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 Rc<1, 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.

Mathematical Models for Typhoid Disease Transmission: A Systematic Literature Review

“…In this selection process, we found that some articles did not contain the content that we were looking for. The references in [25][26][27][28] study typhoid disease using the fractional differential system to represent the transmission of the disease. Another article in [29] studies typhoid disease transmission by considering the spatial data factor in the model.…”

New Results Involving Riemann Zeta Function Using Its Distributional Representation

“…The use of non-integer operators has emerged recently in the different disciplines of engineering and the physical sciences [35][36][37][38][39][40]. For instance, the fractional kinetic equation is important to investigate the theory of gases, aerodynamics, and astrophysics [41][42][43][44][45][46][47][48].…”

Fractional order mathematical modeling of lumpy skin disease

“…Modelling of epidemic diseases is of utmost importance to understand the behaviour of the ailment across time and to devise appropriate safeguards for the same. Numerous epidemic models have been developed for various diseases, including dengue and chikungunya [1], typhoid [2], cholera [10], HIV/AIDS [11], Covid-19 [14], [15], [57], leptospirosis, H1N1, measles [17], and others. But to our surprise there is not enough research on transmission dynamics and LSD control using a compartmental modelling technique; by the time this study was completed, there had only been one work [46], to examine the effects of vaccination on LSD and the spread of the illness in Ethiopia.…”