Impact of Four-Dose Vaccination on the Transmission Dynamics of a Fractional-Order Malaria Model in the Context of Sub-Saharan Africa
This study develops a comprehensive mathematical framework for analyzing the transmission dynamics of malaria, with particular emphasis on a four-dose vaccination strategy combined with treatment interventions as primary control mechanisms. The model stratifies the human population into nine compartments—susceptible, exposed, infected, under treatment, recovered, and four sequential vaccination classes (V 1 through V 4 )—together with susceptible and infected vector populations, all governed by a system of ordinary differential equations. Analysis of the diseasefree equilibrium demonstrates local asymptotic stability when the basic reproduction number R 0 < 1, while a unique endemic equilibrium exists and is stable when R 0 > 1. Sensitivity analysis identifies the mosquito biting rate, transmission probability, first-dose vaccination rate, and treatment rate as the most influential parameters governing malaria dynamics. Numerical simulations confirm that increasing vaccination coverage and completion rates across all four doses, in conjunction with optimized treatment, substantially reduces malaria incidence. To capture long-term memory effects inherent in malaria transmission—particularly non-exponential immunity waning and heterogeneous parasite development—a Caputo fractional-order version of the model is introduced, solved using the Adams–Bashforth predictor-corrector method. Comparison of the integer-order and fractional-order results reveals that memory effects slow the approach to equilibrium and may lead to higher endemic levels under identical intervention parameters, underscoring the value of the fractional extension. These findings emphasize that achieving high completion rates across all four vaccination doses, sustained treatment access, and integrated vector control are essential for meaningful progress toward malaria elimination in Sub- Saharan Africa.