# American Institute of Mathematical Sciences

September  2020, 13(9): 2365-2384. doi: 10.3934/dcdss.2020141

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

 1 Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Omsk Branch, Omsk, 644043, Russian Federation 2 Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, 119333, Russian Federation

* Corresponding author: Nikolay Pertsev

Received  January 2019 Revised  April 2019 Published  November 2019

Fund Project: The first and the last authors are supported by Russian Science Foundation grant 18-11-00171

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.

Citation: Nikolay Pertsev, Konstantin Loginov, Gennady Bocharov. Nonlinear effects in the dynamics of HIV-1 infection predicted by mathematical model with multiple delays. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2365-2384. doi: 10.3934/dcdss.2020141
##### References:

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##### References:
Schematic representation of the model of acute HIV-1 infection
Computational Experiment 1. Dynamics of $y_1(t)$, $y_2(t)$, $y_3(t)$ for $R_0 = 0.9224$, $V^0 = 10^4$
Computational Experiment 2. Dynamics of $y_1(t)$, $y_2(t)$, $y_3(t)$ for $R_0 = 0.9073$, $V^0 = 10^4$
Computational Experiment 3. Dynamics of $y_1(t)$, $y_2(t)$, $y_3(t)$ for $R_0 = 2.2875$. (a) $V^0 = 10^4$, (b) $V^0 = 10^6$
Computational Experiment 4. Dynamics of $y_1(t)$, $y_2(t)$, $y_3(t)$ for $R_0 = 2.2875$. (a) $V^0 = 10^4$, (b) $V^0 = 10^6$
Computational Experiment 5. Dynamics of $y_1(t)$, $y_2(t)$, $y_3(t)$ for $R_0 = 13.2135$. (a) $V^0 = 10^2$, (b) $V^0 = 10^6$
Computational Experiment 6. Dynamics of $y_1(t)$, $y_2(t)$, $y_3(t)$ for $R_0 = 3.3789$. (a) $V^0 = 10^4$, (b) $V^0 = 10^6$
Time post infection needed for a complete elimination of HIV-1 infection
 $V_0$ Computational experiment number, the value $\tau_0$ (day) 1 2 3 4 5 6 $10$ 1 1 $-$ $-$ $-$ 46 $10^2$ 51 46 $-$ 51 $-$ 44 $10^4$ 60 115 $-$ 37 $-$ 42 $10^6$ 67 132 76 78 102 53 $10^7$ 72 140 96 94 $-$ 57
 $V_0$ Computational experiment number, the value $\tau_0$ (day) 1 2 3 4 5 6 $10$ 1 1 $-$ $-$ $-$ 46 $10^2$ 51 46 $-$ 51 $-$ 44 $10^4$ 60 115 $-$ 37 $-$ 42 $10^6$ 67 132 76 78 102 53 $10^7$ 72 140 96 94 $-$ 57
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