Computational and Mathematical Methods in Medicine

Volume 2019, Article ID 2954041, 10 pages

https://doi.org/10.1155/2019/2954041

## Existence of a Conserved Quantity and Stability of *In Vitro* Virus Infection Dynamics Models with Absorption Effect

^{1}Facultad de Matemáticas, Universidad Autónoma de Guerrero Chilpancingo, Av. Lázaro Cárdenas S/N, Cd. Universitaria, 39087 Chilpancingo, Guerrero, Mexico^{2}Escuela Superior de Medicina, Instituto Politécnico Nacional, Plan de San Luis y Díaz Mirón S/N, Col. Casco de Santo Tomas, Del. Miguel Hidalgo, 11340 Ciudad de México, Mexico

Correspondence should be addressed to Cruz Vargas-De-León; xm.moc.oohay@28zurcnoel

Received 29 October 2018; Revised 28 December 2018; Accepted 20 January 2019; Published 3 March 2019

Academic Editor: Konstantin Blyuss

Copyright © 2019 Celia Martínez-Lázaro et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

In mathematical biology models, two of the best-known conserved quantities correspond to the Lotka–Volterra predator-prey and Kermack–McKendrick SIR models. The conserved quantities are also known as first integrals associated with the systems of differential equations that describe the process or phenomenon of interest; particularly, in physics, there are several examples, namely, the Hamiltonian systems, among others. In biology, a conserved quantity can be used as a null hypothesis, compared to values from a real population, to describe statistically significant deviations from the first integral. In ecology, the Lotka–Volterra predator-prey system is used as a null model, and if the biomass is not conserved, it leads to look for factors that are affecting the habitat of the species. In epidemiology, the classical SIR epidemic model without vital dynamics has a first integral that is used to calculate the maximum number of infected individuals () reached in the epidemic. In other words, the conserved quantity is used to predict when the number of infected individuals will begin to decline. It is important to find conserved quantities in mathematical models focused on biology because of their relevance.

Interest has recently increased in the development of methods to find first integrals of biological systems. Nucci and Sanchini [1] applied Lie group analysis to a two-dimensional population model and found that it can be integrated by quadrature under some conditions on the parameters. To derive the first integrals of a two-dimensional epidemic model with nonlinear relapse, Naz et al. [2] use the partial Lagrangian approach developed by Kara et al., in [3, 4]. Also, Naz [5] uses the partial Hamiltonian approach to derive another first integral of the classical Lotka–Volterra model that had not been reported in the literature. Pugliese and collaborators [6] find conserved equations for the epidemic multidimensional systems; among these, a model with heterosexual transmission and another model of Zika virus sexual transmission, which under certain conditions, allow to calculate the basic reproductive number and determine the stability of the disease-free equilibrium state.

At the cellular level, a conserved quantity occurring in an *in vitro* experiment will indicate that a biological material (amino acids, nucleic acids, and even the materials used by a cell to preserve its viability and function) is constant. Kakizoe and collaborators in [7] reported the existence of a conserved quantity in a basic model of viral infection in a cell culture.

In the mathematical theory of *in vivo* and *in vitro* viral infections, the basic models are concentrated on population dynamics of target cells and the interaction between virus particles and target cells, but several of the *in vivo* or *in vitro* models ignore the absorption effect of the virus particle or viral genome. The viral dynamics that considers the absorption effect has been studied by the following authors. Perelson et al. studied a four-dimensional system for the interaction of HIV with CD4^{+} T cells and incorporated the absorption effect when a virus infects an uninfected cell [8]. Berreta and Kuang studied the dynamics of marine bacteriophages. They determined two equilibria, and obtained stability conditions for the infection-free and infected equilibrium states, and derived conditions for Hopf bifurcation to occur [9]. Smith and De Leenheer proposed a family of viral infection models with a generalized function of cell dynamics and effect of absorption. They determined the basic reproductive number and proved the global stability of infection equilibrium by using the theory of competitive systems [10]. Iggidr and collaborators [11] proposed a malaria intrahost model with *k* classes of age for the parasitized erythrocytes and *n* strains for the parasite. They calculated the basic reproductive number and proved the global stability of equilibria by using the Lyapunov function method. 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. Furthermore, they also compared between the models with or without absorption.

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. Our second goal is to estimate the parameters and selection of one of the four models that can describe the experimental data of virus infection in cell cultures. Our last goal is to test the law of conservation experimentally.

The organization of this paper is as follows. In Section 2, we present the extension to two *in vitro* models, which consider the effect of loss of a viral particle, which is called the absorption effect when it infects uninfected cells. In Section 3, we found a first integral of each system, and we discuss and establish the conditions for the stability of each model by using the technology of Lyapunov functions. In Section 4, we estimate the parameters of models with and without absorption effect by least-squares minimization between numerical solution of the system and experimental data, respectively. For the selection between the models, we use the Akaike information criterion () [13], which tells us which model is relatively better. In Section 5, we demonstrate the use of experimental data in order to test the conservation law. In Section 6, we use a biologically realistic range of parameter values to present some numerical simulations. Lastly, in Section 7, we provide a few concluding remarks.

#### 2. An *In Vitro* Viral Infection Model with Absorption Effect

The modelis a modified version of the basic virus dynamics model [14, 15]; this system was proposed in 2008 by Beauchemin et al. [12]. and are the numbers of target (susceptible) and infected (virus-producing) cells per ml of medium, respectively, and is the viral load per ml of the medium.

The parameters are such that *β* is the rate at which virions infect the target cells, *δ* is the rate of death of infected cells, *p* is the production rate of infectious virions by infected cells, and *c* is the virion clearance rate.

Recently, the authors [16, 17] reduced system (1) to a two-dimensional SIR-type model under the assumption that the viral dynamics is much faster than the infected cell dynamics and that a quasi-stationary state at which is attained very quickly. In particular, the authors applied the classical results derived from the SIR epidemic model to the context of the dynamic viral infections, and they calculated the area under the viral load curve, initial viral growth rate, peak viral load, and time to peak viral load, among other quantities.

The process of absorption of a viral particle or its genome is one of the first steps of a viral infection. This process has been modeled by a lot of authors [9–11, 18, 19] and ignored by many others. We introduce in system (1) the absorption effect, which is modeled by incorporating a bilinear term in the third equation, where . The parameter *n* is the average number of viral particles (or their viral genome) that enters a cell (see Figure 1). Consequently, the extended system is given by the following systems of differential equations: