In this work, we replaced the integer derivative with Caputo derivative to model the transmission dynamics of measles in an epidemic situation. We began by recalling some results on the local and global stability of the measles-free equilibrium point as well as the local stability of the endemic equilibrium point. We computed the basic reproduction number of the fractional model and found that is it equal to the one in the integer model when the fractional order ν = 1. We then performed a sensitivity analysis using the global method. Indeed, we computed the partial rank correlation coefficient (PRCC) between each model parameter and the basic reproduction number R₀ as well as each variable state. We then demonstrated that the fractional model admits a unique solution and that it is globally stable using the Ulam–Hyers stability criterion. Simulations using the Adams-type predictor–corrector iterative scheme were conducted to validate our theoretical results and to see the impact of the variation of the fractional order on the quantitative disease dynamics.

In this work, we formulate and mathematically study integer and fractional models of typhoid fever transmission dynamics. The models include vaccination as a control measure. After recalling some preliminary results for the integer model (determination of the epidemiological threshold denoted by ${\mathcal{R}}_c$, asymptotic stability of the equilibrium point without disease whenever ${\mathcal{R}}_c<1$, the existence of an equilibrium point with disease whenever ${\mathcal{R}}_c>1$), we replace the integer derivative with the Caputo derivative. We perform a stability analysis of the disease-free equilibrium and prove the existence and uniqueness of the solution of the fractional model using fixed point theory. We construct the numerical scheme and prove its stability. Simulation results show that when the fractional-order $\eta$ decreases, the peak of infected humans is delayed. To reduce the proliferation of the disease, mass vaccination combined with environmental sanitation is recommended. We then extend the previous model by replacing the mass action incidences with standard incidences. We compute the corresponding epidemiological threshold denoted by ${\mathcal{R}}_{c\star }$ and ensure the uniform stability of the disease-free equilibrium, for both new models, when ${\mathcal{R}}_{c\star }<1$. A new calibration of the new model is conducted with real data of Mbandjock, Cameroon, to estimate ${\mathcal{R}}_{c\star }=1.4348$. We finally perform several numerical simulations that permit us to conclude that such diseases can possibly be tackled through vaccination combined with environmental sanitation.