Memory-Dependent Dynamics of Trachoma with Real Data Analysis from Cameroon via Fractional Framework

Abstract

This study models trachoma transmission in Cameroon using a deterministic approach with integer and fractional-order derivatives, incorporating direct, fly-mediated, and environmental transmission routes. Fitting disease data from 1990–2019, the model forecasts trachoma prevalence until 2035. The research confirms the solution existence and uniqueness, calculates the basic reproduction number R0λ where λ∈(0,1] represents the fractional-order parameter, and analyzes equilibrium stability. A stable trachoma-free equilibrium exists when R0λ<1, while an endemic equilibrium is proven stable for R0λ>1 under specific conditions. Calibration of a fractional model with Cameroon data yielded an R0 of 1.169 (indicating endemicity) and identified an optimal fractional order of λ=0.98. By calculating the strength number, we found that another epidemic wave could occur in 50 years. Global sensitivity analysis highlighted key parameters affecting trachoma dynamics. A numerical scheme of the model based on the Adams–Bashforth–Moulton method is constructed and its stability demonstrated. It is then used to perform several numerical simulations, first to validate the theoretical results obtained, and then to compare the different models (statistical and deterministic). The conclusion is reached that the disease will persist in the population (R0>1), although the statistical model shows that it could disappear by 2030. This proves that, for trachoma dynamics in Cameroon, it is advisable to use a deterministic model.