Amodel exploration of carrier andmovement transmission as potential explanatory causes for the persistence of foot-and-mouth disease in endemic regions

Funding information BiotechnologyandBiological Sciences ResearchCouncil (BBSRC);University of Warwick fundedMidlands IntegrativeBiosciencesTrainingPartnership (MIBTP), Grant/AwardNumber:BB/M01116X/1; BBSRC/EEID,Grant/AwardNumber:Grant NumberBB/T004312/1 Abstract Foot-and-mouth disease (FMD) is a virulent and economically important disease of livestock, still endemic in many areas of Asia and sub-Saharan Africa. Transmission from persistently infected livestock, also known as carriers, has been proposed as a mechanism to support the persistence of FMD in endemic regions. However, whether carrier livestock can infect susceptible animals is controversial; recovered virus is infectious and there are claims of field transmission, but it remains undemonstrated experimentally. Alternate hypotheses for persistence include the movement of livestockwithin andbetween regions, and fomite contaminationof the environment.Using a stochastic compartmental ordinary differential equation (ODE) model, we investigate the minimum rates of carrier transmission necessary to contribute to the maintenance of FMD in a region, and compare this to the alternate mechanism of persistence through cattle shipments. We find that carrier transmission can theoretically support persistence even at transmission rates much lower than the highest realistic rates previously proposed, and that the parameters with the most effect on the feasibility of carrier-mediated persistence are the average duration of both the carrier phase and natural immunity. However, shipment-mediated persistence remains a viable alternate mechanism for persistence without carrier transmission.


INTRODUCTION
Foot-and-mouth disease (FMD) is one of the most important diseases of livestock in the world, causing billions of dollars in economic damage in low and middle income countries (LMIC) annually (Grubman & Baxt et al., 2004;Knight-Jones & Rushton et al., 2013). The causative agent of the disease is a virus, which can infect approximately 70 species of cloven-hoofed animals, including domestic cattle, sheep, goats and pigs (Grubman & Baxt, 2004). Although the disease results in low mortality in infected animals, it causes high morbidity and can result in painful F I G U R E 1 a) The regions of Turkey modelled, made up of sets of provinces. Four different areas were simulated, shown in different colours, relevant statistics for each of the regions being provided in Table 7. ER (red) contains I2, I2 (purple) contains I1, I1 (blue) contains EZ (green).   Research into FMD has historically focused on the disease in Western Europe and North America, regions where the disease was once endemic but is now extinct, and more recently the effect of reintroduction of the disease into these free regions on the agricultural industry (Björnham et al., 2020;Brown et al., 2003;Keeling, 2001;Tildesley et al., 2012). For a variety of reasons including lack of good quality data and the more complex dynamics of multiple circulating serotypes and natural immunity, less research has focused on the disease in endemic areas.
FMD exhibits several possible distinct phases of infection. Upon infection with foot-and-mouth disease virus (FMDV), animals enter a latent (or exposed) phase where the virus replicates within the host animal but the animal is not infectious and no symptoms are exhibited.
In cattle, approximately 2 days before symptoms are shown, the animal begins shedding infectious virus, becoming sub-clinically infectious (Yadav et al., 2019). The vast majority of naïve cattle exhibit clinical symptoms: most commonly lesions around the mouth, tongue and feet, as well as a fever and possible lameness, during which they are clinically infectious . Following this phase, a propor-

F I G U R E 3
The basic disease compartments which animals in the model can be in. (M)aternal, (S)usceptible, (E)xposed, (I)nfectious, (R)ecovered and (C)arrier. Moving between these compartments is done at different rates, dependent on the population of each compartment as well as model inputs. These equations are described in Equation 1 tion of the cattle recover fully and generate immunity against the strain or serotype they were infected with, and the remainder become persistently infected-these animals are known as carriers. Persistently infected cattle display no symptoms, but have detectable levels of virus recoverable from the oro-pharyngeal fluid (OPF) more than 28 days post-infection; this period can potentially last up to 3 years, although most evidence suggests around 6 months to 1 year is likely Tenzin et al., 2008). Experimental evidence suggests that close to 50% of cattle become persistently infected, although field studies find lower proportions Stenfeldt et al., 2016Stenfeldt et al., , 2011Sutmoller et al., 1968). Vaccination does not prevent an animal becoming persistently infected (Stenfeldt et al., 2016).

