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     {\huge СИБИРСКИЕ \ ЭЛЕКТРОННЫЕ} \\[2mm]
     {\huge МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ} \\[2mm]
     {\large Siberian Electronic Mathematical Reports} \\[1mm]
     {\LARGE\tt{http://semr.math.nsc.ru}}\\[0.5mm]
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     {\flushleft\it Том 16, стр. 144--144 (2019) \hspace{65mm}{\rm\small УДК \udcs}}
     \newline
     	{\rm\small DOI~10.33048/semi.2019.16.xxx}\hphantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}{\rm\small MSC\ \ \mscs }
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\title[Multitype Branching Processes on a Graph]{Critical Multitype Branching Processes on a Graph and the Model of the HIV Infection Development}
\author{{V.A. Topchii, N.V. Pertsev}}%
\address{Valentin Alekseevich Topchii  % Контактные данные всех авторов и место работы указываются только на английском языке
\newline\hphantom{iii} Sobolev Institute of Mathematics,
\newline\hphantom{iii} pr. Koptyuga, 4,
\newline\hphantom{iii} 630090, Novosibirsk, Russia}%
\email{topchij@gmail.com}%
\address{Nikolai Viktorovich Pertsev  % Контактные данные всех авторов и место работы указываются только на английском языке
\newline\hphantom{iii} Sobolev Institute of Mathematics,
\newline\hphantom{iii} pr. Koptyuga, 4,
\newline\hphantom{iii} 630090, Novosibirsk, Russia}%
\email{homlab@ya.ru}

\thanks{\sc Topchii, V.A., Pertsev,  N.V.,
Critical Multitype Branching Processes on a Graph and the Model of the HIV Infection Development}
\thanks{\copyright \ 2015 Топчий В.А., Перцев Н.В. }
\thanks{\rm The research was funded in accordance with the state task of the IM SB RAS, project FWNF-2022-0003}
\thanks{\it Поступила 1 января 2015 г., опубликована 31 декабря 2015 г.}%




\semrtop \vspace{1cm}
\maketitle {\small
\begin{quote}
\noindent{\sc Abstract. } We consider the Crump-Mode-Jagers 
branching process on an oriented graph in an application to 
modeling the development of HIV-1 infection in a human 
organism. For all particles of the same global type, located 
at each of the verteces or arcs of the graph, different types 
are assigned. Checking the criticality condition and searching 
for the eigenvectors of an offspring mean matrix in the 
critical case for the original process are reduced to an 
offspring mean matrix for some Galton-Watson process. The last 
has the types of particles corresponding only to the verteces 
of the graph.
\medskip

\noindent{\bf Keywords:} Crump-Mode-Jagers branching process on an oriented graph, Yaglom type limit theorem for critical branching process, eigenvectors for the mean matrix of high dimension, stochastic model of HIV-1 infection.
 \end{quote}
}

\section{Introduction}
Various mathematical models are used to study the development 
of HIV-1 infec\-tion in the human organism (see \cite{BCG12} -- 
\cite{{SVA16}}). If we consider the process of developing HIV-1 
infection within a relatively short period of time after 
infection of a healthy person (from several days to 3-4 weeks), 
then the main components of the process are mature viral 
particles (virions) and productively infected cells. Virions 
come into contact with target cells, for example, CD4+ 
T-lymphocytes, enter these cells and start the process of 
converting target cells into productively infected cells. 
Productively infected cells produce new viral particles, which 
in turn infect new target cells. Each virion and each 
productively infected cell can be located in the lymph nodes 
and move between them, and may also die due to the influence of 
various factors. The transitions of virions and productively 
infected cells between two connected lymph nodes are 
unidirectional, which is due to the specificity of lymph flow in 
the lymphatic vessels. Traditionally, there are two approaches 
to the study of the problem. This is its study using a system of 
differential equations with delay (see \cite{PPL19}) and the 
method of stochastic simulation (see \cite{19LPT} --
\cite{21BLPT}).

We will consider a stochastic model of the development of HIV-1 
infection in the human organism in the form of a branching process 
with several types of particles. Suppose that the particles of type
$A$ are mature viral particles (virions) and of type $B$ are 
productively infected cells. We assume that particles of global types $A$ and $B$ can be 
in any of the $n$ lymph nodes or move between some pairs of lymph 
nodes in one direction. The system of lymph nodes and connections 
between them is interpreted as a one-connected directed graph. The 
evolution of particles of global types $A$ and $B$ in each of the 
nodes and on the vessels (or in terms of graph theory: at each of 
the verteces and on the arcs) are individual, and it is natural 
to assign different types to them. Since the number of lymph nodes 
in the human organism is quite large ($n\approx100$), a very large 
number of particle types  are presented in the model. In addition, the particles ability of moving between nodes is limited 
due to the specific structure of the human lymphatic system. When 
conducting research, we assume that all transformations of  
particles occur independently of the behavior of other particles, 
but significantly depend on their location.


Note that in the general setting, it is possible to simulate the 
development of HIV-1 infection in the human organism within the 
framework of a process that is non-homogeneous in time. 
Non-homogeneity arises when describing the duration of particle 
transitions between lymph nodes using some functions that depend 
on time. Non-homogeneity leads to significant difficulties when 
using classical analytical methods and requires the use of 
simulation methods. The proposed analytical results are important 
for developing efficient algorithms and testing the correctness of 
computational procedures in a simulation model.

Later we will define the model of moving two type particles 
through the directed graph in terms of Crump-Mode-Jagers 
branching process (see \cite{Ja75} or below in section 
\ref{s3}). It will be named as $\mathcal{M}$ model. 
If the graph contains $n$ vertices, then there can be up to 
$n(n-1)/2$ unidirectional edges. On each of these objects, the 
branching process develops according to different laws, i.e. 
we have to define $n(n+1)/2$ two-dimensional branching 
processes with transitions along directed edges.
The result is a Crump-Mode-Jagers process with dimensions up 
to $(n(n+1))\times(n(n+1))$, which we will call the 
$\mathcal{M}$--CMJ process.


A detailed description of the $\mathcal{M}$--CMJ 
process associated with the lifetime of particles, as will be 
shown below, is not essential for our research. Our goal is to 
simplify the calculation of the eigenvalue (Perron root) of 
the offspring mean matrix (the mean numbers of all types of 
offspring born to all parent types until the death of the 
last) for $\mathcal{M}$--CMJ process.
Branching processes are traditionally divided into 
supercritical (the number of all types of particles grows 
exponentially), subcritical (the number of all types of 
particles dies exponentially quickly), and critical, where the 
averages for all types of particles are asymptotically 
constant and the process decays (degenerates) with 
probabilities inversely proportional to time (see \cite{Ha66}, 
\cite{Se71}, \cite{Ja75}). We will concentrate on the 
algorithm for calculating the eigenvectors of the offspring 
mean matrix only in critical case. 

The evolution of individual particles in the multi-type case are 
described in terms of point processes that determine the joint 
distribution of a finite number of pairs: the ages of generation of 
groups of offspring and the number of offspring in these groups 
(see Section \ref{s3}). The maximum age in the pairs is the 
lifetime of the particle. Age-related integer multivariate random 
variables describe the non-zero number of offspring in these groups 
with a distribution by type. The exception is the number of 
offspring at the time of the death of the particle, which may be 
zero. Point processes describing the evolution of particles are 
independent and equally distributed for coincident types of 
particles. The main characteristics of the evolution of individual 
particles are the distributions of their lifetime and the total 
number of generated offspring. We believe that all these 
particle-specific pairs of random variables have finite second 
moments. The explicit forms and asymptotic behavior of the first 
and second moments for the number of particles in the critical 
Crump-Mode-Jagers model are described in \cite{Ho82}. In 
\cite{Ho82} Yaglom type limit Theorem is also given for the number 
of particles of all types under the condition that the process is non-degenerate.
 
These asymptotic results are expressed in terms of the 
eigenvectors of the mean matrix and the functionals associated 
with the age of the parent at the time of generation of the 
offspring. The specificity of the model makes it possible to 
transfer transformations of particles on edges to vertices, find 
eigenvectors for auxiliary processes of small dimension, and 
then return to the original problem.

More precisely, in the critical case for $\mathcal{M}$--CMJ 
process, a simple algorithm of calculating the eigenvectors 
for the matrix $\mathbf{A}$ of dimension up to $(n(n+1))\times 
(n(n+1))$ is proposed in terms of the eigenvectors for some 
matrix $\mathbf{A}_{1}$ of dimension $(2n)\times(2n)$.


The structure of the paper is as follows. 
In section \ref{r2}, we provide a formal description of the 
development of HIV-1 infection in terms of moving of some 
particles on a graph and state the main result.
In section \ref{s3}, the classical multitype models of 
branching processes and offspring mean matrix are reminded. 
In section \ref{s4}, an auxiliary Bellman-Harris process of a 
lower dimension is constructed on the probabilistic space of 
the $\mathcal{M}$--CMJ process. Section \ref{s5} proves the 
main result.
In section \ref{s6}, some generalization of the $\mathcal{M}$--CMJ model and empirical results are discussed. 



\section{Evolution of particles on a graph} \label{r2} 

Let $\Gamma$ be a directed simply connected graph without loops 
with weighted arcs. The verteces of the graph $\Gamma$ are 
denoted by $N_ {i}$, and the arc going from $N_{i}$ to $N_{j}$ -- 
by $N_{i,j}$, $1\leq i,j\leq n$, where $n$ is the number of 
verteces, respectively.  

The graph structure is determined by the stochastic weight matrix $\mathbf{G}=\big(g_{i,j}\big)_{i,j=1}^{n}$, in particular, there are no arcs if their weight is zero. The weights $g_{i,j}$ of the arcs $N_{i,j}$ in the graph $\Gamma$ are satisfying the conditions
$$
0\leq g_{i,j}\leq1,\quad  g_{i,j}g_{j,i}=0,\quad \sum_{j=1}^{n} 
g_{i,j}=1,\quad 1\leq i,j\leq n. 
$$

The stochastic matrix $\mathbf{G}$ can be interpreted as the transitions probabilities from states $N_{i}$ to $N_{j}$, where $j\neq i$,  
for the Markov chain. We assume that all states of this 
chain are communicating.


