Group-theoretic and equivalence-symmetry analysis of a susceptible–infected–AIDS HIV model

Technical Summary: Group-theoretic and Equivalence-Symmetry Analysis of a Susceptible–Infected–AIDS HIV Model

Problem Statement
Mathematical modeling of HIV transmission, particularly within the Susceptible–Infected–AIDS (SICA) framework, often conflates two distinct concepts in symmetry analysis: Lie point symmetries (transformations mapping solutions of a single fixed model to other solutions of the same model) and equivalence transformations (transformations mapping one model of a family to another by adjusting both states and parameters). This conflation can lead to invalid reductions, such as assuming population scalings are symmetries of a fixed-parameter system when they are not. The paper addresses this by rigorously distinguishing between these two concepts for a standard three-compartment HIV model, aiming to establish a reproducible template for symmetry analysis that avoids assuming non-existent first integrals or misidentifying model-family equivalences as dynamical invariants.

Methodology
The author employs a systematic Lie group analysis approach:

1. Model Formulation: A deterministic SICA model is defined with state variables $S$ (susceptible), $I$ (infected), and $A$ (AIDS), governed by recruitment ($\Sigma$), transmission ($\gamma$), natural mortality ($\alpha$), progression ($\omega$), and AIDS-induced mortality ($\delta$).
2. Dynamical Analysis: The author first establishes the forward invariance of the nonnegative octant, the boundedness of solutions, and the existence of disease-free and endemic equilibria separated by the basic reproduction number ($R_0$).
3. Lie Point Symmetry Testing: Using the first prolongation of vector fields, the author tests candidate generators directly against the determining equations. They specifically test time translation and intuitive population scalings (uniform scaling of all states and scaling of infected classes) while holding parameters fixed.
4. Equivalence Group Construction: Recognizing that population scalings fail as point symmetries, the author extends the analysis to the joint state–parameter space. They construct an equivalence group where parameters transform alongside state variables to preserve the form of the differential equations.
5. Dimensionless Reduction: By identifying the joint invariants of the commuting dilation generators within the equivalence group, the author derives a reduced dimensionless model.
6. Numerical Verification: The theoretical findings are validated through independent numerical experiments, comparing trajectories of a base model and a transformed model mapped through the derived invariants.

Key Contributions and Results

• Rejection of Simple Scaling Symmetries: The analysis proves that for a fixed-parameter model, the only admitted elementary Lie point symmetry is time translation ($X_0 = \partial_t$). Common assumptions that population scalings (e.g., $S \to kS, I \to kI$) are symmetries are shown to be false; the determining equations yield non-vanishing residuals due to the constant recruitment term ($\Sigma$) and the bilinear incidence term ($\gamma SI$).
• Identification of the Equivalence Group: The population scalings are rehabilitated as equivalence transformations. When parameters ($\Sigma, \gamma, \alpha, \omega, \delta$) are rescaled in step with the populations, the system forms a solvable three-parameter equivalence group. This group consists of time translation, time dilation (contracting rate parameters), and population dilation (rescaling states and specific parameters).
• Dimensional Reduction: The joint invariants of the two commuting dilations reduce the original five dimensional parameters ($\Sigma, \gamma, \alpha, \omega, \delta$) to three independent dimensionless parameters:
  • $\beta = \gamma\Sigma/\alpha^2$ (transmission strength relative to demographic scale)
  • $q = \omega/\alpha$ (progression rate relative to natural mortality)
  • $d = \delta/\alpha$ (excess AIDS mortality relative to natural mortality)
    The dimensionless time is $\tau = \alpha t$, and populations are scaled by $\Sigma/\alpha$.
• Dynamics of the Reduced Model: The reduced system exactly reproduces the epidemiological dynamics of the full system. It preserves the transcritical bifurcation at $R_0 = 1$, where the disease-free equilibrium loses stability and a unique endemic equilibrium emerges. The basic reproduction number in the reduced form is $R_0 = \beta / (1+q)$.
• Numerical Confirmation: An independent numerical experiment confirms the equivalence group to approximately ten significant figures (maximum discrepancy of $3.9 \times 10^{-11}$), demonstrating that distinct dimensional models collapse onto a single reduced orbit when mapped through the derived invariants.
• Sensitivity Analysis: The paper derives exact analytic sensitivity indices for $R_0$, showing that transmission ($\gamma$) and recruitment ($\Sigma$) have unit elasticity, while natural mortality ($\alpha$) acts as the strongest damping factor.

Significance and Claims
The paper claims its primary contribution is methodological rather than the discovery of a new epidemiological mechanism. Its significance lies in:

1. Correcting a Common Error: It provides a clear, rigorous distinction between point symmetries (dynamical invariants) and equivalence transformations (structural invariants of a model family), preventing the misinterpretation of scaling laws as conserved quantities of a specific epidemic trajectory.
2. Providing a Template: It offers a reproducible framework for analyzing richer HIV models (including those with treatment, age structure, or spatial spread). The author asserts that the discipline of testing determining equations directly and separating point symmetries from equivalences is essential for future extensions.
3. Efficiency in Calibration: By reducing five parameters to three dimensionless groups, the analysis shrinks the parameter space that must be explored for calibration or control studies, without losing the ability to map results back to the original dimensional system.
4. Robustness to Interventions: The paper notes that while time-dependent interventions (e.g., seasonal forcing) break the time-translation symmetry, the equivalence structure remains applicable if coefficient functions transform consistently, allowing the reduction to survive in a parameter-function form.

The author emphasizes that no nonconstant first integral is assumed or found; the reduction relies entirely on the solvable structure of the equivalence algebra. The work serves as a foundational correction and tool for the mathematical epidemiology community.