Quantifying the Threat: A Bayesian Probabilistic Framework for Heavy Metal Health Risks among Children in the Densu Basin, Ghana
Authors: T. Ansah-Narh, G. Y. Afrifa, J. B. Tandoh, E. N. N. Nortey, D. Adjei, R. Opoku-Sarkordie, R. Davis
Organizations: Ghana Space Science and Technology Institute, Ghana Atomic Energy Commission, P. O. Box LG 80, Accra, Ghana. · Department of Statistics and Actuarial Science, University of Ghana, P. O. Box LG 115, Legon, Accra, Ghana. · Institut de Chimie Physique, Universit´e Paris-Saclay / CNRS UMR8000, 91400, Rue Michel Magat, Orsay, France. · Department of Information Technology and Mathematical Sciences, Methodist University Ghana, P. O. Box DC 940, Dansoman, Accra, Ghana. · Department of Mathematics and Actuarial Science, Pentecost University, P. O. Box KN 1739, Kaneshie, Accra, Ghana.
Groundwater in Ghana's Densu River Basin remains a key domestic water source, but multi-metal contamination threatens children's health. Previous deterministic hazard index assessments provide limited treatment of uncertainty, co-exposure, and spatial variability. This study applies a Bayesian probabilistic framework for non-carcinogenic risk assessment using 94 groundwater samples and measurements of Fe, Mn, Ni, Pb, Cd, and As. A Bayesian regression model propagates parameter uncertainty and accounts for contaminant co-occurrence. Adaptive Hamiltonian Monte Carlo is used for posterior inference, while inverse distance weighting generates spatially continuous risk surfaces. Pb, As, and Cd dominate cumulative risk. Posterior predictive results indicate that approximately 31% of sampling sites have a probability greater than 0.5 of exceeding a hazard index (HI) of 1. The risk surface is highly heterogeneous, with a posterior mean of 0.372 and a standard deviation of 0.416. High-risk zones often reflect moderate co-exposure rather than extreme contamination by a single metal. The framework provides uncertainty-aware, spatially explicit information to support groundwater monitoring, risk mitigation, and public health planning in data-limited regions.
Figures & tables
Figure 2.1 : Location and geological setting of the Densu River Basin, Ghana. (A) Geological map of the Densu Basin showing major lithological units, river networks, administrative boundaries, and groundwater sampling locations used in this study. (B) Map of Ghana indicating the position of the Densu Basin within the national context. (C) Regional map of Africa highlighting the location of Ghana. The spatial distribution of lithological units reflects the geological heterogeneity of the basin, which exerts a strong control on groundwater occurrence, flow pathways, and geochemical characteristics.
Figure 2.2 : Univariate distribution of heavy metal concentrations in groundwater samples from the Densu River Basin. Violin plots illustrate the kernel density of concentration values for each metal on a logarithmic scale, with individual observations overlaid as points. Box plots displayed below each violin indicate the median, interquartile range, and extreme values. Red dashed horizontal lines denote relevant regulatory guideline limits for drinking water. The distributions reveal pronounced right skewness, strong concentration at the analytical detection limit, and elevated variability driven by a limited number of high-concentration observations.
Figure 2.3 : Spearman rank correlation matrix of heavy metal concentrations in groundwater samples from the Densu River Basin. Correlation coefficients ( ρs ) quantify monotonic associations between metal pairs based on ranked concentrations, reducing sensitivity to skewness, outliers, and left-censored observations. Statistically significant correlations ( p<0.05 ) are indicated by an asterisk. The matrix highlights coherent co-variation among selected metals, reflecting shared hydrogeochemical controls such as lithological composition and redox-sensitive mobilisation within the basin’s crystalline basement aquifers.
Figure 4.1 : Schematic illustration of the Adaptive Hamiltonian Monte Carlo (HMC) sampling framework employed for Bayesian health risk assessment. Observed groundwater contaminant data are combined with prior distributions to define the posterior density of the model parameters θ=(α,β,σ) as specified in Eq. ( 12 ). Posterior exploration is performed using the No-U-Turn Sampler (NUTS), which adaptively simulates Hamiltonian trajectories by dynamically tuning the step size and trajectory length. Candidate states (θ(t+1),r(t+1)) are accepted with probability min{1,exp(ΔH)} , where ΔH denotes the change in Hamiltonian energy between successive states. Accepted samples are subsequently propagated through the posterior predictive model in Eq. ( 13 ) to estimate the probability of exceedance of the health risk threshold defined in Eq. ( 14 ).
Figure 7.1 : Trace plots and marginal posterior distributions for the intercept ( α ), metal-specific regression coefficients ( βi ), and residual standard deviation ( σ ) from the Bayesian health risk model. The stable trace behaviour and unimodal posterior densities indicate satisfactory convergence and effective exploration of the posterior parameter space.
Figure 7.2 : Corner plot showing the joint posterior distribution of model parameters, including the intercept ( α ), regression coefficients for individual heavy metals ( βFe , βMn , βNi , βPb , βCd , βAs ), and the residual error term ( σ ). Diagonal panels display marginal posterior distributions with 95% credible intervals (red dashed lines), while off-diagonal panels illustrate pairwise parameter dependencies. Annotated values denote the posterior probability of a positive effect, P(βi>0) .
Figure 7.3 : Spatial distribution of dissolved heavy metal concentrations across the Densu Basin derived using IDW interpolation. Panels show interpolated concentrations of Fe, Mn, Ni, Pb, Cd, and As based on groundwater samples collected at discrete locations (circles). The maps highlight pronounced spatial heterogeneity, reflecting the combined influence of aquifer lithology, groundwater flow pathways, redox conditions, and localised anthropogenic inputs. Concentration patterns provide a hydrogeochemical context for subsequent Bayesian health risk integration, but do not, on their own, represent cumulative exposure or probabilistic health risk.
Figure 7.4 : Posterior predictive spatial distribution of cumulative non-carcinogenic health risk for children, expressed as the probability of Health Index exceedance, P(HIchildren>1) . Site-level posterior risk probabilities, computed from the Bayesian health risk model (Eq. ( 16 )), were interpolated using inverse distance weighting to produce a continuous risk surface constrained to the basin boundary. Contours denote equal-risk probability levels, while point symbols indicate sampling locations coloured by their posterior risk estimates. Summary statistics reported on the map (mean, standard deviation, maximum risk, and proportion of high-risk sites) quantify spatial variability and uncertainty in cumulative exposure across the basin.
Institute of Environment and Sanitation Studies University of Ghana Legon, Accra, Ghana · Ghana Space Science and Technology Institute Accra, Ghana · Department of Mathematics Rhodes University, Grahamstown, South Africa +2
Ghana Space Science and Technology Institute, Ghana Atomic Energy Commission, P. O. Box LG 80, Legon, Accra, Ghana · Department of Medical Microbiology, University of Ghana Medical School, University of Ghana, Accra, Ghana