Define two sets of pairs
\begin{equation*} 
\mathcal{S}=\big\{(i,j):1\leq i,j \leq n,\ g_{i,j}>0\big\},
\quad \mathcal{S}_{0}=\mathcal{S}\cup\big\{(i,i),1\leq i \leq 
n\big\}.
\end{equation*}
Assume that a lexico-graphic order has been 
introduced for them. We denote by $\big|S\big|$ the cardinality (number of 
elements) of an arbitrary finite set $S$.

Two kinds of particles $A_{i,j}$ and $B_{i,j}$, 
$(i,j)\in\mathcal{S}_{0}$, can be located at the vertex 
$N_{i}$ of the graph $\Gamma$ for $j=i$ or on arcs $N_{i,j}$ for 
$(i,j)\in\mathcal{S}$. In our conditions on the graph $\Gamma$ we 
have $2n+2\big|\mathcal{S}\big|$ types of particles. Obviously, $4n\leq 2n
+2\big|\mathcal{S}\big|\leq(n+1)n$.

From the moment of birth, the evolution of any particle and its 
offspring  does not depend on the further behavior of the 
particles and their offspring that were present at the moment.


Define the model $\mathcal{M}$. The evolution of 
$A_{i,i}$ and $B_{i,i}$ in verteces $N_{i}$, $i=1,\cdots,n$ 
are defined through traditional two-dimensional 
Crump-Mode-Jagers bran\-ching process and are kept 
as $A_{i,j}$ and $B_{i,j}$ in the arcs $N_{i,j}$, $1\leq 
i,j\leq n$. The main property in $\mathcal{M}$ for 
verteces $N_{i}$ is that sometime the death of the particle is 
interpre\-ted as emigration to one of arcs. Set the 
probabilities of emigration for $A_{i,i}$ and $B_{i,i}$ are 
equal to $p(A_{i,i})>0$ and $p(B_{i,i})>0$, correspondingly.
The transition of the particles $A_{i,i}$ and $B_{i,i}$ to the 
arcs $N_{i,j}$, $(i,j)\in\mathcal {S}$, occurs with  
probabilities $g_{i,j}$. It means that the particle $A_{i,i}$ 
(or $B_{i,i}$) at the death time generates exactly one particle 
of type $A_{i,j}$ (or $B_{i,j}$) with probability $g_{i,j}
p(A_{i,i})$ (or $g_{i,j}p(B_{i,i})$). 


In a number of applied publications the evolution of particles 
$A_{i,j}$ (or $B_{i,j}$), $(i,j)\in\mathcal{S}$, located on 
arcs $N_{i,j}$ depends on their type 
and the birth times of these particles. We suppose that our 
process is  homogeneous and the evolution of particles 
$A_{i,j}$ and $B_{i,j}$ is independent of their birth times.
In the model of HIV-1 infection the arcs $N_{i,j}$ correspond to 
lymphatic vessels of various lengths with lymphatic fluid 
moving along them. The initial 
particle $A_{i,j}$  (or $B_{i,j}$) that appears in $N_{i,j}$ 
from $N_{i}$ and all of its descendants simultaneously leave 
the vessel. In this case, we cannot control the offspring on 
$N_{i,j}$ within the branching process on all graph.
The problem of particles evolution on arcs $N_{i,j}$ will be 
investigated separately. As a result we define that all 
particles $A_{i,j}$ and $B_{i,j}$ are kept on the arcs 
$N_{i,j}$ (do not produced offspring and no die) and only at 
the moment of moving of the particles $A_{i,j}$ and 
$B_{i,j}$ to $N_{j}$ they die and, depending on age, spawn a 
random number of offspring $A_{j,j}$ and $B_{j,j}$. This 
determines the evolution of $A_{i,j}$ and $B_{i,j}$ on the arcs 
$N_{i,j}$ in terms of the Crump-Mode-Jagers processes in the 
model $\mathcal{M}$.



Denote $\mathcal{M}$--CMJ branching process by $\mathbf{Z}(t)$. 
Counting processes describing the evolution of particles of the
type $A_{i,j}$ specify a representation for $\xi(A_{i,j})$ -- the 
total number of its offspring
$$
\xi(A_{i,j})=\sum_{(s,k)\in\mathcal {S}}\big(\xi(A_{i,j},A_{s,k})+
\xi(A_{i,j},B_{s,k})\big),
$$
where $\xi(A_{i,j},A_{s,k})$ is the random number of offspring of 
the $A_{s,k}$ type generated by the $A_{i,j}$ type particle.
Denote the mean of offspring of the $A_{s,k}$ type number
generated by the $A_{i,j}$ type particle through $m(A_{i,j}; 
A_{s,k})=\mathbb{E}\xi(A_{i,j},A_{s,k})$ and the same for $B_{s,k}$ 
through $m(A_{i,j};B_{s,k})=\mathbb{E}\xi(A_{i,j},B_{s,k})$.

Similar representations are true when replacing the particle type 
of the parent $A_{i,j}$ with the type $B_{i,j}$. In this case, we 
denote $m(B_{i,j};A_{s,k})=\mathbb{E}\xi(B_{i,j},A_{s,k})$ and
$m(B_{i,j};B_{s,k})=\mathbb{E}\xi(B_{i,j},B_{s,k})$. 

