Projections and fractional dynamics of the typhoid fever: A case study of Mbandjock in the Centre Region of Cameroon

Abstract 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.

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