INTRODUCTION
Foot-and-mouth disease (FMD) is a viral disease in cloven-hoofed animals caused by foot-and-mouth disease virus (FMDV). Transmission of FMDV is difficult to control. The magnitude of transmission of any infection is assessed using the reproduction ratio R [Reference Anderson and May1, Reference Heffernan, Smith and Wahl2]. R is defined as the average number of new cases arising from a typical infected individual during its whole infectious period in a fully susceptible population. An infectious agent is able to cause major outbreaks only if R is > 1 [Reference Kermack and McKendrick3]. For FMDV, R has been quantified using field data [Reference Hagenaars4] and experimental data [Reference Cox5–Reference Van Roermund12]. Using experimental data, R has been quantified for both vaccinated and non-vaccinated sheep-to-sheep transmission [Reference Eblé, Orsel and Dekker7, Reference Orsel11] and for vaccinated and non-vaccinated cattle-to-cattle transmission [Reference Orsel9, Reference Orsel10] (and in Bravo de Rueda et al., unpublished observations). In addition, a partial R for non-vaccinated sheep to non-vaccinated cattle [Reference Bravo de Rueda13] has been quantified; however, this estimate alone is not sufficient to assess the magnitude of transmission of FMDV in a mixed population of cattle and sheep. In order to understand the transmission of FMDV in field conditions where different species co-exist, it is necessary to quantify R for heterogeneous populations (i.e. consisting of sheep and cattle).
Vaccination against FMDV has been recognized as an important tool for the control of FMDV. Vaccination against FMDV can prevent transmission of the virus both in field conditions [Reference Leforban14–Reference Sutmoller16] and experimentally [Reference Cox5, Reference Eblé, Orsel and Dekker7, Reference Orsel9–11, 17, Reference Parida18]. In mainland Europe, FMDV was eradicated by prophylactic vaccination of cattle only [Reference Leforban15]. In parts of South America, FMDV was successfully eradicated by vaccination of cattle only [Reference Sutmoller16]. For example, in Uruguay, where cattle and sheep, mingle freely and where (before 2002) the proportion of sheep in the population was slightly higher than that of cattle, vaccination of cattle only was sufficient to eradicate FMDV [Reference Sutmoller16, Reference Donaldson19]. In the European Union, emergency vaccination of (all) susceptible species is an option during an FMDV outbreak (EU directive 2003/85). The Netherlands used emergency vaccination in the 2001 outbreak with vaccination of all FMDV-susceptible animals. However, it is unclear whether emergency vaccination of all susceptible species is necessary to control an epidemic or if targeting vaccination to certain species (e.g. only cattle) could be sufficient.
In the current study we developed a method to determine R for mixed populations consisting of cattle and sheep by using experimental transmission data and a technique known as the next-generation matrix (NGM) [Reference Diekmann, Heesterbeek and Metz20]. The method allows analysis of different vaccination strategies in different mixed populations consisting of cattle and sheep.
METHODS
Data source from experimental studies
Data available from direct contact transmission studies [Reference Eblé, Orsel and Dekker7, Reference Orsel9–Reference Orsel11, Reference Bravo de Rueda13] (and in Bravo de Rueda et al. unpublished observations) were used. These data were collected from three cattle-to-cattle transmission studies (26 experimental groups), two sheep-to-sheep transmission studies (24 experimental groups) and one sheep-to-cattle transmission study (10 experimental groups). These transmission studies were selected because the raw data on the number of susceptible, infectious, and recovered animals were readily available, and because comparable methods were used in the experiments. The donors in all these studies were inoculated via the intranasal route using either FMDV O/NET/2001 or Asia-1 TUR/11/2000. Five of these six studies also contained data on transmission of FMDV after vaccination, using either FMDV O Manisa or FMDV Asia-1 Shamir as vaccine strains.
Quantification of (partial) R values by using experimental data
The SIR model [Reference Becker21] was used for the quantification of R cattle to cattle (R c–c), sheep to sheep (R s–s), and partial R sheep to cattle (partial R s–c) for non-vaccinated animals and of R vac c–c and R vac s–s for vaccinated animals. The animals from the direct contact transmission studies [Reference Eblé, Orsel and Dekker7, Reference Orsel9–11, Reference Bravo de Rueda13] (and in Bravo de Rueda et al., unpublished observations) were classified as susceptible, infectious, or recovered (S-I-R, respectively), at the start (S0, I0) and at the end (St, Rt) of the experiment. It was assumed that the animals were infectious if they tested positive by virus isolation (on secondary lamb kidney cells) or if they developed infection specific antibodies (detected by NS-ELISA). Contact animals were considered infected if they tested positive for FMDV or FMDV-specific antibodies during the experiment. Animals that were infectious during the experiment were considered as recovered at the end of the experiment (Rt). Data originating from experiments with the same donor species and same contact animal species were pooled for the calculation of the reproduction ratio R.
The recorded data as S0, I0, St and Rt and the frequencies at which they occurred (see Tables 1 and 2), were used to estimate the reproduction ratio R [Reference Anderson and May22] for non-vaccinated and/or vaccinated groups by using the final size method [Reference Velthuis23, Reference De Jong and Kimman24]. The R c–c, R vac c–c, R s–s, R vac s–s and partial R s–c, and their 95% confidence intervals (CI) were calculated from the final sizes using the maximum likelihood estimation and exact confidence bounds [Reference Kroese and De Jong25] in Mathematica® (http://www.wolfram.com/mathematica/).
Table 1. Final outcome from the transmission experiments with non-vaccinated animals
S0 and St represent the number of susceptible animals at the start and at the end of the experiment; I0 represents the number of infectious animals at the start of the experiment; and Rt represents the number of recovered animals at the end of the experiment. Frequency represents the number of experimental groups with the same outcome. Dashed lines separate the experimental groups of animals.
Table 2. Final outcome from the transmission experiments with vaccinated animals
S0 and St represent the number of susceptible animals at the start and at the end of the experiment; I0 represents the number of infectious animals at the start of the experiment; and Rt represents the number of recovered animals at the end of the experiment. Frequency represents the number of experimental groups with the same outcome. Dashed lines separate the experimental groups of animals.
The null hypothesis that no difference existed between the estimates of R c–c and R s–s, R s–s and partial R s–c, and R c–c and partial R s–c was tested by using a two-sided test with a significance level of 0·05 [Reference Kroese and De Jong25].
Estimation of relative infectivities, susceptibilities and the unknown (partial) R values by using the separable mixing assumption
We built a 4 × 4 table using the (partial) R's between non-vaccinated/vaccinated cattle and non-vaccinated/vaccinated sheep. In this 4 × 4 table only five out of the possible 16 values were quantified using the experimental data. By assuming separable mixing, i.e. assuming that the (partial) R's are the product of a relative infectivity f i [where i is either non-vaccinated cattle (nc), vaccinated cattle (vc), non-vaccinated sheep (ns), or vaccinated sheep (vs)] and a relative susceptibility g i [where i is either non-vaccinated cattle (nc), vaccinated cattle (vc), non-vaccinated sheep (ns), or vaccinated sheep (vs)], we calculated the missing values in the table. Without loss of generality we chose non-vaccinated cattle to have a susceptibility of 1. Further, we assumed the relative susceptibility of vaccinated animals also to be 1, the same as the relative susceptibility of non-vaccinated animals. This assumption might seem counterintuitive, but local virus replication is often detected in vaccinated animals after challenge [Reference Cox26], indicating that vaccinated animals are still susceptible. In our calculations, the value is only necessary for filling the table. It does not influence the results on the diagonal, which are the only numbers that will be used (as will be explained below) in the calculation of the R values for the different strategies. Note that the reduction in transmission due to vaccination is now assumed to be due to the lower infectivity of the vaccinated and then infected animals (Table 3).
Table 3. (Partial) R values as estimated from infected non-vaccinated (NV) or vaccinated (V) animals to non-vaccinated (NV) or vaccinated (V) contact animals
The values in bold were estimated from experimental data using the final size method. The other R values were based on the product of the relative infectivity (f i ) and relative susceptibility (g j ). We assumed that the relative susceptibility of both non-vaccinated and vaccinated cattle and sheep are equal. Without any loss of generality we took non-vaccinated cattle to have susceptibility equal to 1. Note: this table is not yet the NGM as the proportion of the different animal species and the proportion of vaccinated animals are still missing.
Construction of a NGM
A NGM allows the analysis of the effect of different categories of individuals on the overall transmission, i.e. in a mixed population [Reference Diekmann, Heesterbeek and Roberts27]. In our case, R for a mixed population of cattle and sheep depends on the proportion of each animal species in the population. In the matrix p c is the proportion of cattle (i.e. the total number of cattle divided by the total number of cattle and sheep in a population) and 1 – p c is the proportion of sheep in the same population. In the matrix the proportion of vaccinated animals per species are indicated by pvc and pvs, where pvc and pvs represent the proportion of vaccinated cattle and the proportion of vaccinated sheep, respectively. The relative infectivity f i and relative susceptibility g i from the above 4 × 4 table were added to the NGM. Thus, the elements of our matrix are functions of the relative infectivity (f i ), relative susceptibility (g i ), the proportion of cattle (p c), and the proportion of vaccinated cattle and that of sheep (pvc or pvs).
Evaluation of the influence of different proportions of cattle (p c) and sheep (1 – p c)
We studied the influence of different proportions of cattle and sheep on the transmission of FMDV in our NGM. To illustrate this we used five different populations: (1) a population consisting of cattle only, (2) a population with a higher number of cattle than sheep, (3) a population with a relatively similar number of cattle and sheep, (4) a population with a higher number of sheep than cattle, and (5) a population consisting of sheep only. For defining the different mixed populations consisting of cattle and sheep we used proportions of known livestock populations from the FAOSTAT database [28]. In 2011 these p c values were: 0·78 in The Netherlands (for population 2), 0·61 in Uruguay (for population 3), and 0·24 in New Zealand (for population 4). The proportions of the population of cattle (p c per population) were included in the NGM. Finally, the dominant eigenvalue of the NGM, i.e. the reproduction ratio for the mixed populations, was determined for all five populations.
Evaluation of the effect of different vaccination strategies
We used the five above-mentioned populations to evaluate the effect of three different vaccination strategies for the control of FMD transmission. These strategies were: (1) vaccinating both species equally, thus pvc = pvs, (2) vaccinating all cattle with additional vaccination of sheep (pvc = 1 and pvs ≠ 0) and, (3) vaccinating all sheep with additional vaccination of cattle (pvc ≠ 0 and pvs = 1). The obtained results were plotted for each strategy. Because R depends on p c, pvc and pvs, we calculated the proportion of animals that has to be vaccinated (or has to be present in a population) at which R reached the value of 1.
RESULTS
Quantification of (partial) R values by using experimental data
In groups where no vaccination was applied, R c–c was estimated as 5·3 (95% CI 3·0–42) and R s–s was estimated as 1·1 (95% CI 0·44–2·4). The partial R s–c was estimated as 0·87 (95% CI 0·20–2·9) (bold values in Table 3). R c–c was found to be significantly higher than R s–s (P = 0·002). Moreover, R c–c was significantly higher than partial R s–c (P = 0·005). R s–s was not significantly different from partial R s–c (P = 0·56) and therefore based on these results the susceptibility of cattle and sheep are considered similar.
In groups where vaccination was applied, the R vac c–c was estimated as 0·13 (95% CI 0·0032–0·83) and R vac s–s was estimated as 0·098 (95% CI 0·0026–0·65). The estimated relative infectivities (f i ), relative susceptibilities (g i ), and the (partial) R's are shown in Table 3.
Construction of the NGM
Equation (1) shows the 4 × 4 NGM in which the proportions of cattle and sheep and the proportion of vaccinated animals are included. In our matrix, because of the assumption of separable mixing, the dominant eigenvalue equals the sum of the elements on the diagonal (from top left to bottom right) which is called the trace of the matrix [Reference Diekmann, Heesterbeek and Roberts27]. This dominant eigenvalue is the R for the mixed population described by the NGM [Reference Diekmann, Heesterbeek and Metz20, Reference Diekmann, Heesterbeek and Roberts27]. Thus
$$\eqalign{ R\left( {\,p_{\rm c}, p{\rm v}_{\rm c}, p{\rm v}_{\rm s}} \right) &= p_{\rm c} \left( {\left( {1-p{\rm v}_{\rm c}}\right)f_{\rm c} g_{\rm c} + p{\rm v}_{\rm c} f_{\rm vc} g_{\rm vc}} \right) \cr &\quad + \left( {1-p_{\rm c}} \right)\left( {\left( {1-p{\rm v}_{\rm s}} \right)f_{\rm s} g_{\rm s} + p{\rm v}_{\rm s} f_{\rm vs} g_{\rm vs}} \right).} $$
For example, for a population consisting of non-vaccinated cattle only, the dominant eigenvalue of that matrix is R(1, 0, 0) = f c g c = R c–c and, for a population consisting of only vaccinated cattle, the dominant eigenvalue of that matrix is R(1, 1, 0) = f vc g vc = R vac c–c.