Define the offspring mean matrices of the average number 
of offspring $A_{s,k}$ and $B_{s,k}$ from the particles 
$A_{i,j}$ throughout their life in the first row and from the 
particles $B_{i,j}$ throughout their life in the second line 
through $\mathbf{N}_{(i,j)(s,k)}$. In the general case, we will 
use the notation
\begin{equation*}
\mathbf{N}_{(i,j)(s,k)}=\left(\begin{array}{cc}
m(A_{i,j};A_{s,k})&m(A_{i,j};B_{s,k})\\
m(B_{i,j};A_{s,k})&m(B_{i,j};B_{s,k})
\end{array}\right).
\end{equation*}
But many of these matrices can be written explicitly for the model $\mathcal{M}$.


Write the offspring mean matrix $\mathbf{A}$ for 
$\mathcal{M}$--CMJ process in terms of block matrices 
$\mathbf{N}_{(i,j)(s,k)}$ of dimension $2\times2$
\begin{equation} \label{A01}
\mathbf{A}=\left(\mathbf{N}_{(i,j)(s,k)} \right)_{(i,j),(s,k)
\in \mathcal{S}_{0}}.
\end{equation}

The previously introduced definitions and conditions lead to the relations
\begin{eqnarray} \label{1svAbl1}
&\mathbf{N}_{(i,i)(i,i)}=\left(\begin{array}{cc}
m(A_{i,i};A_{i,i})&m(A_{i,i};B_{i,i})\\
m(B_{i,i};A_{i,i})&m(B_{i,i};B_{i,i})
\end{array}\right),
\\ \label{1svAbl2}
&\mathbf{N}_{(i,i)(i,j)}=g_{i,j}\left( \begin{array}{cc}
p(A_{i,i})&0\\
0&p(B_{i,i})
\end{array}\right), \mbox{ for } (i,j)\in 
\mathcal{S},
\\ \label{1svAbl3}
& \mathbf{N}_{(i,j)(j,j)}=\left( \begin{array}{cc}
m(A_{i,j};A_{j,j})&m(A_{i,j};B_{j,j})\\
m(B_{i,j};A_{j,j})&m(B_{i,j};B_{j,j})
\end{array}\right),  \mbox{ for } (i,j)\in 
\mathcal{S},\\ \label{1svAbl4}
&\mathbf{N}_{(i,j)(s,k)}=\left( \begin{array}{cc}
0&0\\
0&0
\end{array}\right),  \forall \,(i,j),(s,k)\in \mathcal{S}, 
\, \mbox{or } \{(i,j),(s,k)\in\mathcal{S}_{0}\}\& \{j\neq s\}.
\end{eqnarray}

Suppose that the following conditions are true
\begin{equation} \label{dc}
\begin{array}{c}
m(A_{i,i};A_{i,i})m(B_{i,i};B_{i,i})>0,\
m(A_{i,i};B_{i,i})+m(B_{i,i};A_{i,i})>0\\
m(A_{i,j};A_{j,j}) m(B_{i,j};B_{j,j})>0
\end{array}
\end{equation}

Note that in a lot of applications, the $A_{i,j}$ and $B_{i,j}$ 
types of particles can only perish on the arcs $N_{i,j}$ or 
traverse the entire arc $N_{i,j}$. In such a case the evolution 
of particles is described in terms of Crump-Mode-Jagers 
process and we are dealing with real moments of particles  
$A_{i,j}$ and $B_{i,j}$ death. For such a model, the 
transformations of the matrices of means used by us are 
preserved after replacing the matrices $\mathbf{N}_{(i,j)(j,j)}
$ from (\ref{1svAbl3}) with diagonal ones with the probabilities of passing the corresponding arcs on the diagonal.

Define accompanying mean matrix  $\mathbf{A}_{1}=\left(
\mathbf{N}_{i,j}\right)_{i,j=1}^{n}$ of dimension $2n\times2n$ 
in block form, where
\begin{equation} \label{2i,i} 
\mathbf{N}_{i,i}=\left(\begin{array}{cc}
m(A_{i,i};A_{i,i})&m(A_{i,i};B_{i,i})\\
m(B_{i,i};A_{i,i})&m(B_{i,i};B_{i,i})
\end{array}\right)=\mathbf{N}_{(i,i)(i,i)}
\end{equation}
and $\mathbf{N}_{i,j}=\mathbf{N}_{(i,i)(i,j)}
\mathbf{N}_{(i,j)(j,j)}$, for $i\neq j$, or
\begin{equation} \label{2i,j}
\mathbf{N}_{i,j}=
g_{i,j}\left( \begin{array}{cc}
p(A_{i,i})m(A_{i,j};A_{j,j})&p(A_{i,i})m(A_{i,j};B_{j,j})\\
p(B_{i,i})m(B_{i,j};A_{j,j})&p(B_{i,i})m(B_{i,j};B_{j,j})
\end{array}\right). 
\end{equation}

The selection of these matrices will be explained when proving 
the main result.


Define positive $2\big|\mathcal{S}_{0}\big|$ -- dimensional 
left and right eigenvectors $\mathbf{v}$ and $\mathbf{u}$ for 
the matrix $\mathbf{A}$ 
\begin{equation} \label{svA} 
\mathbf{v}\mathbf{A}=\mathbf{v},\quad \mathbf{A}
\mathbf{u}^{\top}=\mathbf{u}^{\top}.
\end{equation}
where
$$
\mathbf{v}=\left(v_{(i,j)}\right)_{(i,j)\in \mathcal{S}_{0}},\quad 
\mathbf{u}=\left(u_{(i,j)}\right)_{(i,j)\in \mathcal{S}_{0}},
$$
and $v_{(i,j)}=\left(v_{(i,j),1},v_{(i,j),2}\right)$,\ \ 
$u_{(i,j)}=\left(u_{(i,j),1},u_{(i,j),2}\right)$.

