Fractional-order analysis of temperature- and rainfall-dependent mathematical model for malaria transmission dynamics - Gizaw, Ademe Kebede; Deressa, Chernet Tuge (Frontiers in Applied Mathematics and Statistics, 2024)
This study proposes a temperature and rainfall fractional-order compartmental model for malaria transmission dynamics using the AB fractional operators in the Caputo sense. For the model to be stable, have a steady state, and possess biological significance, its solutions are proven to be positive. The existence and uniqueness of the model's solutions are established using the Banach fixed-point theorem. The next-generation matrix method is employed to determine the basic reproduction number of the model, whose value depends on the threshold quantity, denoted by ℳ. When ℳ ≤ 1, the basic number of the proposed fractional order model is zero, while it is shown as an expression when ℳ > 1. The model system's equilibria (both disease-free and endemic) are identified, and lemma and theorems are developed to prove their stability. Furthermore, different temperature ranges and rainfall data were used and validated with the existing literature. Numerical simulations using the Toufik-Atangana schemes with different fractional-order alpha values revealed that as the value of the fractional-order alpha approaches 1, the value of the classical model resembles that of the fractional-order model. The numerical results are promising and are certain to be helpful for future research related to fractional order models.