Fractional Dynamics of Typhoid Fever Transmission Models with Mass Vaccination Perspectives

Abstract

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 Rc, asymptotic stability of the equilibrium point without disease whenever Rc1, the existence of an equilibrium point with disease whenever Rc>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 η 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 Rc⋆ and ensure the uniform stability of the disease-free equilibrium, for both new models, when Rc⋆1. A new calibration of the new model is conducted with real data of Mbandjock, Cameroon, to estimate Rc⋆=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.

Keywords

• typhoid fever disease
• vaccination
• model calibration
• Caputo derivative
• asymptotic stability
• fixed point theory