TA B L E 2
Relevant parameters of the ordinary differential equations (ODEs) and model, with their value(s)

TA B L E 5
The parameter values which were investigated for shipment-induced persistence, shipment-1. For all model simulations runs with these parameters, k c = 0.0 (i.e. there were no carriers simulated)

TA B L E 6
The parameter values which were investigated for shipment-induced persistence, shipment-2. For all model simulations with these parameters long range shipments were simulated, but no carriers were simulated (k c = 0). Each combination of these parameters was simulated. Area farm density refers to the percentage of farms randomly selected from each area to be included in the simulation

Parameter Values
Shipments Simulated A subset of these farm data was used, shown in Figure 1 and outlined in Table 1. As an example of the shipment records, a subset of the records is displayed in Figure 2. Incorporating these records allowed for the seasonality of such shipments to be explicitly modelled. After cleaning and cross-referencing, only those data where a location and a headcount could be matched were used, reducing the number of farms to 40,208.

F I G U R E 4
The observed relationship between the probability of persistence and carrier transmission for four different values of carrier duration (indicated on the right). Each type of line represents a given immunity duration (λ r ). The red vertical line indicates the estimated carrier transmission value by Tenzin et al. (2008). The values explored in carrier-1 lie in the grey area to the right. For any given value of immunity duration (λ r ), an increase in the duration of the carrier state (λ c ) increases the probability of persistence. For any given duration of the carrier state (λ c ), an increase in the duration of immunity decreases the probability of persistence In order to attain reasonable model running times, only a geographical subset of Turkey was used, corresponding to those data in the 13 easternmost provinces, into nested regions ( Figure 1, Table 1

Model
In this study, we utilize a metapopulation model where each farm is considered a separate population and the within-farm and betweenfarm dynamics are modelled interdependently. This provides an advantage in modelling potential carrier transmission-any such transmission would almost certainly be constrained to those animals closest to the carrier. Figure 3 describes the progression of disease states for each infected animal. At the beginning of the model timeline, it is assumed that all animals are in the susceptible compartment, and infection is seeded at a random farm or farms. Within each population, progression to the exposed/latent stage is dependent on contact rates with infectious cattle (β a ) and carrier animals (the much smaller β c ).
Once exposed, cattle proceed to become infectious dependent on rate σ, and then either recover or become carrier animals. Recovered animals are considered immune to the disease, but this immunity decays over time to become susceptible again. Carrier animals are modelled if the proportion of carriers (k c ) is above 0, and will gradually proceed to the recovered compartment, simulating the final clearance of the virus.
Natural mortality and natality are modelled, with maternally immune offspring dependent on the proportion of the population with some immunity to the disease. For infected stages (exposed/latent and infectious), disease mortality is also simulated. Progression between these compartments is described by ordinary differential equations (ODEs) outlined in Equation (1). The meaning of each term and the values used are described in Table 2. Stochastic simulation of these ODEs was done via the τ-leap approximation (Keeling & Rohani et al., 2008). (2009), but has been used to flexibly describe the spread of FMD in many different regions such as Japan and the United States (Probert et al., 2018;Tsao et al., 2020). To explore uncertainty in the kernel parameterization for Turkey, an assessment of sensitivity of the results to the kernel parameters can be found in the supporting information.
For the purpose of computational speed, local spread is done in a grid, using the algorithm outlined in Sellman et al. (2018). If infection was adjudged to happen, the susceptible farm has a number of susceptible animals proceed to the exposed/latent stage, drawn from a binomial distribution.
Livestock shipments are modelled on a daily basis utilizing the ani-

Investigating carriers
The model was run using different sets of parameters to investigate each hypothesis, referred to as carrier-1. For each combination of the parameter values outlined in Table 3, the model simulated a 5-year period (2007-2012) 100 times. Five years was chosen as the maximum time-period for which full data were available, and for which it could be reasonably assumed that disease persistence indicated endemicity.
The population multiple indicates whether the population of each farm had been multiplied, as a test of the effects of farm and overall population on persistence. Shipments not being simulated and carriers not being simulated (k c = 0) are included as null tests to be certain that the effect seen can be attributed to the shipments or carriers being simulated. The proportion of these simulations where FMD was still present at the end of the year was assumed to approximate the probability of persistence given those parameters.
Subsequent to the carrier-1 parameters being simulated, and as an extension to it, the values of β c investigated were extended by repeatedly dividing by 2 until all parameter sets no longer exhibited persistence, extending down to 7.951400e-11; these values are outlined in F I G U R E 6 Simulated prevalence of foot-and-mouth disease (FMD) over time in simulations where the disease does persist over the 5-year period, organized by carrier (columns) and immunity (rows) duration, taken from the carrier-1 and carrier-2 results. Carrier duration and immunity duration were used as two of the most important parameters to illustrate the trends. Prevalence is defined as the number of farms where at least one animal is acutely infected. The black line indicates the average prevalence for simulations at that time point, the grey area is the interquartile range, and the light grey indicates the 5%-95% range of results at that timepoint. Blank plots indicate combinations of carrier and immunity duration where no simulation exhibited persistence. Where persistence occurred, there is a clear oscillation around a long-term endemic equilibrium, and there is a large time lag between the initial outbreak before the carrier animals re-seed the outbreak and it proceeds towards endemicity. The size of the time lag depends on the duration of immunity Table 4. Values of β c below 4.17e-5 were investigated with the population multiple set to 'x1' , k c at 0.5, and no shipments modelled due to the results obtained for carrier-1. λ r and λ c were investigated using the same values used in Table 3. This set of parameters is referred to as carrier-2.

