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In this work, we formulate a mathematical model with a non-integer order derivative to investigate typhoid fever transmission dynamics. To combat the spread of this disease in the human community, control measures like vaccination are included in the proposed model. We calculate the epidemiological threshold called the control reproduction number, R c , and perform the asymptotic stability of the typhoid-free equilibrium point. We prove that the typhoid-free equilibrium for both integer and non-integer models is locally and globally asymptotically stable whenever R c is less than one. We also prove that both models admit only one endemic equilibrium point which is globally asymptotically stable whenever R c > 1 and no endemic equilibrium point otherwise. This means that the backward bifurcation phenomenon does not occur. In absence of vaccination, R c is equal to the basic reproduction number R 0. We found out that R c < R 0 which means that vaccination can permit to decrease of the spread of typhoid fever in the human community. Using fixed point theory, we perform existence and uniqueness analysis of solutions of the fractional model. We use the Adams-Bashforth-Moulton method to construct a numerical scheme of the fractional model. We prove the stability of the proposed numerical scheme. To calibrate our model, we estimate model parameters on clinical data of Mbandjock District Hospital in the Centre Region of Cameroon, using the Non-linear Least-Square method. This permits us to find R c = 1.3722 , which means that we are in an endemic state (since R c > 1), and then to predict new cases of typhoid fever per month at Mbandjock in the next new year. To determine model parameters that are responsible for disease spread in the human community, we perform sensitivity analysis (SA). This analysis shows that the vaccination rate, the human-bacteria contact rate, as well as the recovery rate, are the most important parameters in the disease spread. To validate our analytical results, and to see the impact of some control measures in the spread of typhoid fever in the human community, as well as the impact of the fractional-order on typhoid transmission dynamics, we perform several numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we derive and analyze a model for the control of typhoid fever which takes into account an imperfect vaccine combined with protection, environment sanitation, and treatment as control mechanisms. The analysis of the autonomous model passes through the computation of the control reproduction number R c , the proof of the local and global stability of the disease-free equilibrium whenever R c is less than one using Lyapunov's theory. When R c is greater than one, we prove the local asymptotic stability of the unique endemic equilibrium through the Center Manifold Theory and we find that the model exhibits a forward bifurcation. Using clinical data from Mbandjock, a town in the Centre Region of Cameroon, we calibrate the model by estimating model parameters. We find that the control reproduction number is approximatively equal to 2.4750, which means that we are in an endemic state (R c > 1). We also performed a sensitivity analysis by calculating the Partial Rank Correlation Coefficient (PRCC) of R c and of infected compartments classes. Then, we extend the model by reformulating it as an optimal control problem, with the use of three time-dependent controls, namely vaccination, individual protection/environment sanitation, and treatment. Optimal control theory is used to analyze our optimal control model. Numerical simulations and efficiency analysis are performed to show the impact of each control strategy on the decrease of the disease burden. [ABSTRACT FROM AUTHOR]