Modeling Typhoid Fever Dynamics: Stability Analysis and Periodic Solutions in Epidemic Model with Partial Susceptibility

Mathematical models play a crucial role in predicting disease dynamics and estimating key quantities. Non-autonomous models offer the advantage of capturing temporal variations and changes in the system. In this study, we analyzed the transmission of typhoid fever in a population using a compartmental model that accounted for dynamic changes occurring periodically in the environment. First, we determined the basic reproduction number, R0, for the periodic model and derived the time-average reproduction rate, [R0], for the non-autonomous model as well as the basic reproduction number, R0A, for the autonomous model. We conducted an analysis to examine the global stability of the disease-free solution and endemic periodic solutions. Our findings demonstrated that when R0<1, the disease-free solution was globally asymptotically stable, indicating the extinction of typhoid fever. Conversely, when R0>1, the disease became endemic in the population, confirming the existence of positive periodic solutions. We also presented numerical simulations that supported these theoretical results. Furthermore, we conducted a sensitivity analysis of R0A, [R0] and the infected compartments, aiming to assess the impact of model parameters on these quantities. Our results showed that the human-to-human infection rate has a significant impact on the reproduction number, while the environment-to-human infection rate and the bacteria excretion rate affect long-cycle infections. Moreover, we examined the effects of parameter modifications and how they impact the implementing of efficient control strategies to combat TyF. Although our model is limited by the lack of precise parameter values, the qualitative results remain consistent even with alternative parameter settings.