Investigating shipments
The ability of shipments to allow for FMD to persist over the 5-year period was investigated in a similar manner by running the model 100 times with every combination of the parameter values outlined in  Table 5 are referred to as shipment-1.
As an extension of shipment-1, and to disentangle the effect of the area modelled from the number of farms modelled (which are correlated) on the probability of persistence, the area was split into four areas as previously defined (ER, I2, I1 and EZ), and simulated with either 25%, 50%, 75% and 100% of the farms in that area included.
Farms were selected randomly with probability equal to 25%, 50%, F I G U R E 7 Simulated number of farms infected with foot-and-mouth disease (FMD) over time in simulations where the disease does not persist over the 5-year period, organized by carrier (columns) and immunity (rows) duration, taken from the carrier-1 and carrier-2 results. Carrier duration and immunity duration were used as two of the most important parameters to illustrate the trends. Prevalence is defined as the number of farms where at least one animal is acutely infected. The black line indicates the average prevalence for simulations at that time point, the grey area is the interquartile range, and the light grey indicates the 5%-95% range of results at that timepoint. Blank plots indicate combinations of carrier and immunity duration where no simulation exhibited persistence. In these cases, after the initial outbreak burned itself out, there was often no revival of the disease. With some combinations of parameters, there was occasionally a small outbreak following the decline in immunity, but this did not last to the end of the 5-year period 75% or 100%; a set of farms was only accepted if the convex hull area was within 1% of the actual approximate area covered by the 100% set of farms. For each parameter set with density <100%, four replicates were taken and simulated to eliminate the effect of randomly missing possibly important nodes in the shipment network. Each combination of area and density was simulated for 5 years, and the probability of persistence assessed as previously and averaged for the replicates.
These parameter sets were referred to as shipment-2 and are shown in Table 6.

Carrier-induced persistence
Simulating the model with the combinations of parameters found in Table 3, it was found that combinations of parameters with infectious carriers could lead to the persistence of FMD in the population over the 5-year period simulated. This was true with values of β c (carrier transmission) much smaller than estimated by Tenzin et al. in 2008. In the carrier-1 parameter set, β c took the values on the furthest right in Figure 4, extending from 2.67e-3 down to 4.17e-5, whereas carrier-2 included values down to 7.95e-11. Due to the range restricted values, no association of carrier transmission with persistence was found for the carrier-1 parameters, as shown in Figure 5a. As shown in Figure 7, this is due to all the values chosen resulting in persistence being certain. However, with values of β c extended in the carrier-2 parameter set, a clear relationship between β c and the probability of F I G U R E 8 The observed relationship between the probability of fomite transmission and the probability of persistence for shipment-1. The top half of the plot contains results when no long-range shipments were simulated, the bottom half when long range shipments were simulated. Each line type indicates a different duration of immunity (λ r ), and each point type the area being simulated. As the probability of fomite transmission increases, the probability of persistence also increases. Erzurum Province (EZ) requires much a higher probability of fomite transmission for persistence than Eastern Region (EA), as it has both a smaller area and fewer farms. Within each area, increasing immune duration increased the probability of fomite transmission required for persistence by a small amount. There is little difference whether long range shipments are simulated or not persistence is visible. Additionally, a pattern is visible in the relationship between λ c and λ r and the probability of persistence in Figures 4 and 5b. Population and the presence/absence of shipments remain uncorrelated with persistence. λ r is weakly negatively correlated with persistence, as shown in Figure 4 where after holding λ c constant, a longer duration of immunity decreases the probability of persistence.
β c and λ c are moderately positively correlated with persistence, also shown in Figure 4.
PRCC analysis was performed on these data, as shown in Figure 5a.
It was found that the parameter most strongly associated with changes to the probability of persistence is the presence or absence of carriers in the population (k c ). Significantly associated but weakly correlated are λ r (immunity duration) and λ c (carrier duration). Figures 6   and 7 show the average prevalence of the disease in the population over the 5-year period simulated, organized by these two parameters and demonstrating the relatively minor effects they have on the probability of persistence. No significant association was found between persistence and population size, the presence/absence of shipments or β c . These results remain when restricting analysis to parameter sets where k c = 0.5.

