Mathematical Analysis and Control of Typhoid Fever Dynamics Using a SEITRB Fractional Model

Abstract

The seitrb compartmental model was formulated to investigate the transmission dynamics and control of typhoid fever, incorporating the susceptible, exposed, infected, treatment, recovered, and environmental bacteria populations. The model was analyzed for well-posedness, establishing existence, uniqueness, positivity, and boundedness of solutions. Equilibrium states were examined under both disease-free and endemic conditions, with the basic reproduction number  derived as the threshold parameter. The analysis showed that typhoid infection dies out when  but persists when . Local and global stability analyses were established, while sensitivity analysis identified treatment rate, bacterial decay, and vaccination efficacy as the most influential parameters on . Numerical simulations, carried out using the Laplace Adomian Decomposition Method in conjunction with Caputo fractional derivatives, illustrated the impact of control measures. Findings revealed that optimal treatment effectiveness, sufficient treatment coverage, and improvements in sanitation act synergistically to minimize infection and reinfection risks. Over a multi-year horizon, these combined interventions significantly reduced disease prevalence in endemic populations. This study demonstrates that integrating mathematical analysis with practical interventions provides a robust model for understanding typhoid dynamics. By identifying the parameters most critical to disease reduction, the seitrb model offers evidence-based guidance for health practitioners in designing localized and sustainable typhoid control strategies. Overall, the model highlights the transformative role of coordinated vaccination, treatment, and sanitation in achieving effective prevention and long-term community health improvement.