Stochastic compartmental model of HIV-1 infection

Konstantin Loginov,Nikolai Pertsev

doi:10.1051/itmconf/20203102003

Abstract

Stochastic model of the dynamics of HIV-1 infection describing the interaction of target cells and viral particles in the lymphatic nodes and their movement between the lymphatic nodes is constructed. The lymphatic system is represented as a graph, vertices of which are the lymphatic nodes and edges are the lymphatic vessels. The novelty of the model consists in the description of populations of cells and viral particles in terms of a multidimensional birth and death process with the random point-distributions. The random pointdistributions describe the duration of the transition of cells and viral particles between the lymph nodes and the duration of the stages of their development. The durations of transitions of viral particles and cells between the lymphatic nodes are not random and based on the rate of lymph flow. The durations of the developmental stages of infected target cells are assume to be constant. The graph theory for the formalization and compact representation of the model is used. An algorithm for modelling the dynamics of the studied populations is constructed basing on the Monte-Carlo method. The results of computational experiments for a system consisting of five lymphatic nodes are presented.

Highlights

An important direction in mathematical modeling in immunology is the development of mathematical models of the dynamics of HIV-1 infection in the human lymphatic system [1, 2]
A stochastic model with integer variables describing the dynamics of HIV-1 infection in a single lymph node in the form of random process with particle interaction is proposed at [6]
Following [2, 4, 6, 10], we introduce the set of assumptions that determine dynamics of HIV-1 infection:

Summary

Introduction

An important direction in mathematical modeling in immunology is the development of mathematical models of the dynamics of HIV-1 infection in the human lymphatic system [1, 2]. Xii(t) = Tii(t), Cii(t), Iii(t), Uii(t), Vii(t), Kii(t), Eii(t) , Xi j(t) = Ti j(t), Ci j(t), Ii j(t), Ui j(t), Vi j(t), Ki j(t), Ei j(t) vectors containing the number of system components at the vertex Ni and at the edge Ni j respectively (for each fixed t the components of the vectors are non-negative integer random variables) Note that in this model Ci j(t) = Ui j(t) = Ki j(t) = 0 for each t 0, i j. Assume t0 = 0; Xi j(t0) 0 are initial numbers of system components at vertices and edges of graph G; Ω(iiC)(t0), Ω(iiU)(t0), Ω(iiK)(t0), Ω(iTj )(t0), Ω(iIj)(t0), Ω(iEj )(t0), Ω(iVj )(t0) are fixed random point-distributions, 1 i, j n, i j. Beginning of the transition of cells and viral particles from vertex Ni to vertex N j along the edge Ni j

Special case of the model

Computational experiments

Conclusions