Mathematical modeling of the Coronavirus (Covid-19) transmission dynamics using classical and fractional derivatives

Abstract

The goal of this work is the formulation and analysis of a Covid-19 transmission dynamics model which takes into account two doses of the vaccination process, confinement, and treatment with limited resources, using both integer and fractional derivatives in the Caputo sense. After the model formulation with classical derivative, we start by establishing the positivity, boundedness, existence, and uniqueness of solutions. Then, we compute the control reproduction number Rc and perform the local and global asymptotic stability of the disease-free equilibrium whenever R_c < 1. We also prove the existence of at least one endemic equilibrium point whenever R_c > 1. Using real data from Germany, we calibrate our models by performing parameter estimations. We find that the control reproduction number is approximately equal to 1.90, which shows that we are in an endemic state. We also perform global sensitivity analysis by computing partial rank correlation (PRCC) coefficients between R_c (respectively infected states) and each model parameter. After that, we formulate the corresponding fractional model in the Caputo sense, proving positivity, boundedness, existence, and uniqueness of solutions. We also compute the control reproduction number of the fractional model, which depends on the fractional order ϕ. We prove the local and global asymptotic stability of the disease-free equilibrium whenever the control reproduction number is less than one, as well as the existence of an endemic equilibrium point whenever the control reproduction number is greater than one. To validate our theoretical analysis of both models, and compare the two types of derivatives, we perform several numerical simulations. We find that for long-term forecasting, the fractional model, with a fractional order ϕ ≤ 0.87 is better than the model with integer derivative.