Abstract
Spatial heterogeneity of a host population of mobile agents has been shown to be a crucial determinant of many aspects of disease dynamics, ranging from the proliferation of diseases to their persistence and to vaccination strategies. In addition, the importance of regional and structural differences grows in our modern world. Little is known, though, about the consequences when traits of a disease vary regionally. In this paper, we study the effect of a spatially varying per capita infection rate on the behaviour of livestock diseases. We show that the prevalence of an infectious livestock disease in a community of animals can paradoxically decrease owing to transport connections to other communities in which the risk of infection is higher. We study the consequences for the design of livestock transportation restriction measures and establish exact criteria to discriminate those connections that increase the level of infection in the community from those that decrease it.
1. Introduction
In today's world, connections between communities become ever more important for epidemiological modelling. In this respect, the incorporation of spatial heterogeneity has considerably improved the understanding of the behaviour of diseases. While the spread of a disease can naturally be described only by explicitly allowing for space [1,2], spatial heterogeneity has also proved to have a significant impact on vaccination strategies [3–5] and the persistence of diseases [6–9]. The underlying question in this field is how communities are affected by the global dissemination of a disease [10]. For livestock diseases, understanding the relevant mechanisms is essential for the design of containment strategies, such as livestock transportation restrictions that are capable of stemming the progression of a disease. Such considerations become increasingly important because endemic livestock diseases can have severe economic consequences owing to a significant decline of the commodities that are produced by the livestock, such as food or fibre [11–13]. Additionally, zoonotic diseases such as brucellosis and particular influenza variants have the potential to cause infections in humans, leading to the threat of starting a subsequent outbreak in the human population [14].
While some studies have considered the case where parameters of the disease such as the rate of infection β depend on the individuals' locations [4,15–19], spatial heterogeneity is usually associated with a spatial segregation of the host population where the characteristics of the disease remain homogeneous in space [1–3,8,9]. However, spatially heterogeneous infection rates can occur owing to climatic and environmental differences [4]. In particular, the per capita infection rates for livestock diseases may crucially depend on the housing and husbandry conditions, which may impact possible containment strategies. The effects, however, remain largely unexplored.
Here we introduce a metapopulation model where the animal communities differ only by the local infection rate β and the population size N. A community can be thought of as a farm that is connected to other farms via a livestock transportation network, but our model is not restricted to a specific agricultural setting. We first analyse how the endemic fraction of infectious animals in two coupled communities depend on the coupling strength (i.e. transportation rate between them). We find that coupling a community to another community with a higher infection rate can both increase and decrease the endemic fraction of infectious animals, depending on the coupling strength between them. We subsequently study the consequences of these findings for the design of livestock transportation restrictions in a more complex scenario of multiple interconnected communities. We show that imposing transportation restrictions to communities with higher infection rates commonly has the opposite of the intended effect and increases the fraction of infectious animals.
We finally study the stability of the globally diseasefree state. We find that there is a threshold for the transportation rate above which coupling leads to a global eradication of the disease under appropriate conditions.
2. Model
We consider a SIR metapopulation model, where animals are located in one of M populations of size N_{k} (k∈{1, … , M}) and are randomly transported from population to population regardless of their state (susceptible, infectious or recovered). Every population is thus partitioned into three subgroups: susceptible (S_{k}), infectious (I_{k}) and recovered (R_{k}) animals. Susceptible animals contract the disease with a rate β_{k} only from infectious animals who reside in the same population. The infection rate β_{k} depends on the common location of the susceptible and the infectious animal. Recovery from the disease occurs with rate γ and elicits lifelong immunity. We assume that there is no diseaserelated mortality, such that all animals die with rate μ regardless of their current state. To ensure that the population size of any isolated population is constant, we require the rate at which animals produce offspring to equal the death rate μ. All animals are born susceptible. Finally, animals are transported from population j to k with rate ɛ_{kj}. Since we also require the populations to have on average a constant population when they are coupled, we require a detailed balanced net flow of animals (i.e. ɛ_{kj}N_{j} = ɛ_{jk}N_{k}). This condition uniquely determines ɛ_{kj} from ɛ_{jk} so that we define the symmetric transportation rate between community j and k as (figure 1).
When we consider the fractions (X∈{S, I, R}) of susceptible, infectious and recovered animals in two spatially segregated communities (M = 2) their respective temporal evolution is governed by2.1For brevity, we have omitted the equations for the second community, which are obtained by exchanging the subscripts 1 and 2. The fraction of the recovered class is determined by the identity s_{k} + i_{k} + r_{k} = 1 with k∈{1, 2}. The derivation of equation (2.1) for an arbitrary number of communities M is given in the electronic supplementary material, appendix I.
3. Results
(a) Paradoxical effect of coupling
In a fully susceptible and isolated (i.e. uncoupled) population, an infinitesimal fraction of infectious animals grows with the rate β_{k} − γ − μ. The basic reproductive ratio must therefore be larger than unity to allow an infection to spread. For the isolated populations (ɛ_{1–2} = 0), it is well known [15,20] that the solutions of equation (2.1) exhibit damped oscillations towards the endemic states3.1If , the second relation indicates that all endemic states of a homogeneously mixing population that can be obtained by varying with given and fixed γ and μ lie on a straight line with slope μ/γ + μ in the s − i plane.
We assume in the following without loss of generality that Community 1 has a higher basic reproductive ratio than community 2 (i.e. ). For the uncoupled communities, the endemic level of infection is hence higher in community 1 than in community 2, . Upon coupling, we accordingly expect because of mixing that the endemic level of infection decreases in community 1, , and increases in community 2, .
To study the effect of coupling on the endemic states for small transportation rates ɛ_{1–2}, we determine the slope of the endemic levels of infection with respect to the coupling strength. We find that the intuition that decreases and increases upon coupling is not generally confirmed. In fact, we find that there is a threshold for the basic reproductive ratio above which the endemic level of infection in community 1 increases upon coupling:3.2
Analogously, if , the endemic level of infection in community 2 startlingly decreases upon coupling. The derivation of these results can be found in the electronic supplementary material, appendix II. Similar thresholds can be derived for a model where immunity is only temporal (SIRS model) and for a model where animals go through a latent phase upon infection (SEIR model), given in the electronic supplementary material, appendix III. Note that apart from the restriction the threshold can be exceeded independently for the two communities. From this follows that there are three distinct cases.

(1) Intuitive case, : upon coupling, the endemic level of infection decreases in community 1 and increases in community 2 (threshold is exceeded in neither community).

(2) Detrimental case, : upon coupling, the endemic level of infection increases both in community 1 and in community 2 (threshold is exceeded in one community).

(3) Paradoxical case, : upon coupling, the endemic level of infection increases in community 1 and decreases in community 2 (threshold is exceeded in both communities).
Figure 2 exemplifies the three cases. The endemic states of the two communities were calculated by numerically integrating equation (2.1) until the stationary state had been reached. First, the left part of figure 2 shows that the effect of coupling on the animal communities cannot be described as a simple average of the animal communities, since this would imply that the coupled states lie on the straight line connecting the two uncoupled states. Second, we see from the right part of figure 2 that the paradoxical effect extends over a broad range of transportation rates ɛ_{1–2}. For large ɛ_{1–2}, however, the intuitive expectation that the endemic fraction of infectious animals of the two communities adjust to an intermediate value is confirmed, irrespective of the choice of parameters. This implies that there exists a neutral transportation rate for which coincides with the fraction of the uncoupled state, i_{k}(0). This transportation rate can be determined by equating the fraction of infectious animals in the coupled case, , with the fraction of the uncoupled case, , in equation (2.1), and solving for :3.3Surprisingly enough, the neutral transportation rate is different in the two communities and its value in one community is independent of the other community.
