ABSTRACT
In this work, we formulate and analyze a new mathematical model for COVID-19 epidemic with isolated class in fractional order. This model is described by a system of fractional-order differential equations model and includes five classes, namely, S (susceptible class), E (exposed class), I (infected class), Q (isolated class), and R (recovered class). Dynamics and numerical approximations for the proposed fractional-order model are studied. Firstly, positivity and boundedness of the model are established. Secondly, the basic reproduction number of the model is calculated by using the next generation matrix approach. Then, asymptotic stability of the model is investigated. Lastly, we apply the adaptive predictor-corrector algorithm and fourth-order Runge-Kutta (RK4) method to simulate the proposed model. Consequently, a set of numerical simulations are performed to support the validity of the theoretical results. The numerical simulations indicate that there is a good agreement between theoretical results and numerical ones.
ABSTRACT
Complex nature of the analytical solutions to 3-compartment pharmacokinetic models leads to the discrete approximation of the continuous differential equation been mostly used. In this paper, we applied nonstandard finite difference method to 3-compartment pharmacokinetic models. This method was introduced to compensate for the weaknesses of methods such as the standard finite difference methods, numerical instabilities being a prime example of such. Three-compartment pharmacokinetic model with 2 different routes of administration (IV bolus injection and IV bolus infusion) is considered for simulations. For the case when the system is homogeneous (models arising from IV bolus injection mode of administration), "exact" finite difference scheme is obtained for any step size while in the case of nonhomogeneous (models arising from IV bolus infusion route of administration), scheme that has the same qualitative behavior as the analytical solution for all step sizes is obtained. It was shown computationally that the nonstandard finite difference scheme for 3-compartment pharmacokinetic model with IV bolus mode of administration is exact while 3-compartment pharmacokinetic model with IV infusion is dynamically consistent with the continuous model for all step sizes.
