Stochastic compartmental model of HIV-1 infection

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.


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]. Mostly, these models are divided into deterministic and stochastic one. Deterministic models take the form of the systems of delay differential equations [3,4]. The equations of such models describe concentration of target cells, infected cells, viral particles, etc. An example of a stochastic model is constructed at [5] in the form of multidimensional random birth and death process. 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]. The lymphatic system is a collection of lymph nodes interconnected by lymphatic vessels through which cells and viral particles can move. Modelling movements of cells in the lymphatic system is needed the construction of a stochastic compartment model with pipes or its deterministic analogue. One of the first deterministic models in this direction presented at [7]. An important feature of the model described at [7] is the presence of transport delays, which reflect the duration of particle movement through the pipes. Other models of the lymphatic system based on partial differential equations including the Navier-Stokes equations are presented, for example at [8]. Mathematical model of the dynamics of HIV-1 infection in the form of high-dimensional system of ordinary differential equations is presented at [9]. The model described at [9] contains detailed information regarding the structure of the human lymphatic system, however, there are no transfer delays in the model equations. It is important to note that the absence of transport delays leads to the possibility of instantaneous and simultaneous changes in the number of populations in various lymph nodes caused by processes occurring in only one specific lymph node. Stochastic model with one type of particles moving in a compartment system with pipes is considered at [10].
The purpose of this article is to build a stochastic compartmental model of the dynamics of HIV-1 infection which combines the approaches proposed at [6,10].

Assumptions on the dynamics of HIV-1 infection
In describing the dynamics of HIV-1 infection, we will use the following notation: • T denotes T-lymphocytes (target cells for mature viral particles); • C is infected cells, i.e. cells in preparation for the production of viral particles; • I is productively infected cells (cells producing viral particles); • U denotes immature viral particles; • V corresponds to mature viral particles (or virions); • K is precursor cells of effector lymphocytes; • E is related to cytotoxic T-lymphocytes (effectors).
Moreover, let S denotes the cell source in bone marrow and lymphoid tissue generating the target cells T and cells K; D is all dead cells and viral particles.
We will describe the lymphatic system in the form of graph G. The notations are as follows: • G is oriented graph without loops with weighted edges; • n is number of graph vertices G; • N i denotes vertices of the graph G (lymph nodes), 1 i n; • N i j corresponds edges of the graph G (lymph vessels), 1 i, j n, i j; • Q = ||q i j || is matrix defining relationships between vertices of the graph G; constants 0 q i j 1 for all i = 1, ..., n satisfy the conditions: If q i j > 0, then vertices N i and N j interconnected by edge N i j but connection between N j and N i is absent. For q i j = 0 there is no connection between N i and N j but there may be a connection between N j and N i .
• ∆ i j (t) is positive and bounded above real-valued function that defines the weight of the edge N i j , t ∈ (−∞; ∞), q i j > 0. We assume that ∆ i j (t) + t is strictly increasing.
Example of the graph G is presented on the figure 1. Denote  ( t ) Figure 1. Example of the graph G with five vertices Following [2,4,6,10], we introduce the set of assumptions that determine dynamics of HIV-1 infection: • Inflow of target cells T to vertex N i with rate r (T ) ii > 0: S • Natural death of cells T , C, I, E and particles U, V: • Interaction of T -cells with virions V (cell infection) with rate γ (T,V) ii > 0 per one pair of (T, V): • Infection of cells T as a result of interaction with productively infected cells I with rate ii > 0 per one pair of (T, I): T ii + I ii • Production by cells I of immature viral particles of U with rate ν (U) ii > 0: I ii • Destruction of infected cells I by cytotoxic cells E with rate γ (I,E) ii > 0 per one pair of (I, E): • Cell death I due to the destructive process of production of viral particles with rate σ U ν (U) ii , σ U > 0: I ii • Formation of precursor cells K cytotoxic lymphocytes E with rate ν (K) ii > 0: I ii • The cell C which appeared at time t and did not die during the time interval (t, t + ω C ), ω C > 0, at the time t + ω C transforms into a productively infected cell I: • The immature virus particle U that arose at time t and did not die during the time interval • The cell K which arose at time t, at time t + ω K , ω K > 0, transforms into a collection of n E > 0 cytotoxic T-lymphocytes E: ii respectively: i j 0 is probability that the transition of cells and virions will occur from vertex ii > 0 is rate at which cells and virions begin to transition from vertex N i to vertex N j along edge N i j ; 1 i, j n, i j.
The assumptions made are shown in figure 2. Figure 2. Scheme of the model for a single lymph node N i Let us introduce a set of assumptions that determine the movement of cells T , I, E and virions V along the edges of the graph G: • The cells C, K and viral particles U don't move along the edges (lymphatic vessels) of the graph G.
• Interactions of cells and virions in edges of G is absent.
• Natural death of cells T i j , I i j , E i j and virions V i j : • The cells T i j , I i j , E i j and virions V i j that started transition from vertex N i to vertex N j along edge N i j at the moment t and didn't die during the transition, at the time The above assumptions have the following probabilistic interpretation. Fix the indexes 1 i, j n, i j. Assume, that h → +0.
vectors containing the number of system components at the vertex N i and at the edge N i j respectively (for each fixed t the components of the vectors are non-negative integer random variables). Note that in this model C i j (t) = U i j (t) = K i j (t) = 0 for each t 0, i j. To account for the stages of development of cells and viral particles, we introduce random point-distributions. Random point-distributions contain information about the moments of the appearance of maturing cells and particles, as well as the moments of their transformation into the corresponding cells and particles. For instance, random point-distribution, using to describe C ii cells, has the structure ii t, ii is the moment of the appearance of another infected cell C; τ (Ck) ii + ω C is the moment of transformation of an infected cell into a productively infected cell I (without taking into account death). If C ii (t) = 0 for some t, then Ω (C) ii (t) = ∅. For particles U ii and cells K ii random point-distributions are constructed similarly.
To describe a population of cells T i j , I i j , E i j and virions V i j we also use random pointdistributions reflecting the movement of particles from vertex N i to vertex N j , i j. For instance, random point-distribution, using to describe T i j cells, has the structure where τ (T ) i j,k is the moment of the beginning of the transition of a certain particle T i j from vertex N i to vertex N j along the edge N i j ; and is the moment of completion of the transition provided that the specified particle does not die during the transition. If T i j (t) = 0 for some t, then Ω (T ) i j (t) = ∅. For I i j , E i j and V i j random pointdistributions are constructed similarly.
Next, we show the state change of the studied random process at discrete moments of time. Assume t 0 = 0; X i j (t 0 ) 0 are initial numbers of system components at vertices and edges of graph G; ,