The paradoxical effect thus occurs in communities in which the threshold (equation (3.2)), is exceeded and for connections with a transportation rate in the interval .
(b) Transportation restrictions
The movement of livestock is governed by a transportation network that represents the routes by which the animals are traded and marketed. Examples include the network of sheep movements in Great Britain [21] and the network of cattle trade movements in the United States [22]. This motivates the consideration of a more complex scenario in which we study the effect of imposing selective transport restrictions. For the sake of generality, we consider a nonspecific disease and extend the presented model to the SEIR model, which includes a latent period of the disease (σ^{−1}). We consider a focal community with R_{0} = 16 that is connected with a transportation rate ɛ to six communities of different sizes whereof three communities have lower basic reproductive ratios (R_{0}∈{14, 14.5, 15}) and three communities have higher basic reproductive ratios (R_{0}∈{17, 17.5, 18}), as shown in figure 3a. The three high R_{0} communities are coupled twice as strongly to each other and so are the three low R_{0} communities. All communities are initially in their uncoupled endemic state and settle to their coupled equilibrium after a sufficiently long transient time. Figure 3b,c shows time series of the fractions of infectious animals for two different transportation restriction measures. The initial transportation rate was chosen to be ɛ = 1.5a^{−1}, but other values (smaller than ɛ_{n}) yield qualitatively the same results. Figure 3b shows the effect of transportation restrictions imposed at t = 0 that suspend all transportation between the focal population (blue) and the three communities with a higher R_{0} (red). After a steep increase of the fraction of infectious animals in the focal population, it settles to a new equilibrium that is higher than the equilibrium before the transportation restrictions were imposed. Contrary to this, when transportation restrictions are imposed that suspend all transportation between the focal population and the three communities with a lower R_{0} (green) (figure 3c), the fraction of infectious animals in the focal population steeply decreases and settles to a new equilibrium that is lower than before. These observations are in accord with the analytical predictions of the previous section. Our model suggests that the R_{0} threshold for the paradoxical effect is generically small. In particular, for figure 3, we chose μ = 0.03a^{−1}, γ^{−1} = 7d and σ^{−1} = 7d resulting in a threshold value of R_{0} = 1.001. All seven communities are hence in the paradoxical regime.
The connection of the focal community to the communities with higher (lower) R_{0} thus reduced (increased) the fraction of infectious animals. Imposing transportation restrictions between them removes that reduction (increase) and as a consequence the fraction of infectious animals increases (decreases).
In order to better visualize the effect of transportation restrictions, we have disregarded the effects of seasonally varying infection rates. The inclusion of seasonal variation, however, does not qualitatively change the observed behaviour. The results for the same scenario as above but including seasonal variation are shown in the electronic supplementary material, appendix IV.
(c) Disease eradication
In the previous sections, we have considered the case in which R_{0} > 1 in both communities. In that case the disease is endemic in both communities irrespective of coupling and only the level of infection is of interest. If on the contrary R_{0} < 1 in both communities, the endemic state is unstable in both communities and the disease dies out globally irrespective of the transportation rate. A pertinent question is therefore under which conditions the disease dies out globally if R_{0} > 1 in one community but R_{0} < 1 in the other.
A linear stability analysis of the globally diseasefree state of the SIR model defined by equation (2.1) reveals that the disease can only die out globally if the condition3.4is fulfilled. This condition is intuitively appealing for the following reason. In the limit of large transportation rates, the two communities behave as one large community with a basic reproductive ratio that equals the weighted average of the two communities. Equation (3.4) is thus equivalent to the requirement that R_{0} of the fully coupled system fulfils the condition for the diseasefree state to be stable.
