⚙️ Core: Refactor Newton-Raphson Solver

[fderuiter]

🔧 The Violation:
The calculate_final_size function in math_explorer/src/epidemiology/analytics.rs contained a hardcoded implementation of the Newton-Raphson method, violating DRY (logic duplication of a standard algorithm) and SRP (mixing domain logic with numerical solving). It also used string-based error handling.

🛠️ The Fix:

1. Extracted Logic: Implemented a robust NewtonRaphson solver in math_explorer/src/pure_math/analysis/roots.rs. This struct implements the existing RootFinder trait (using numerical differentiation) and exposes a specialized find_root_with_derivative method for high-precision use cases.
2. Composability: The solver supports optional bounds handling (with backtracking) and configurable iterations/tolerance.
3. Refactoring: Updated calculate_final_size to use the new NewtonRaphson component, injecting the analytical derivative of the final size equation.
4. Error Handling: Enhanced EpidemiologyError to wrap AnalysisError, moving away from stringly-typed errors.

📉 Debt Reduction:

• Removed ad-hoc solver loop from epidemiology domain code.
• Centralized root-finding logic in the analysis module.
• Improved error safety with typed errors.

🧪 Integrity:

• Added unit tests for NewtonRaphson covering linear, quadratic, and bounded scenarios.
• Verified test_final_size_high_r0 passes with the new implementation.

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PR created automatically by Jules for task 18224543845005302773 started by @fderuiter

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