This paper analyzes a novel use of the shifted Fibonacci polynomials (SFPs) to treat the time-fractional Burgers equation (TFBE). We first develop the fundamental formulas of these polynomials, which include their power series representation and the inversion formula. We establish other new formulas for the SFPs, including integer and fractional derivatives, in order to design the collocation approach for treating the TFBE. These derivative formulas serve as tools that aid in constructing the operational metrics for the integer and fractional derivatives of the SFPs. We use these matrices to transform the problem and its underlying conditions into a system of nonlinear equations that can be treated numerically. An error analysis is analyzed in detail. We also present three illustrative numerical examples and comparisons to test our proposed algorithm. These results showed that the proposed algorithm is advantageous since highly accurate approximate solutions can be obtained by choosing a few terms of retained modes of SFPs.

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, ${\mathcal{R}}_0$, for the periodic model and derived the time-average reproduction rate, $\left[{\mathcal{R}}_0\right]$, for the non-autonomous model as well as the basic reproduction number, ${\mathcal{R}}_0^A$, 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 ${\mathcal{R}}_0<1$, the disease-free solution was globally asymptotically stable, indicating the extinction of typhoid fever. Conversely, when ${\mathcal{R}}_0>1$, 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 ${\mathcal{R}}_0^A$, $\left[{\mathcal{R}}_0\right]$ 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.

This paper introduces two numerical algorithms for solving the fractional and classical Rosenau–Hyman equations. Certain shifted Pell–Lucas polynomials (SPLPs) are employed. New formulas for these polynomials are developed and used to analyze our proposed algorithms, together with the application of the typical collocation method. The established operational matrices for both integer and fractional derivatives are used to convert the problems, along with their governing conditions, into matrix systems that can be treated effectively. The convergence of the shifted Pell–Lucas expansion is thoroughly investigated by developing new estimates. The high accuracy and efficiency of the proposed numerical algorithms are tested through numerical experiments supported by comparisons with some other techniques in the literature.

We present a nonautonomous compartmental model that incorporates vaccination and accounts for the seasonal transmission of typhoid fever. The dynamics of the system are governed by the basic reproductive number ${\mathcal{R}}_0$. This demonstrates the global stability of the disease-free solution if ${\mathcal{R}}_0<1$. On the contrary, if ${\mathcal{R}}_0>1$, the disease persists and positive periodic solutions exist. Numerical simulations validate our theoretical findings. To accurately fit typhoid fever data in Taiwan from 2008 to 2023, we use the model and estimate its parameters using Latin hypercube sampling and least squares techniques. A sensitivity analysis reveals the significant influence of parameters such as infection rates on the reproduction number. Increasing vaccination coverage, despite challenges in developing countries, reduces typhoid cases. Accessible and highly effective vaccines play a critical role in suppressing the epidemic, outweighing concerns about the efficacy of the vaccine. Investigating possible parameter changes in Taiwan highlights the importance of monitoring and managing transmission rates to prevent recurring annual epidemics.