If equation (3.4) is fulfilled, the globally diseasefree state is stable (i.e. the disease dies out globally) when the transportation rate exceeds the threshold ɛ_{1–2} ≥ ɛ_{s} with3.5
A detailed derivation of these results is given in the electronic supplementary material, appendix V. Figure 4 shows the endemic levels of a hypothetical disease in two communities that fulfil equation (3.4). For ɛ_{1–2} < ɛ_{s}, the disease is endemic in both communities although R_{0} < 1 in one of them. Above ɛ_{s}, however, the influence of the formerly diseasefree community becomes predominant and the disease dies out in both communities. Similar results were found by Burton et al. [23].
Although these findings show that mere coupling to a community with R_{0} < 1 can be used to eradicate a disease globally, it is important to note that the necessary transportation rate can be unattainably high and an intermediate increase of the transportation rate can increase the level of infection owing to the occurrence of the paradoxical effect.
4. Discussion
In this work, we have shown that a connection to a community with a higher risk of infection can paradoxically decrease the level of infection. Correspondingly, a connection to a community with a lower risk of infection can unfortunately increase the level of infection. This paradoxical effect can intuitively be understood by recalling that a disease flourishes when many potential hosts are susceptible. A higher endemic level of infectious hosts can only be sustained when there is an additional source of susceptible hosts. Coupling to a community with a lower risk of infection results in exactly such a steady import of susceptible hosts, and it is this additional source of susceptible hosts that increases the fraction of infectious animals. This stands in contrast to the widespread belief that transportation restrictions are unconditionally suitable to reduce the level of infection as they stop the import of the pathogen. While the efficacy of travel restrictions in case of human diseases has already been questioned [24–28], our findings are the first to show that transport restrictions for livestock can even have detrimental effects.
Our analysis of a more complex metapopulation shows that—given sufficiently small transportation rates—transport restrictions should be imposed on connections to communities with lower infection rates rather than to communities with higher infection rates. In our example, sufficiently small transportation rates mean that every animal in the population is transported to another equally sized population less than every day. This exceptionally high upper limit shows that the paradoxical effect affects virtually all except for very tight connections. More generally, the R_{0} threshold for the paradoxical regime is typically exceeded for diseases that are commonly modelled by compartmental models with an endemic state. For infectious diseases with an infectious period and/or latent period on the scale of weeks, the threshold is around 1.001 and hence very close to the critical threshold value R_{0} = 1, which is exceeded by the majority of endemic livestock diseases [29].
In the case where the weighted average of R_{0} in the different communities is below 1, we have shown that a global eradication is possible through mere coupling. For two communities in that situation it is likely, though, that only one community exceeds the threshold for the paradoxical effect, corresponding to the detrimental case. A policy to increase the transport rate is therefore likely to be detrimental for a broad range of values.
Our results depend on how coupling is accounted for. The commonly used metapopulation model in epidemiological studies assumes in contrast to our model that individuals are not transported but that the effect of coupling is captured by an effective force of infection that depends on the constituents of the entire metapopulation [3,4,8,9,17–19,30]. We show in the electronic supplementary material, appendix VI, that in such a direct transmission model the fraction of infectious animals in a community behaves as intuition suggests and decreases (increases) when coupled to a community with a lower (higher) R_{0}. The difference between this direct transmission model and the model used here, equation (2.1), consists in the manner in which the coupling of the populations is accounted for. The explicit modelling of the transport in contrast to the effective force of infection is therefore responsible for the paradoxical effect.
Our description of the host's transport as memoryless is also appropriate for the migration of many nondomesticated animal populations [20,31], while human movement follows more complex patterns (e.g. [32–36]). It remains an open and interesting question how our results extend to human diseases where hosts follow these more complex patterns of movement. We expect the present work to trigger research in this direction.
In short, we have demonstrated that conventional prevention and control strategies for livestock may be harmful when disease properties vary spatially. By contrast, disease models that properly account for the spatial variation may suggest prevention and control strategies that appear to be counterintuitive but are capable to significantly reduce the prevalence of diseases in a livestock population.
 Received November 17, 2014.
 Accepted December 2, 2014.
 © 2014 The Author(s) Published by the Royal Society. All rights reserved.