Tempered fractional rabies dynamics with deep neural surrogate modeling: Analysis, computation, and prediction

Rabies is a fatal zoonotic disease for which domestic dogs constitute the main reservoir; its long and variable incubation period motivates transmission models that account for memory effects. This paper proposes a tempered fractional rabies model in which the classical time derivative is replaced by the left tempered Caputo derivative of order ξ ∈ (0, 1] and tempering parameter λ ≥ 0, so that the dynamics remain nonlocal while the influence of past states decays exponentially. We first establish the qualitative properties of the model: solutions remain nonnegative for nonnegative initial data, total populations are bounded on finite time intervals, and a unique continuous solution exists, which we prove by combining an exponential transformation with a contraction argument. We further derive the basic reproduction number R₀ by the next-generation-matrix approach and prove local asymptotic stability of the disease-free equilibrium whenever R₀<1, uniformly in the fractional order and the tempering parameter. A predictor-corrector product-integration scheme is then developed for the tempered system, and a mesh-refinement study confirms an empirical convergence order close to one. To accelerate repeated simulations, we train a deep neural surrogate that maps a short history window of log-transformed states to log-state increments, using a two-stage procedure consisting of one-step pretraining followed by differentiable rollout fine-tuning. Ablation experiments show that the logarithmic state representation and the increment-based targets are the decisive design choices, while rollout fine-tuning further improves worst-case robustness of free-running predictions. The selected surrogate attains high one-step accuracy (RMSE 0.4121, MAE 0.1303, R²=0.999998) and, in fully free-running closed-loop mode, reproduces unseen tempered test scenarios with mean R²=0.9990 (mean RMSE 3.32) on the benchmark cases and R²=0.9973±0.0031 over a random evaluation suite of 18 unseen tempered scenarios; accuracy degrades only in the near-untempered long-memory limit λ → 0, which we characterize explicitly. Parameter-matched LSTM and GRU baselines trained under the identical protocol do not outperform the proposed feed-forward surrogate, and in batched operation the surrogate is about two orders of magnitude faster than the numerical solver. The resulting framework provides a fast, accurate, and stable proxy of the tempered fractional rabies model that supports rapid parameter sweeps and forecasting-oriented scenario analysis.