Hereinafter symbol $^{\top}$ means transposing of a matrix 
(vector).


In the same way define positive $2n$ -- dimensional left and right eigenvectors $\mathbf{v}_{1}$ and $\mathbf{u}_{1}$ for the matrix $\mathbf{A}_{1}$ 
\begin{equation} \label{svA0} 
\mathbf{v}_{1}\mathbf{A}_{1}=\mathbf{v}_{1},\quad 
\mathbf{A}_{1}\mathbf{u}_{1}^{\top}=\mathbf{u}_{1}^{\top}.
\end{equation}
where
$$
\mathbf{v}_{1}=\left(v_{i}\right)_{1\leq i\leq n},\quad 
\mathbf{u}_{1}=\left(u_{i}\right)_{1\leq i\leq n},
$$
and $v_{i}=\left(v_{i,1},v_{i,2}\right)$,\ \ $u_{i}
=\left(u_{i,1},u_{i,2}\right)$.

An essential condition for applying of asymptotic results for 
critical branching processes of all types is that the offspring 
mean matrix $\mathbf{A}$ is indecomposable. This means the 
simultaneous positiveness of all elements of 
$\mathbf{A}^{n_{0}}$ for some $n_{0}\in\mathbb{N}$.




\begin{theorem} \label{th3}
Fix $\mathcal{M}$--CMJ branching process $\mathbf{Z}(t)$ 
with the offspring mean matrix $\mathbf{A}$, defined in 
(\ref{A01}) and (\ref{1svAbl1})--(\ref{1svAbl4}). Let lifetime  
distribution in terms of point processes with
finite second moments for this branching process, be also 
fixed.

Then, under the conditions (\ref{dc}) the matrix $\mathbf{A}$ 
and the accompanying matrix $\mathbf{A}_{1}$, defined in 
(\ref{2i,i}) and (\ref{2i,j}) are indecomposable  and  have 
equal Perron roots and in critical case the eigenvectors of the 
matrices $\mathbf{A}$ and $\mathbf{A}_{1}$ are linked by formulas
\begin{equation} \label{Abl9}
v_{(i,i)}=v_{i},\quad v_{(i,k)}=v_{i}\mathbf{N}_{(i,i)(i,k)},
\quad 1\leq i,k\leq n,
\end{equation}
\begin{equation} \label{Abl10}
u_{(i,i)}=u_{i},\quad u_{(i,j)}=u_{j}\mathbf{N}_{(i,j)(j,j)}^{\top},
\quad 1\leq i,j\leq n.
\end{equation}

\end{theorem}



\section{Branching process models and an accompanying Galton-Watson process} \label{s3}

Let us give brief definitions of abstract models of branching processes with 
$N$ types of particles. All processes $\mathbf{Z}(t)\in 
\mathbb{N}_{0}^{N}$ will be interpreted as a number of particles of 
all types at time $t$, where $\mathbb{N}_{0}=\mathbb{N}\cup
\{0\}$. In all cases will be defined $\mathbf{Z}(0)$ and stochastic evolution of each of the particles. From the moment of particle birth its evolution does not depend on the behavior of the other particles (branching condition).

In the Galton-Watson processes $\mathbf{Z}(t)$ (see \cite{Ha66}) the lifetime of all particles is equal 
to 1. Particles of the $i$-th type at death produce a 
random number of offspring with the generating function $F^{(i)} 
(\mathbf{s})$, where $\mathbf{s}=(s_{1},\cdots,s_{N})
\in[0,1]^{N}$.

In the Bellman-Harris processes $\mathbf{Z}(t)$ (see \cite{Ha66}) the lifetime of 
particles of the $i$-th type have the distribution $G_{i}(t)$ and   
at time of death (upon death) it produces a random number of offspring with the generating function $F^{(i)} 
(\mathbf{s})$, where $\mathbf{s}=(s_{1},\cdots,s_{N})\in[0,1]^{N} 
$. 

The asymptotic properties of critical Bellman-Harris processes 
under the additi\-onal condition of indecomposability of the 
matrix of means are described in terms of $\mathbf{Z}_{a}(t)$ 
-- critical Galton-Watson processes (we call it the 
accompanying Galton-Watson process for $\mathbf{Z}(t)$), with 
generating functions for the number of offspring $F^{(i)}
(\mathbf{s})$ for the particles of the $i$-th type and some 
coefficients expressed in terms of 
moments for $G_{i}(t)$. Formally, the accompanying 
Galton-Watson process is obtained by replacing all particles 
lifetimes with a single one without changing the number of 
offspring. See Theorem 1 \cite[Ch. V, sec. 5]{72AN} for exact asymptotic formulas. 


