Nonlinear effects in the dynamics of HIV-1 infection predicted by mathematical model with multiple delays

Abstract

We formulate a novel mathematical model to describe the development of acute HIV-1 infection and the antiviral immune response. The model is formulated using a system of delay differential-and integral equations. The model is applied to study the possibility of eradication of HIV infection during the primary acute phase of the disease. To this end a combination of analytical examination and computational treatment is used. The model belongs to the Wazewski differential equation systems with delay. The conditions of asymptotic stability of a trivial steady state solution are expressed in terms of the algebraic Sevastyanov-Kotelyanskii criterion. The results of the computational experiments with the model calibrated according to the available estimates of parameters suggest that a complete elimination of HIV-1 infection after acute phase of infection is feasible.

Stochastic Modeling in Immunology Based On a Stage-Dependent Framework with Non-Markov Constraints for Individual Cell and Pathogen Dynamics

“…To build the model, we used publications listed in the references of [11] (articles No. 7,8,11,12,26,28,29,30), monograph [24] and articles [25], [26].…”

“…Next, in parentheses, the dimension of the model parameters is indicated. Based on the [7], [8], [10], [11], we assume that T * 0 = 5•10 8 , A * = 2•10 6 , µ I 0 = 2.5 (day −1 ), β I 1 = 0.8 (day −1 ), ω I 2 = 0.75 (day), η V = 150 (day −1 ), µ V = 3.5 (day −1 ), γ T 0 ,V = 1.2 • 10 −9 (day −1 ), γ T 0 ,I 4 = 2.5 • 10 −7 (day −1 ). Additionally, we assume that p V = 0.7, ω I 3 = 1.25 (day), the random variable ω I 4 is given by the expression ω I 4 = 0.75 + 0.6 ξ 2.5 , where ξ is uniformly distributed over the interval [0; 1], [a; b] = [0.75; 1.35] (day), ω * I 4 = 0.921 (day).…”

“…Consider the dynamics of (3.1) on the interval [0; T mod ] = [0; 10] days for model parameters that include parameters from the reference set and three additional sets: The results of numerical simulation are presented in Figures 5,6,7 and in Tables 2, 3, 4. Figures 5, 6, 7 show ten typical realizations of the auxiliary variable log 10 (X S (t) + 1) for the parameter sets (3.55), (3.56), (3.57), respectively, and V 0 = 10, 50, 100, where X S (t) -total number of all X(t) components:…”

Construction of exponentially decreasing estimates of solutions to a Cauchy problem for some nonlinear systems of delay differential equations

“…The case R 0 < 1 provides exponential decreasing of a part of components of solution of the models and allows to evaluate the time interval until the decreasing of these components from the initial level to some favorable or unfavorable one. The ways to study the considered systems of dierential equations are transferred to equations that are close in structure to the former ones, that emerge in models of epidemiology and immunology, see, for example, [11], [12].…”

Mathematical Modeling of the Initial Period of Spread of HIV-1 Infection in the Lymphatic Node

“…Обзор основных подходов к разработке математических моделей динамики ВИЧ-1 инфекции в организме человека и методов анализа реальных данных представлен в [1], [2]. Большинство из известных математических моделей динамики ВИЧ-1 инфекции в организме человека строится на основе систем дифференциальных уравнений, включая уравнения с запаздыванием [3]- [6].…”