probability that an insect became infected (β) multiplied by the probability that it was still infected when tested (exp(-γt i ). The probability (4), is the probability that at least one insect (out of the n j feeding) transmitted LSDV, where the probability that an individual insect will transmit LSDV is the product of the probabilities that it became infected (β), that it was still infected when refeeding occurred (exp(-γt j )) and that it subsequently transmitted LSDV to the animal during refeeding (b). Non-informative priors were assumed for all three parameters: exponential with mean 100 for γ and uniform with range (0,1) for b and β.
Samples from the joint posterior density were generated using an adaptive Metropolis scheme (Haario, Saksman, & Tamminen, 2001), modified so that the scaling factor was tuned during burn-in to ensure an acceptance rate of between 20% and 40% for more efficient sampling of the target distribution (Andrieu & Thoms, 2008). Two chains of 50,000 iterations were run, with the preceding 10,000 iterations discarded to allow for burn-in of the chain. The chains were then thinned (taking every fifth sample) to reduce autocorrelation amongst the samples. The adaptive Metropolis scheme was implemented in Matlab (version R2018a; The Mathworks Inc.).
Convergence of the scheme was assessed visually and by examining the Gelman-Rubin statistic provided in the coda package (Plummer, Best, Cowles, & Vines, 2006) in R (R Core Team, 2018).

| Latent and infectious periods for LSDV in cattle
The mean infectious period (1/r I ) for LSDV in cattle was estimated using data from experimental infections (Babiuk et al., 2008;Tuppurainen et al., 2005). Three proxy measures of infectiousness were considered: detection of viral DNA in blood by PCR; detection of virus in blood by virus isolation (VI) in cell culture; and detection of virus (by transmission electron microscopy) or viral DNA (by PCR) in skin lesions. For each animal, the minimum infectious period was calculated as the time between the first positive and last positive samples, while the maximum infectious period was calculated as the time between the last negative and first subsequent negative sample for each measure (i.e. accounting for sampling frequency; Table S2).
Although it is not needed for the calculation of R 0 (see Eq 1), the mean latent period was also estimated for each proxy measure using data from experimental infections of cattle (Babiuk et al., 2008;Chihota et al., 2001;Tuppurainen et al., 2005). In this case, the shortest latent period was the time of the last negative sample and the longest latent period was the time of the first positive sample for each measure (i.e. detection of viral DNA in blood, detection of virus in blood and appearance of skin lesions) ( Table S2).
The infectious period was assumed to follow a gamma distribution with mean duration 1/r I and shape parameter n I , while the latent period was assumed to follow a gamma distribution with mean duration 1/r E and shape parameter n E (see Text S1). Parameters (i.e. mean and shape parameter) for each proxy measure were estimated using Bayesian methods, with the likelihood given by, where t min and t max are the minimum and maximum infectious period (i = I) or latent period (i = E) for an animal, respectively, and f is the probability density function for the gamma distribution. Non-informative priors (exponential with mean 100) were assumed for both the mean and the shape parameter.
Samples from the joint posterior distribution were generated using a random walk Metropolis-Hastings algorithm (Andrieu & Thoms, 2008), with each parameter updated in turn. Two chains of 20,000 iterations were run, with the preceding 5,000 iterations discarded to allow for burn-in of the chain. The chains were then thinned (taking every other sample) to reduce autocorrelation amongst the samples. The Metropolis-Hastings scheme was implemented in Matlab (version R2018a; The Mathworks Inc.). Convergence of the scheme was assessed visually and by examining the Gelman-Rubin statistic provided in the coda package (Plummer et al., 2006) in R (R Core Team, 2018).

| Vector life history parameters
Life history parameters, specifically the reciprocal of the time interval between blood meals (a), the vector to host ratio (m), and the vector mortality rate (μ), were estimated for S. calcitrans, C. nubeculosus, Ae. aegypti, An. stephensi, and Cx. quinquefasciatus. For each parameter, plausible ranges were derived from the published literature (Table 1), so they could be incorporated in the uncertainty and sensitivity analysis.