Random variable η (m) is the nearest to t m time when any of cells T i j , I i j , E i j or virions
ii +ω K , are the minimal (first) elements of point-distributions is random variable representing the duration of time until one of the events at N i : • inflow of cell T from S to the vertex N i ; • death of cells or viral particles at the vertex N i ; • interaction of cells and viral particles at the vertex N i ; • beginning of the transition of cells and viral particles from vertex N i to vertex N j along the edge N i j .

Special case of the model
Without taking into account the transfer of cells and virions, the stochastic model includes the trivial special case.
Using the postulates of the random birth and death process, we can derive the differential equation for the mathematical expectation T (i) (t) = E T ii (t): The other components of vector E X ii (t) are identically zero. One can see, that Following [6], we introduce the quotient basic reproductive number for a single lymph node N i : .

Computational experiments
The purpose of computational experiments is to demonstrate the dynamics of mathematical expectations of variables depending on the variation of values of some model parameters.
Number of vertices n = 5, structure of the graph G is presented in figure 1. The values of the model parameters were chosen so that R (0) i < 1 for all vertices or R (0) i > 1 for at least one of the vertices. The simulation was carried out using the Monte-Carlo method on the time interval [0; W]. The number of independent realizations of the studied random process is N = 1000. Components of mathematical expectation E X ii (t) were estimated by standard formulas of mathematical statistics. We used pointX ii (t) and intervalX ii (t) ± 1.96 σ X ii (t) estimates at a confidence level P = 0.95. For each of two experiments the maximum value of σ X ii (t) for the components E X ii (t) does not exceed 1.62. For computational experiments we used a reference set of parameters, the numerical values of which were selected from the articles [2-4, 8, 11, 12]: The initial conditions for all experiments:    In the figures 9, 10 we can see that number of infected cells I and virions V after some fluctuations go to some stationary levels. The presence of oscillations is explained by cosines in representation of ∆ (I) i j (t), ∆ (V) i j (t) functions.

Conclusions
A continuously-discrete compartmental stochastic model of the dynamics of HIV-1 infection has been developed taking into account the interaction of cells and viral particles in a separate lymph node, the maturation of cells and viral particles, as well as the movements of cells and viral particles through the human lymphatic system. An important feature of the developed continuously-discrete stochastic model is the parameterization of the duration of the movement of particles between the compartments using the specified time functions.
Further development of the model may be associated with a complication of the processes occurring in the vessels connecting the lymph nodes. In particular, infection of target cells and the production of viral particles by infected cells in the vessels will be considered.