The explicit formulation by Hicken and Kaur uses all points in the point cloud to define the implicit function and shows favorable decay in surface reconstruction error as the number of points in the point cloud NΓ increases. This structure has been used in combination with RBFs for hole-filling in [37] and anisotropic basis functions for representing sharp corners in [40]. Another approach is to construct a uniform grid of points to control the implicit function. Unlike the aforementioned approaches, the distribution of points is decoupled from the resolution of the point cloud. As a result, deformations to the geometric shape can be represented without loss in accuracy near the surface as shown by Zhao et al.. This makes it a popular structure in partial differential equation based reconstruction methods that evolve the surface during reconstruction, such as in [47, 48]. In general, more points representing the implicit function are required to achieve the same level of accuracy to other approaches. As a result, implicit functions defined by a uniform grid are more computationally expensive to solve for in both time and memory usage than the aforementioned approaches, as experienced by Sibley and Taubin, but can be reduced by a GPU-based multi-grid approach as implemented by Jakobsen et al.. The signed distance function presents an ideal candidate for implicit surface reconstruction and geometric non-interference constraints. It is known that the zero level set of the SDF is a smooth representation of the points in a point cloud, indoor grow light shelves and its gradient field is a smooth representation of the normal vector field from the normal vectors in a point cloud. As a result, many formulations to approximate the SDF have been researched for implicit surface reconstruction.

We note that there exists other methodologies, such as wavelets and a Fast Fourier Transform based method, that fit a smooth indicator function instead, but are less applicable for non-interfernce constraints where a measurement of distance is desired. We identify four categories that approximate the SDF in some way: explicit formulations, interpolation formulations with RBFs, PDE-based formulations, and energy minimization formulations. These formulations then apply smoothing to these linear approximations in order to define the level set function. Risco et al. present the simplest approach which uses the nearest edge and normal vector to define the function explicitly. The resultant constraint function is piece wise continuous but non-differentiable at points where the nearest edge switches. Belyaev et al. derive a special smoothing method for defining signed Lp-distance functions, which is a continuous and smooth transition between piece wise functions. Hicken and Kaur use modified constraint aggregation methods to define the function in a smooth and differentiable way. Upon the investigation of Hicken and Kaur, the signed Lp-distance functions give poor approximations of the surface. Additionally, Hicken and Kaur’s formulation is shown to increase in accuracy as the data in the point cloud, number of points NΓ, increases. We identify Hicken and Kaur’s explicit formulation as a good candidate for enforcing non-interference constraints,as it is continuous and differentiable with good accuracy. Another method to construct the level set function is to solve an interpolation problem given an oriented point cloud P. Because the data points of P always lie on the zero contour, nonzero interpolation points for the implicit function can be defined on the interior and exterior, as originally done by Turk and O’Brien. Radial basis functions are then formulated to interpolate the data. To avoid overfitting, thinplate splines can be used to formulate the smoothest interpolator for the data, as noted in [37, 45].

Solving for the weights of a RBF involves solving a linear system, which is often dense and very computationally expensive due to their global support. Turk and O’Brien solve up to 3,000 RBF centers, and improvements by Carr et al. allow up to 594,000 RBF centers to be constructed in reasonable time . On top of the significant computational expense, interpolating RBFs have been criticized for having blobby reconstructions which poorly represent sharp features in the geometric shapes.The vector field is then integrated and fit, usually by a least squares fitting, to make the zero level set fit the point cloud. We classify the methods that solve for the vector field as a solution to a partial differential equations as PDE-based methods. Poisson’s method uses variational techniques to Poisson’s equation to construct a vector field. Improvements to this method add penalization weights to better fit the zero contour to the point cloud in [54]. Tasdizen et al. prioritize minimal curvature and minimal error in the vector field by solving a set of coupled second order PDEs to derive their level set function. Zhao et al. use the level set method, originally introduced by Osher and Sethian, for surface reconstruction, with the advantage of modeling deformable shapes. In the aformentioned PDE-based methods, the setup for the implicit function reduces to solving a PDE by time-stepping or a sparse linear system in the case of Poisson’s equation.In the analysis done by Calakli and Taubin, they found that Poisson’s method often over-smooths some surfaces. We also note that solutions to PDEs are more difficult to implement than other methods in practice. Aquaculture is an important contributor to the Irish economy, producing products to the value of e167 million in 2016, including e105 million from farmed Atlantic salmon . The industry is particularly important along the western seaboard of Ireland.