The offspring mean matrix for the Bellman-Harris process and the same for the accompanying Galton-Watson process (in general form without reference to the original problem) is written as
\begin{equation*} 
\mathbf{A}=\big(a_{i,j}\big)_{i,j=1}^{N}=
\left(\dfrac{\partial F^{(i)}(\mathbf{s})}{\partial s_{j}}
\bigg|_{\mathbf{s}=\mathbf{1}}\right)_{i,j=1}^{N}.
\end{equation*}


Sevast'yanov processes $\mathbf{Z}(t)$ with $N$ types of particles (see  
\cite[Ch. VIII]{Se71}) differ from Bellman-Harris processes in that 
the number of offspring of a particle may depend on the age of its 
death. In other words, the generating function $ F^{(i)}
(\mathbf{s},u)$ of the number of offspring of a particle of the 
$i$-th type that perishes at the age of $u$ has the first moments
$$
a_{i,j}(u)=\dfrac{\partial F^{(i)}(\mathbf{s},u)}
{\partial s_{j}}\bigg|_{\mathbf{s}=\mathbf{1}}.
$$

Consequently the accompanying Galton-Watson process $\mathbf{Z}_{a}(t)$ for $\mathbf{Z}(t)$ is given by the generating functions
$$
F^{(i)}(\mathbf{s})=\int_{0}^{\infty}F^{(i)}(\mathbf{s},u)d
G_{i}(u)
$$ 
and the offspring mean matrix $\mathbf{A}$ consists of elements 
$$
a_{i,j}=\int_{0}^{\infty}a_{i,j}(u)dG_{i}(u).
$$

The Crump-Mode-Jagers processes $\mathbf{Z}(t)$ are a 
generalization of Sevast'yanov processes and differ from them 
in that particles can repeatedly generate random numbers of 
offspring throughout their life, while the number of groups of 
offspring, the moments of their generation and the lifetime of 
the particles themselves may depend on each other. Formally, 
the offspring generation process is specified using a 
multidimensional counting process. We will not go into details, 
but the asymptotic properties of such processes are described 
in terms of the offspring mean matrix $\mathbf{A}$ for 
$\mathbf{Z}(t)$ (or $\mathbf{Z}_{a}(t)$ -- its the 
accompanying  Galton-Watson) and some functionals such as 
moments from counting processes (from the evolution of 
particles), see \cite{Ja75} and \cite{Ho82}. 

The asymptotic properties of multidimensional critical 
Galton-Watson processes are described in \cite[Ch.\,VI,\,\S\,3]
{Se71}. In \cite{Se71}, the criticality condition are expressed 
in terms of the properties of the offspring mean matrix 
$\mathbf{A}$. Its Perron root will be equal 1. Exact asymptotic 
formulas for the nonextinction probability  of the process and 
conditional distributions for the normalized number of 
particles are related to the second moments (provided that they 
are finite) and the right and left eigenvectors $u>0$ and $v>0$ 
are defined by the relations $\mathbf{A}u^{\top}=u^{\top}$, 
$v\mathbf{A}=v$, $(u,v)=1$. 



\section{$\mathcal{M}$--CMJ process and its accompanying process}  \label{s4}


For multitype Crump-Mode-Jagers model in \cite{Ho82} are 
presented integral equations for the first two moments in an 
explicit form and asymptotic representations and 
conditional Yaglom type limit Theorems for the number of 
particles of all types. The generating vector-functions for 
the number of offspring produced by particle of the type $i$   
throughout their life was denoted by $F^{(i)}(\mathbf{s})$.
These generating functions define multitype accompanying  
Galton-Watson process $\mathbf{Z}_{a}(t)$. The critical type 
for both processes is the same. 

In our $\mathcal{M}$--CMJ process we have $2n+2\big|
\mathcal{S}\big|$ particle types $A_{i,j}$ and $B_{i,j}$, 
$(i,j)\in\mathcal{S}_{0}$ with offspring mean matrix $\mathbf{A}$ defined in (\ref{A01}) and 
(\ref{1svAbl1})--(\ref{1svAbl4}). 
If any of the means $m(A_{i,j};A_{s,k})$, $m(A_{i,j};
B_{s,k})$, $m(B_{i,j};A_{s,k})$, $m(B_{i,j};B_{s,k})$ is equal 
to zero, then the particles of the first type from the 
argument do not produce particles of the second type. In $
\mathbf{Z}_{a}(t)$ the particle of the $A_{i,i}$ (or $B_{i,i}$) 
type produce a random number of offspring $A_{i,i}$ and 
$B_{i,i}$ and in some step  with probability $g_{i,j}
p(A_{i,i})$ (or $g_{i,j}p(B_{i,i})$) gives rise to exactly one 
offspring of the $A_{i,j}$ (or $B_{i,j}$) type. After that, at 
the next step, the $A_{i,j}$ (or $B_{i,j}$) type particle  
produces a random number of offspring of the 
types $A_{j,j}$ and $B_{j,j}$ with the generating function 
$F^{(i)}_{A}(s_{1},s_{2})$ (or $F^{(i)}_{B}(s_{1},s_{2})$). 