Equation 1: NGM with non-vaccinated and vaccinated animals. f c and f s correspond to the infectivity of cattle and of sheep, respectively. g c and g s correspond to the susceptibility of cattle and of sheep, respectively. The proportion of the population of cattle p c and of sheep 1 – p c depends on the characteristics of a mixed population. pvc represents the proportion of vaccinated cattle and pvs, the proportion of vaccinated sheep:
$$\eqalign{& \left[ {\matrix{ {{\,f_{\rm c}}{g_{\rm c}}{\,p_{\rm c}}(1 - P{{\rm v}_{\rm c}})} & {{\,f_{\rm s}}{g_{\rm c}}{\,p_{\rm c}}(1 - P{{\rm v}_{\rm c}})} \cr {{\,f_{\rm c}}{g_{\rm s}}(1 - {\,p_{\rm c}})(1 - p{v_{\rm s}})} & {{\,f_{\rm s}}{g_{\rm s}}(1 - {\,p_{\rm c}})(1 - p{v_{\rm s}})} \cr {{\,f_{\rm c}}{g_{{\rm vc}}}{\,p_{\rm c}}p{v_{\rm c}}} & {{\,f_{\rm s}}{g_{{\rm vc}}}{\,p_{\rm c}}p{v_{\rm c}}} \cr {{\,f_{\rm c}}{g_{{\rm vs}}}(1 - {\,p_{\rm c}})p{v_{\rm s}}} & {{\,f_{\rm c}}{g_{{\rm vs}}}(1 - {\,p_{\rm c}})p{v_{\rm s}}} \cr } } {\matrix{ {{\,f_{{\rm vc}}}{g_{\rm c}}{\,p_{\rm c}}(1 - p{v_{\rm c}})} & {{\,f_{{\rm vs}}}{g_{\rm c}}{\,p_{\rm c}}(1 - p{v_{\rm c}})} \cr \hskip6pt{{\,f_{vc}}{g_s}(1 - {\,p_c})(1 - p{v_{\rm s}})} & {{\,f_{{\rm vs}}}{g_{\rm s}}(1 - {\,p_{\rm c}})(1 - p{v_{\rm s}})} \cr {{\,f_{{\rm vc}}}{g_{{\rm vc}}}{\,p_{\rm c}}p{v_c}} & {{\,f_{{\rm vs}}}{g_{{\rm vc}}}{\,p_{\rm c}}p{v_{\rm c}}} \cr {{\,f_{{\rm vc}}}{g_{{\rm vs}}}(1 - {\,p_{\rm c}})p{v_{\rm s}}} & {{\,f_{{\rm vs}}}{g_{{\rm vs}}}(1 - {\,p_{\rm c}})p{v_{\rm s}}} } } \right]} $$
Evaluation of the influence of different proportions of cattle (p c) and sheep (1 – p c)
In the different non-vaccinated mixed populations, for populations with 0%, 24%, 61%, 78% and 100% cattle, R was estimated to be 1·1, 2·1, 3·7, 4·4 and 5·3, respectively.
Evaluation of the effect of different vaccination strategies
Strategy 1: vaccination of both cattle and sheep
In Figure 1a we show the effect of vaccination when we vaccinate (the same proportion of) both cattle and sheep (so when pv = pvc = pvs) for populations consisting of cattle or sheep in different proportions. The R for a fully vaccinated mixed population with 0%, 24%, 61%, 78% and 100% cattle was 0·1, 0·11, 0·12, 0·12 and 0·13, respectively, i.e. always <1.
Fig. 1. The effect of different vaccination strategies on the reduction of R in mixed populations. (a) The effect of vaccination of both cattle and sheep (in equal proportions) on the reduction of R in different mixed populations with cattle and sheep. (b) The effect of vaccination of all cattle and additional vaccination of sheep on the reduction of R in different mixed populations with cattle and sheep. (c) The effect of vaccination of all sheep and additional vaccination of cattle on the reduction of R in different mixed populations with cattle and sheep. p c represents the proportion of cattle of the mixed population. The threshold value of R = 1 is indicated by a grey line. The percentage of the population of (a) cattle and sheep, (b) sheep, or (c) cattle that needs to be (additionally) vaccinated to reach the threshold value of 1 is indicated.
The percentage of the population that has to be vaccinated to achieve R = 1 is: 14%, 56%, 75%, 79% and 83% for populations with 0%, 24%, 61%, 78% and 100% cattle, respectively.
Strategy 2: vaccination of all cattle with additional vaccination of sheep
When in the populations no sheep, but only all cattle (thus 100% of the cattle) are vaccinated, R was 1·1, 0·90, 0·52, 0·35 and 0·13 for populations with 0%, 24%, 61%, 78% and 100% cattle, respectively (see Fig. 1b ). The percentage of cattle in the population that has to be present to reach R = 1 (when all cattle are vaccinated) was 14%.