Shipment-induced persistence
Investigating scenarios where shipments can spread disease and fomite transmission is simulated, we see that no persistence appears to  Province, there is a large difference, with persistence rarely exhibited until the probability of fomite transmission >0.5. A small reduction in persistence is seen from removing long range transmission, as well as a slightly larger reduction from increasing the duration of immunity.
PRCC analysis of the shipment-1 parameter set suggests that the area simulated and the probability of fomite transmission are significantly positively associated with the persistence measure when shipments are simulated, as shown in Figure 9. The presence of long-range movements is not correlated, although the coefficient is significantly different from 0. The duration of immunity was not significantly associated with persistence in this analysis.

DISCUSSION
The results presented here suggest that persistence of FMD in populations is possible even with very small per-capita probabilities of transmission and no other pro-persistence factor, and that carriers therefore cannot yet be set aside as a possible cause for the persistence of FMD. This effect seemed to be independent of greater population size, suggesting that current realistic farm sizes are already large enough for this effect to take place.
The values of β c (carrier transmission) investigated are much smaller than the value estimated by Tenzin et al. (2008), itself an overestimate due to an inability to calculate species-specific β c due to a lack of experiments. Although that analysis could not explicitly rule out carrier transmission occurring, more studies have been carried out since that might be able to provide greater statistical certainty (Bertram et al., 2018;Hayer et al., 2018;Parthiban et al., 2015). It is unlikely, however, that statistical certainty could be provided for values of β c as low as seen in these experiments.
Focusing on carriers, the two main factors relating to carrierinduced persistence appear to be the average duration of immunity to FMD, and the average duration of the carrier state. Realistic estimates of these parameters may therefore be important in determining whether carrier-induced persistence is a realistic proposition.
Many different studies support different values, with some supporting shorter durations of 6-12 months and others supporting longer durations of up to several years (Bertram et al., 2020;Hayer et al., 2018;Moonen & Schrijver, 2000). Assuming a shorter carrier duration of 6 months, durations of natural immunity longer than 1.5 years appear to rule out undetected transmission from carrier animals, and evidence suggests immunity can last much longer (Doel et al., 2005). However, assuming a longer duration of the carrier state relaxes this, with realistic durations of immunity theoretically allowing both carrier transmission to be happening and to have remain undiscovered by experiments to date.
An important assumption of this study is that carrier transmission is homogenous through time, meaning that a persistently infected animal is as likely to infect a nearby susceptible animal 1 day before it clears the infection as it is 28 days after infection. This is to some extent unavoidable through the use of the τ-leap algorithm, which is memoryless and so has difficulties achieving this. Additionally, since we have difficulty demonstrating carrier transmission at all, there is no evidence that might inform whether or how transmission changes over time. Further work is needed here and it is important to establish whether our results hold under a more pessimistic assumption.
In the absence of explicitly modelled fomite transmission, the shipment of potentially infected cattle from infected farms to susceptible farms did not lead to persistence. This suggests either that the mechanism of persistence in this case is not asynchronous outbreaks in spatially separated areas, or that fomite transmission is necessary to achieve those asynchronous outbreaks.
The minimum probability of fomite transmission necessary for the probability of persistence to be greater than 0 declined as the number of farms modelled increased, suggesting that even small probabilities of fomite transmission would be sufficient for persistence to happen in regions with greater numbers of farms, or larger regions. Shipments therefore appear to represent a viable alternate mechanism for supporting persistence when fomite transmission is explicitly modelled.
This model's estimation of the effect of fomite transmission via this route is also likely an underestimate, as it assumes that fomites are only transmitted when the source farm is actively infected. In reality, fomites can survive for up to 6-9 months in the environment given favourable conditions, expanding the time period where fomites might contaminate vehicles and likely reducing further the necessary to contribute to persistence (Mielke & Garabed et al., 2020).
In conclusion, this study suggests that carrier-induced persistence cannot yet be discounted as a possibility, with our modelling approach demonstrating the ability of even very sporadic carrier transmission events that are unlikely to be detected to support persistence within a greater population. The main factors that affect the plausibility of carrier transmission being epidemiologically relevant to persistence are the duration of the carrier state and the immune state-further work on elucidating those are likely to narrow the range of values at which potential carrier transmission can be epidemiologically relevant and simultaneously undetected. However, shipment-induced persistence is a viable alternate mechanism by which persistence might occur and requires only small probabilities of fomite transmission. As fomite transmission is a recognized and well-studied mechanism, whereas carrier transmission has still not been shown to occur in the field, this study suggests that shipment-induced persistence remains the more likely of the two hypotheses to be occurring.