Let us construct a new process $\mathbf{Z}_{1}(t)$ on the 
probabilistic space of the Galton-Watson process 
$\mathbf{Z}_{a}(t)$. We divide all one-step transformations of 
$A_{i,j}$ and $B_{i,j}$ in $\mathbf{Z}_{a}(t)$ from one state 
to other states into three groups: inside the vertices, from 
the vertices to the arcs, and from the arcs to the vertices. In the first group, 
offspring particles can stay at the same vertex for several 
steps in a row, while in the rest they leave the groups in one 
step. In terms of process $\mathbf{Z}_{a}(t)$ implementations, 
the transformations of the first type are preserved for the new 
process $\mathbf{Z}_{1}(t)$, and the transformations of the 
second and third types are combined into one of duration 2. 
Instead of two transitions from $A_{i,i}$ (or $B_{i,i}$) to 
$A_{i,j}$ (or $B_{i,j}$) and to $N_{j}$ with generating 
function $F^{(i)}_{A}(s_{1},s_{2})$ (or $F^{(i)}_{B}
(s_{1},s_{2})$) for number of offspring, we define that in 
two units of time $A_{i,i}$ (or $B_{i,i}$) generates a random 
number of offspring $A_{j,j}$ and $B_{j,j}$ with the same  
generating function $F^{(i)}_{A}(s_{1},s_{2})$ (or $F^{(i)}_{B}
(s_{1},s_{2})$).

New process $\mathbf{Z}_{1}(t)$ is the Sevast'yanov process
with $2n$ particles of the types $A_{i,i}$ and $B_{i,i}$, 
$1\leq i\leq n$, with lifetime 1 or 2, with the lifetime dependent of offspring number and with offspring mean 
matrix $\mathbf{A}_{1}$ are defined in (\ref{2i,i}) and 
(\ref{2i,j}). 



Recall that the condition of subcriticality, criticality and 
supercriticality branching processes corresponds to 
exponential decay, convergence to some positive constant or 
exponential growth of the average number of all components 
simultaneously with unlimited growth of the time parameter see 
\cite{Ha66}, \cite{Se71}). So $\mathbf{Z}_{a}(t)$ and 
$\mathbf{Z}_{1}(t)$ have one and the same critical class.
Denote by $\mathbf{Z}_{1a}(t)$ the accompanying Galton-Watson 
process for the $\mathbf{Z}_{1}(t)$. These two processes share 
the common offspring mean matrix $\mathbf{A}_{1}$.


\section{Proof of Theorem \ref{th3}} \label{s5} 


For our analysis, the initial distributions of $\mathcal{M}
$--CMJ process $\mathbf{Z}(t)$ and its accom\-panying of the 
second order Galton-Watson process $\mathbf{Z}_{1a}(t)$ are not 
important. We assume that the distributions of the number of 
offspring and the lifetime of particles for $\mathbf{Z}(t)$ are given, i.e. the 
process $\mathbf{Z}(t)$ is formally defined. The critical type 
of accompanying Galton-Watson process $\mathbf{Z}_{1a}(t)$ 
coincides with the critical type of the original 
$\mathcal{M}$--CMJ process $\mathbf{Z}(t)$. This means that 
the offspring mean matrices $\mathbf{A}$ and $\mathbf{A}_{1}$ 
have the same Perron roots.

First, let us prove that the offspring mean matrices $\mathbf{A}$ and 
$\mathbf{A}_{1}$ are indecomposable. 

Let us construct an auxiliary Markov chain $\mathbf{M}(n)$ for 
the $\mathcal{M}$--CMJ process with the states $A_{i,j}$, 
$B_{i,j}$, $(i,j)\in\mathcal{S}_{0}$, with a matrix of 
transition probabilities
\begin{equation*} 
\tilde{\mathbf{A}}=\left(\mathbf{p}_{(i,j)(s,k)}\right)_{(i,j),
(s,k)\in \mathcal{S}_{0}},
\end{equation*}
where the elements of matrices
\begin{equation*} 
\mathbf{p}_{(i,j)(s,k)}=\left(\begin{array}{cc}
p(A_{i,j};A_{s,k})&p(A_{i,j};B_{s,k})\\
p(B_{i,j};A_{s,k})&p(B_{i,j};B_{s,k})
\end{array}\right),
\end{equation*}
are satisfying the conditions $p(\ast;\star)>0$ if and only if
$m(\ast;\star)>0$ with the same arguments.

By virtue of the matrix $\mathbf{G}$ properties and conditions
(\ref{dc}), the Markov chain has finite number of communicating states and it 
is not periodic. Therefore, there exists $n_{0}$ such that all 
elements of $\tilde{\mathbf{A}}^{n_{0}}$ are positive (see 
Lemma 2 \cite [Ch.IV, \S5]{Se71}). 

By definition there exists some constant $c>0$ such that for all 
combinations of arguments the inequalities 
$$
m(\ast;\star)\geq c p(\ast;\star)
$$ 
are true.
In matrix form, the last inequalities are written as
$$
\mathbf{A}\geq c \tilde{\mathbf{A}}.
$$
This implies that all elements of the matrix $\mathbf{A}^{n_{0}}$ are positive.

We proved that the offspring mean matrix $\mathbf{A}$ is indecomposable. By analogy, the same can be proved for the offspring mean matrix $\mathbf{A}_{1}$.

 We are 
interested in a critical case. This means that the maximal root of 
the equation 
\begin{equation*} 
\det(\mathbf{A}_{1}-\lambda I_{2n})=0
\end{equation*}
is equal to 1. In this case, there are positive $2n$-dimensional 
left and right eigenvectors $\mathbf{v}_{1}$ and $\mathbf{u}_{1}$ 
satisfying the conditions (\ref{svA0}).