Strategy 3: vaccination of all sheep with additional vaccination of cattle
When in the populations no cattle, but only all sheep (thus 100% of the sheep) are vaccinated, R was 0·1, 1·4, 3·3, 4·1 and 5·3 for populations with 0%, 24%, 61%, 78% and 100% cattle, respectively (see Fig. 1c ).
The additional percentage of the cattle population that has to be vaccinated to reach R = 1 was 0%, 29%, 72% and 78% and 83%, respectively, for populations with 0%, 24%, 61%, 78%, and 100% cattle, respectively.
DISCUSSION
In the current study we quantified the transmission of FMDV in mixed cattle-sheep populations and evaluated the effect of different vaccination strategies. The evaluation of different vaccination strategies was based on the transmission estimates from experimental transmission studies. The higher the proportion of cattle in a mixed cattle-sheep population, the higher the R for the mixed population is. Thus, the impact of vaccination of the cattle is higher. When the whole population is vaccinated, R < 1 regardless of the population composition. In mixed cattle-sheep populations with at least 14% of cattle, vaccination of cattle only is sufficient to reduce R to <1. The strategy of vaccinating cattle only for eradication purposes has been used in the past in continental Europe [Reference Leforban15] and South America [Reference Sutmoller16, Reference Donaldson19] with success. Previous studies using mathematical modelling also predicted that for emergency vaccination, targeting cattle only is much more efficient than using other vaccination strategies [Reference Keeling29]. Therefore, this strategy will be more cost-effective in countries with mixed populations of cattle and sheep where prophylactic vaccination is applied [Reference Yadin30], as it would mean a reduction in the number of vaccine doses needed and in required manpower. Moreover, when using it as an emergency vaccination strategy, it would also mean a reduction in the time needed to immunize all the animals.
While our conclusions are valid for mixed cattle-sheep populations, different results might be expected for mixed populations where other FMDV-susceptible species are present. For instance, in The Netherlands, where routine annual vaccination of cattle only was used from 1953 to 1991, FMD outbreaks occurred between 1961 and 1967 in mixed cattle and pig farms. At that time, additional vaccination of pig herds was used effectively to control the outbreaks [Reference Van Bekkum, Bool and Vermeulen31]. Additionally, in Asian countries, where the Asian buffalo (Bubalus bubalis) is a host of epidemiological importance [Reference Maroudam32], a vaccination strategy that includes (additional) vaccination of the Asian buffalo is probably advisable. Thus depending on the different species and percentages of these species in a population, different vaccination strategies might be needed. When quantitative data of transmission of FMDV for other animal species are known, this could be included in the NGM and then similar analyses can be performed for other heterogeneous populations.
In our analysis, we used data from transmission studies in which good quality vaccines, containing >6 PD50 per dose, were used. Experience in South America [Reference La Torre33] shows that the strategy of vaccinating cattle only is only effective when good quality vaccines are used. The use of good quality vaccines is of course a prerequisite when using vaccination to control a disease. We used data from within-pen transmission studies in which cattle and/or sheep were mingling in one animal room, thus within-pen transmission occurred. However, in many situations, cattle and sheep within a population will have less intensive contact. Other studies show that transmission between pens is in general lower than within a pen [Reference Van Roermund12, Reference Klinkenberg34] and that between-herd transmission will be even lower [Reference Van35]. Thus, the effect of targeting vaccination towards cattle will probably be even better under these circumstances than predicted in the current study. In the current study, we used a mathematical approach to calculate which vaccination strategies would be effective. In field situations, other aspects, e.g. vaccine quality, might influence the results. However, even if the quality of the vaccine batch is low, it is probably better to use it in cattle only than spread the available capacity over both species. Although we did use different serotypes in the current study, which produced similar results, there might be a different outcome for other virus strains. Moreover, our approach looks only at the scenario where eradication of FMDV is the goal, there may be an interest to consider scenarios where intermediate situations (FMDV still endemic) have also to be considered, but this has not been studied here.
We developed an NGM that can be used to evaluate the transmission of FMDV for mixed populations of cattle and sheep and we analysed the effect of a targeted vaccination strategy. We conclude that vaccination of cattle only in mixed populations consisting of sheep and cattle will in most cases be sufficient for controlling FMDV epidemics.
ACKNOWLEDGEMENTS
The research leading to these results have received funding from the European Community's Seventh Framework Programme (FP7/2007-–013) under grant agreement no. 226556 (FMD-DISCONVAC) and the Dutch Ministry of Economic affairs (WOT-01-003-11).
DECLARATION OF INTEREST
None.