| Uncertainty and sensitivity analyses
Replicated Latin hypercube sampling (LHS) was used to explore the parameters influencing the basic reproduction number, R 0 for each insect species and proxy measure of infectiousness (Blower & Dowlatabadi, 1994;Gubbins, Carpenter, Baylis, Wood, & Mellor, 2008;Luz, Codeco, Massad, & Struichner, 2003). Parameters were sampled either from their marginal posterior distributions (b, β, γ, and 1/r I ) or uniformly from plausible ranges (a, m, and μ). The LHS results were used to compute the median and 95% prediction interval for R 0 . The sensitivity of R 0 to changes in each parameter was assessed by calculating the partial rank correlation coefficients (PRCCs). The uncertainty and sensitivity analyses were implemented in Matlab (version R2018a; The Mathworks Inc.).

| Mechanical transmission of LSDV by insects
The model, (3) adequately captured the data for all five species of biting insect, with the observed number of positive insects lying within the 95% credible intervals for the posterior predictive distribution   Figure 2).
There is considerable uncertainty in the estimates for the probability of transmission from insect to bovine for all five insect species ( Figure 2; Table 2). The posterior mode was non-zero only for Ae. aegypti, though the probability of transmission from insect to bovine could not be precisely estimated for this species (Figure 2).
The posterior median for the probability of transmission from insect to bovine was low for An. stephensi (0.03) and Cx. quinquefasciatus (0.04), though the upper 95% credible limit for both species is an order of magnitude higher ( Figure 2; Table 2). Finally, the probability of transmission from insect to bovine could not be reliably estimated for either S. calcitrans or C. nubeculosus (Figure 2; Figure S1). This is, in part, because the number of insects refeeding when attempting transmission to cattle was not reported for these species (Chihota et al., 2003). For C. nubeculosus, the posterior distribution for the probability of transmission from insect to bovine was identical to the prior distribution ( Figure S1), regardless of assumptions made about the numbers of insects that refed (either 1, 5, 10, 20, 50 or 100). For

Culex quinquefasciatus
Biting rate (day −1 ) a 0.08-0.25 Time interval between blood meals (1/a) of 4-12 days Griffith and Turner (1996); Reisen, Fang, and Martinez (2006) Vector to host ratio m 0-80 Modelled ratio of Culex to cattle Gachohi et al. (2016) Mortality rate (day −1 ) μ 0.07-0.84 - Gad, Feinsod, Soliman, and Said (1989); Jones, Lounibos, Marra, and Kilpatrick (2012) S. calcitrans the posterior mass for the probability of transmission from insect to bovine was shifted towards zero if a larger number of insects was assumed to refeed, though the 95% credible interval remained large ( Figure S1). For both species, estimates for the virus inactivation rate and probability of transmission from bovine to insect were not affected by the number of insects assumed to refeed ( Figure S1). In the uncertainty and sensitivity analyses for these species, the posterior distributions for parameters related to mechanical transmission (b, β and γ) were those obtained when 100 insects were assumed to refeed, as this provides the most conservative assessment of the risk they pose.

| Latent and infectious periods for LSDV in cattle
The mean infectious period depends on the proxy measure used to determine when an animal is infectious (

| Uncertainty and sensitivity analyses
The basic reproduction number for LSDV was highest for trans-  Figure 3; Table 4). Indeed, the median R 0 for Cx. quinquefasciatus was below the threshold for an outbreak to occur (i.e. R 0 = 1). The

