A memory-driven pneumonia dynamics model validated against Ethiopian mortality data: a fractional-order differential equation framework

Every year, pneumonia kills millions across the globe, disproportionately claiming the lives of children under five and immunocompromised adults in low-resource settings, yet the mathematical tools deployed to understand and contain it remain fundamentally inadequate. Classical integer-order epidemic models assume that disease dynamics depend solely on the present state of a system, a biologically untenable assumption for pneumonia, where cumulative immune history, pathogen exposure memory, and treatment delays collectively govern transmission trajectories. This study deviates from that tradition by introducing a well-stablished Caputo fractional-order SEIHR model including hereditary effects and epidemiological memory as part of the mathematical framework governing the disease progression. Biological validity is guaranteed through comprehensive well-posedness analysis, with solution positivity and boundedness established via fractional comparison theorems and Mittag -Leffler function theory within a positively invariant feasible region. In this work, the basic reproduction number \(\:{R}_{0}\) is calculated by the next-generation matrix method and, by employing the fractional linearization theory and Lyapunov functional construction, both the local and global asymptotic stability of the disease-free equilibrium and the endemic equilibrium are formally proved, which are the first complete stability proofs ever obtained for a fractional-order pneumonia mode. A landmark theoretical finding reveals that the fractional-order parameter α operates as a precise memory-strength regulator, whereby lower values generate slower convergence trajectories and prolonged infectious periods that far more faithfully capture the biological reality of pneumonia than any integer-order model achieves. Sensitivity analysis demonstrates that parameters β and Λ should be targeted for intervention first since they are most crucial. On the other hand, by carrying out bifurcation analysis, we could observe that there is a threshold that divides the system between elimination and endemicity, namely \(\:{R}_{0}\) =1. The proposed model yields a good fitting to the WHO pneumonia mortality in Ethiopia for 2018–2020, with an \(\:{R}^{2}\) of 0.995 and maximum relative error below 0.55%. The conclusions drawn are obvious and decisive: memory effect is neither an artistic detail, nor a mathematically luxury, but an indispensable element of epidemic modeling.