The optimal placement of Patient Assistance Terminals (PATs) in remote terrains constitutes a multi-objective optimization problem on non-Euclidean manifolds. Let T ⊂ ℝ 3 represent the physical terrain and S = { s 1 , s 2 , ⋯ , s n } denote settlement locations with associated population weights { w i } i = 1 n .

The placement optimization seeks to determine PAT positions p = { p 1 , p 2 , ⋯ , p k } ⊂ T that optimize:

where A ( p ) represents accessibility cost, ℒ ( p ) denotes latency penalty, and ℛ ( p ) encodes robustness constraints.

Traditional Euclidean models fail to capture the exponential growth patterns and hierarchical structures inherent in remote settlement networks [[1], [2]]. The hyperbolic plane ℍ 2 provides a natural framework for modeling such tree-like topologies [[3]]. PAT candidate positions are optimized in the latent hyperbolic space, but the final PAT locations belong to the physical terrain.

The Poincaré disk model represents ℍ 2 as:

with metric tensor:

The hyperbolic distance between points z 1 , z 2 ∈ D is:

The embedding process maps settlement networks to the Poincaré disk while preserving local community structures [[3], [4]]. Let G = ( V , ℰ , W ) represent the settlement graph where W : ℰ → ℝ + assigns edge weights based on travel difficulty.

The embedding function ϕ : V → D minimizes distortion:

where d G ( i , j ) represents graph distance and d ℍ is hyperbolic distance.

The embedding optimization employs gradient descent on the Poincaré disk:

where Exp is the exponential map and grad is the Riemannian gradient.

The accessibility function A ( p ) incorporates population-weighted distances:

The latency function ℒ ( p ) models communication delays:

where c eff is the effective communication speed accounting for terrain obstacles.

The robustness function ℛ ( p ) ensures network connectivity:

where L † is the Moore-Penrose pseudoinverse of the graph Laplacian.

To operationalize the integration of qualitative community knowledge, each response q i ∈ Q is first encoded using a fixed transformer-based text embedding model, embed ( ⋅ ) , producing a dense vector representation x i = embed ( q i ) ∈ ℝ d . In our implementation, we use a 384-dimensional multilingual Sentence-BERT encoder [[5]], chosen for its robustness in capturing semantic similarity across low-resource languages and mixed-dialect inputs common in remote terrains.

The integration function

is implemented as a calibrated logistic model. Its parameters ( w Ψ , b Ψ ) are learned from a labeled subset of the survey responses. During a preliminary field calibration phase, domain experts assigned importance scores [[6]] in the range [ 0 , 1 ] to selected responses, reflecting their relevance to accessibility constraints (e.g., seasonal road closures, culturally protected areas, mobility bottlenecks). A supervised logistic regression model was then fitted to the pairs ( x i , y i ) , where y i denotes the expert-assigned score. A k-fold cross-validation procedure was used to ensure generalization and prevent overfitting.

The resulting values Ψ ( q i ) are aggregated at the settlement or route level and incorporated into the edge weights and node priorities of the optimization process. This ensures that local human knowledge influences PAT placement in a quantitatively defined and reproducible manner while preserving the nuanced contextual constraints provided by communities.

The embedding preserves hub-centric patterns through hierarchical constraints [[7]]. For a hub h with satellites S h , the embedding satisfies:

where ρ controls the hierarchical radius.

The curvature constraint ensures proper hyperbolic structure:

where R is the radius of curvature adapted to local terrain complexity.

The multi-objective optimization employs Pareto-efficient solutions. The scalarized objective is:

The optimization uses hyperbolic gradient descent [[8]]:

The Riemannian gradient in the Poincaré disk is:

where ∇ E f is the Euclidean gradient.

Travel Difficulty (Edge Weights W ): For each settlement edge ( i , j ) ∈ E , the travel difficulty weight is defined as:

where:

d euclid ( i , j ) = physical Euclidean distance, Δ h i j = elevation difference obtained from the data, O i j = binary terrain obstacle indicator (river crossing, marshland, dense forest), α 1 , α 2 , α 3 = normalization constants.

Effective Communication Speed c eff :

where:

c 0 = manufacturer-specified mesh link throughput, δ obst ∈ [ 0 , 1 ] = empirically estimated terrain obstruction factor incorporating foliage density, humidity, and line-of-sight occlusion.

Effectiveness:

where A , L , and R correspond to accessibility, latency, and robustness respectively.

Validation was conducted in three remote regions: sub-Saharan Africa (mountainous terrain), Southeast Asia (dense forest), and South America (river delta). Each region included 15 - 30 settlement hubs with populations ranging from 50 - 2000 individuals.

The hyperbolic geometry framework achieved:

Average accessibility improvement: 34.7% over Euclidean models Latency reduction: 28.3% compared to grid-based placement Network robustness: 91.2% connectivity under single-node failures Local knowledge integration accuracy: 96.8% validation by communities

Table 1 presents comparative results:

Table 1.Comparative performance analysis for PAT placement methods.

Performance across different terrain types:

Mountainous regions: 88.4% effectiveness (elevation variance >500m) Dense forest: 92.1% effectiveness (canopy coverage >80%) River systems: 94.7% effectiveness (seasonal flooding areas) Desert regions: 89.3% effectiveness (sand dune migration)

Community validation through focus group interviews (FGI), seasonal mobility mapping workshops, and cross-validation demonstrated:

Cultural site preservation: 100% compliance Seasonal accessibility accuracy: 94.2% Traditional route integration: 91.8% Resource constraint recognition: 96.5%