Most Irish salmon farming is certified organic . Salmon farming in Ireland is associated with an intricate network of fish movements within and between the different types of salmon farms. There are three different farm types, including brood stock, freshwater, and seawater farms. In earlier work , social network analysis was used in combination with spatial epidemiological methods to characterize the network structure of live farmed salmonid movements in Ireland. It was demonstrated that characteristics of the network of live salmonid fish movements in Ireland would facilitate infection spread processes. These included a power-law degree distribution [that is, “scale free”], short average path length and high clustering coefficients [that is, “small world”], with the presence of farms that could potentially act as super-spreaders or super-receivers of infection, with few intermediaries of fish movement between farms, where infectious agents could easily spread, provided no effective barriers are placed within these farms . A small proportion of sites play a central role in the trade of live fish in the country. Similarly, we demonstrated that highly central farms are more likely to have a number of different diseases affecting the farm during a year, diminishing the effectiveness of in-farm bio-security measures , and that this effect might be explained by an increased chance of new pathogens entering into the farm environment . This is a very important area of research in aquaculture, especially considering that the spread of infection via fish movement is considered one of the main routes of transmission . Mathematical models and computer simulations offer the potential to study the spread of infectious diseases and to critically evaluate different intervention strategies . Through access to real fish movement data, these models can be programmed to incorporate both the time-varying contact network and data-driven population demographics. However, there are considerable computational challenges when stochastic simulations are conducted using livestock data, rolling tables both computationally, including the need for efficient algorithms, and also with model selection and parameter inference . An efficient modeling framework for event-based epidemiological simulations of infectious diseases has recently been developed , including the use of a framework that integrates within farm infection dynamics as continuous-time Markov chains and livestock data as scheduled events. This approach was recently used to model the spread of Verotoxigenic Escherichia coli O157:H7 in Swedish cattle . Cardiomyopathy syndrome is a severe cardiac disease of Atlantic salmon. It was first reported in the mid-1980s in farmed salmon in Norway and later detected in several other European countries, including the Faroe Islands , Scotland and, in 2012, in Ireland . CMS generally presents as a chronic disease, leading to long-lasting, low-level mortality, although some individuals experience sudden death. At times, however, CMS can present as an acute, dramatic increase in mortality associated with stress . A recent Norwegian study has identified risk factors for developing clinical CMS, including stocking time, time at sea, a previous outbreak of pancreatic disease or Heart and Skeletal Muscle Inflammation , and hatchery of origin . The economic impact of CMS is particularly serious as it occurs late in the life cycle, primarily during the second year at sea, by which time the incurred expenditure is high. No effective preventive measures are known, and there is no treatment available . In 2009, CMS was identified as a transmissible disease , and has been linked, in 2010 and 2011, to a virus resembling viruses of the Totiviridae family .

The discovery of this virus, piscine myocarditis virus , has contributed to increased knowledge about the disease including the development of new diagnostic, research and monitoring tools . The agent is spread horizontally, between farms at sea, although there is some indication of a possible vertical transmission pathway . Recent Norwegian research has shown that PMCV is relatively widespread, including in geographic regions and fish groups without any evidence of CMS . The mechanisms leading to progression from PMCV infection to CMS are currently unclear . CMS is present in Ireland. The first recorded outbreak of CMS occurred in 2012, associated with low-level mortalities over a period of 4–5 weeks followed by increased mortalities during bath treatment for sea lice . CMS is not a notifiable disease in Ireland, and there are no systematic records of its occurrence. Nonetheless, anecdotal information from field veterinarians and farmers suggest that CMS occurrence has steadily increased over the years. A retrospective study was recently conducted, using real-time RT-PCR with archived broodstock samples dating back to 2006, which suggests that PMCV may have been introduced into Ireland in two different waves, both from the southern part of the range for PMCV in Norway . PMCV was found to be largely homogenous in Irish samples, with limited genetic diversity. Further, the majority of PMCV strains had been sequenced from fish that were not exhibiting any clinical signs of CMS, which suggests possible changes in agent virulent and/or the development of immunity in Irish farmed Atlantic salmon . This paper describes the use of data-driven network modeling as a framework to evaluate the transmission of PMCV in the Irish farmed Atlantic salmon population and the impact of targeted intervention strategies. This approach can be used to inform control policies for PMCV in Ireland, as well as other infectious diseases in the future. Model parameters were estimated from a previous study, which had been conducted in 2016 and 2017 to determine the prevalence of PMCV infection in Irish salmon farms by real-time RT-PCR . The sampling strategy was replicated to ascertain the status that could have been found if simulated farms had been sampled. In this study, sample collection was conducted on 22 farms from 30 May 2016 to 19 December 2017. A ranching farm is a freshwater broodstock farm that releases juvenile fish to the environment for conservation purposes. Some farms were sampled more than once over the course of the study, with the median samplings per farm in this group being 3.5 . A total of 1,201 fish were sampled during the study. Samples consisted of heart tissue across all fish age classes and ova. In this study, PMCV was detected at a low level in most sites, with only one clinical case of CMS occurring during the study period. We simulated sampling at each time point by randomly sampling fish within each farm and age category, as in the observed data set, from the number of susceptible and infected individuals at the time for the sampling point in the simulated farms. The aforementioned observational study also looked for PMCV in archived samples of Atlantic salmon broodstock from 2006 to 2016, seeking to determine whether the agent had been present in the country prior to the first case report in 2012 . For this, archived samples of broodstock Atlantic salmon were tested for each year from 2006 through 2016, using 60 archived pools per year.