Abstract
The estimation of parameters in biomathematical models is useful to characterize quantitatively the dynamics of biological processes. In this paper, we consider some systems of ordinary differential equations (ODEs) modelling the viral dynamics in a cell culture. These models incorporate the loss of viral particles due to the absorption into target cells. We estimated the parameters of models by least-squares minimization between numerical solution of the system and experimental data of cell cultures. We derived a first integral or conserved quantity, and we proved the use of experimental data in order to test the conservation law. The systems have nonhyperbolic equilibrium points, and the conditions for their stability are obtained by using a Lyapunov function. We complemented these theoretical results with some numerical simulations.
Highlights
In mathematical biology models, two of the best-known conserved quantities correspond to the Lotka–Volterra predator-prey and Kermack–McKendrick SIR models. e conserved quantities are known as first integrals associated with the systems of differential equations that describe the process or phenomenon of interest; in physics, there are several examples, namely, the Hamiltonian systems, among others
We introduce in system (1) the absorption effect, which is modeled by incorporating a bilinear term nβx(t)v(t) in the third equation, where 0 < n < p/δ. e parameter n is the average number of viral particles that enters a cell
We studied two models of an in vitro viral infection that incorporated the absorption effect of viral particles
Summary
Two of the best-known conserved quantities correspond to the Lotka–Volterra predator-prey and Kermack–McKendrick SIR models. e conserved quantities are known as first integrals associated with the systems of differential equations that describe the process or phenomenon of interest; in physics, there are several examples, namely, the Hamiltonian systems, among others. Kakizoe and collaborators in [7] reported the existence of a conserved quantity in a basic model of viral infection in a cell culture. Beauchemin and collaborators [12] developed two influenza viral infection models in vitro, where they estimated the parameters of models based on in vitro virological data under various constant concentrations of amantadine. One of such models includes the absorption effect and a discrete intracellular delay. Our first goal in this paper is to prove the existence a conserved quantity of in vitro virus infection dynamics models with absorption effect, as well as demonstrate the stability of the nonhyperbolic equilibrium points. When n 0, it corresponds to systems (2) and (4) without absorption effect
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