| D ISCUSS I ON
The results of the uncertainty analysis for the basic reproduction number (Figure 3; A certain amount of care should be taken in interpreting these results, however, because there is substantial uncertainty associated with the estimates of R 0 for all five species, but especially for S. calcitrans and C. nubeculosus. One of the major sources of uncertainty is the probability of transmission from insect to bovine ( Figure 2; Table 2).
Although transmission to naïve cattle was attempted (Chihota et al., 2003), it was done at times when S. calcitrans and C. nubeculosus were unlikely to still be infectious (see Figure 1; refeeding was at 1-3 days post infection for S. calcitrans and 3-5 days post infection for C. nubeculosus). Consequently, there is little information in the data to estimate this parameter, which is reflected in the posterior distributions being almost the same as the prior distributions, especially in the case of C. nubeculosus (Figure 2; Figure S1). Yet the probability of transmission from insect to bovine was identified as one of the parameters to which the basic reproduction number is most sensitive for all five vector species considered (Figure 4).
The influence of the virus inactivation rate and vector mortality rate on R 0 differed amongst the species (Figure 4) As with all uncertainty and sensitivity analyses, the conclusions drawn from them are valid only over the parameter ranges considered. For parameters related to mechanical transmission, these ranges were estimated from the available data on the outcome of transmission experiments (Chihota, Rennie, Kitching, & Mellor, 2001 and so represent the best current estimates. Given the uncertainty in these parameters (which reflects the small numbers of animals and insects tested at each time point in the studies; Table   S1), a focus of future experimental work should be to more precisely measure parameters related to mechanical transmission and, in particular, the probability of transmission from insect to bovine and the virus inactivation rate.
All three vector life history parameters were identified as influencing the magnitude of R 0 (Figure 4). Plausible ranges for these parameters were drawn from the published literature (Table 1).
Wherever possible, ranges were derived using data relating to the species themselves. If suitable data were not available, however, ranges were derived using data for related species. In addition, the vector life parameters were assumed to be constant, but they will depend on environmental factors, including temperature and rainfall. Consequently, the basic reproduction number, and so the risk of transmission, is unlikely to be constant over space or time, but will vary both geographically and seasonally.
This study considered the potential of five species of biting insects to transmit LSDV. This was primarily because their ability to transmit LSDV had been assessed experimentally for each of the species. However, the present analysis did not consider the effect of vector feeding preferences on their potential role in transmission.
The stable fly, S. calcitrans, is one of the most damaging arthropod pests of cattle worldwide, both through its impact on production (Taylor, Moon, & Mark, 2012) and its ability to transmit a number of pathogens (Baldacchino et al., 2013). Consequently, this species' feeding preferences are unlikely to affect its efficiency as a vector of LSDV. Similarly, C. nubeculosus and other Culicoides species for which C. nubeculosus can be considered a model are livestock-associated (Lassen, Nielson, & Kristensen, 2012;Purse, Carpenter, Venter, Bellis, & Mullens, 2015), suggesting host preference will not reduce their efficiency as a vector of LSDV. By contrast, Ae. aegypti feeds primarily on humans (Reiter, 2010), potentially limiting its role in the transmission of LSDV, despite its apparent efficiency as a vector.
However, any preference for non-bovine hosts in this species (or other Culex species for which it is a suitable model) will only further reduce its importance in the transmission of LSDV.
The magnitude of the basic reproduction number depends on the proxy measure used for infectiousness ( Figure 3). Viral titres in blood can be low and detection of virus intermittent (Babiuk et al., 2008;Tuppurainen et al., 2005), suggesting this may not be the best proxy measure. By contrast, skin nodules have high titres of virus (Babiuk et al., 2008) and insects which feed on nodules become infected (Chihota, Rennie, Kitching, & Mellor, 2001, indicating that this may be a reasonable proxy measure. This could be tested experimentally by feeding insects on areas with and without lesions of clinically affected cattle and on infected, but subclinical cattle and relating viral titres in skin and blood to the probability of insects becoming infected.  (Figure 3; Table 4). By contrast, the second study estimated R 0 to be around 1.1 for outbreaks in eight cattle herds in Ethiopia in 2014-2015, using viraemia as a proxy for infectiousness . This is substantially lower than the ranges estimated for either S. calcitrans or Ae. aegypti, but is consistent with those for C. nubeculosus, An. stephensi or Cx. quinquefasciatus (Figure 3; Table 4). However, the vector species involved in transmission of LSDV in Ethiopia are not known .
In this paper, the risk of transmission of LSDV was assessed for five species of biting insect using the basic reproduction number.
The results suggest that S. calcitrans and Ae. aegypti are likely to be efficient vectors (i.e. R 0 is substantially above one), while C. nubeculosus, An. stephensi and Cx. quinquefasciatus are likely to be inefficient vectors (i.e. R 0 is close to or below one). However, there is considerable uncertainty associated with the estimates of R 0 for LSDV and future work should focus in particular on estimating the probability of transmission from insect to bovine and the virus inactivation rate. Finally, using the basic reproduction number has demonstrated that any assessment of the risk posed by an insect vector needs to consider all factors which influence its ability to transmit the virus, including life history parameters.

ACK N OWLED G EM ENTS
The author is grateful to Pip Beard, Beatriz Sanz-Bernardo, and Christopher Sanders (all at The Pirbright Institute) for helpful comments on the manuscript.

CO N FLI C T O F I NTE R E S T
The authors declare that there is no conflict of interest.