Rewrite the relations (\ref{svA0}) in block form
\begin{equation} \label{svAbl} 
\sum^{n}_{i=1}v_{i}\mathbf{N}_{i,j}=v_{j},\quad 
\sum^{n}_{j=1}\mathbf{N}_{i,j}u_{j}^{\top} =u_{i}^{\top}.
\end{equation}

 

For $2\big|\mathcal{S}_{0}\big|$ - dimensional offspring 
mean matrix $\mathbf{A}$ left and right eigenvectors $\mathbf{v}
$ and $\mathbf{u}$ are defined in (\ref{svA}).

After rewriting the relations (\ref{svA}) in block form, one obtains the relations
\begin{equation*}  
\sum_{(i,j)\in \mathcal{S}_{0}}v_{(i,j)}\mathbf{N}_{(i,j)(s,k)}
=v_{(s,k)},
\end{equation*}
or in view of (\ref{1svAbl4})
\begin{equation} \label{svAbl6}
v_{(s,k)}=\sum_{(i,j)\in\mathcal{S}_{0},\,j=s}v_{(i,j)}
\mathbf{N}_{(i,j)(s,k)}=\sum_{\{i:(i,s)\in \mathcal{S}_{0}\}}v_{(i,s)}
\mathbf{N}_{(i,s)(s,k)}.
\end{equation}

In the case $s=k$ (\ref{svAbl6}) implies $\sum_{\{i:(i,k)\in \mathcal{S}_{0}\}}v_{(i,k)}\mathbf{N}_{(i,k)(k,k)}=v_{(k,k)}$ or taking into account (\ref{2i,i})
\begin{equation} \label{svAbl7}
v_{(k,k)}\mathbf{N}_{k,k}+\sum_{\{i:(i,k)\in \mathcal{S}\}}v_{(i,k)}
\mathbf{N}_{(i,k)(k,k)}=v_{(k,k)}. 
\end{equation}

If $k\neq s$, then $s=i$ and for $(i,k)\in\mathcal{S}$
\begin{equation} \label{svAbl8}
v_{(i,i)}\mathbf{N}_{(i,i)(i,k)}=v_{(i,k)}. 
\end{equation}
From (\ref{1svAbl2}) and (\ref{1svAbl3}) it follows
\begin{equation} \label{svl}
\mathbf{N}_{(i,i)(i,k)}\mathbf{N}_{(i,k)(k,k)}
=\mathbf{N}_{i,k},
\end{equation} 
therefore (\ref{svAbl7}) and (\ref{svAbl8}) imply
\begin{equation} \label{s0vl}
v_{(k,k)}\mathbf{N}_{k,k}+\sum_{\{i:(i,k)\in \mathcal{S}\}}v_{(i,i)}
\mathbf{N}_{i,k}=\sum_{i=1}^{n}v_{(i,i)}\mathbf{N}_{i,k}=v_{(k,k)}.
\end{equation}

From (\ref{svAbl8})--(\ref{s0vl}) it follows that the left 
eigenvector (without normalization by unity) can be chosen in the form
\begin{equation*} 
v_{(i,i)}=v_{i},\quad v_{(i,k)}=v_{(i,i)}\mathbf{N}_{(i,i)(i,k)}
\end{equation*}
where $v_{k}$ is an eigenvector from (\ref{svAbl}). The representation (\ref{Abl9}) is proved.

Similar calculations can be performed for the right eigenvector, 
namely:
\begin{equation} \label{svAb20} 
\sum_{(s,k)\in \mathcal{S}_{0}}\mathbf{N}_{(i,j)(s,k)}
u_{(s,k)}^{\top} =u_{(i,j)}^{\top}.
\end{equation}

Considering the property (\ref{1svAbl4}), we rewrite (\ref{svAb20}) in the form 
\begin{equation} \label{svAb21} 
u_{(i,j)}^{\top}=
\sum_{\{k:(j,k)\in \mathcal{S}_{0}\}}
\mathbf{N}_{(i,j)(j,k)}u_{(j,k)}^{\top}
\end{equation}

In case $i=j$ (\ref{svAb21}) implies $\sum_{\{k:(i,k)\in 
\mathcal{S}_{0}\}}\mathbf{N}_{(i,i)(i,k)}u_{(i,k)}^{\top}=
u_{(i,i)}^{\top}$ or taking into account (\ref{1svAbl1}) and 
(\ref{1svAbl2}) we have
\begin{equation} \label{svAb22} 
\mathbf{N}_{i,i}u_{(i,i)}^{\top}+\sum_{\{k:(i,k)\in
\mathcal{S}\}}
\mathbf{N}_{(i,i)(i,k)}u_{(i,k)}^{\top}=u_{(i,i)}^{\top}.
\end{equation}

In case $i\neq j$ (\ref{svAb21}) implies
\begin{equation*}  
\mathbf{N}_{(i,j)(j,j)}u_{(j,j)}^{\top}=u_{(i,j)}^{\top}.
\end{equation*}
or taking into account (\ref{svl})
\begin{equation} \label{svAb24} 
\sum_{\{k:(i,k)\in\mathcal{S}_{0}\}}\mathbf{N}_{i,k}u_{(k,k)}^{\top}=
\sum_{k=1}^{n}\mathbf{N}_{i,k}u_{(k,k)}^{\top}=u_{(i,i)}^{\top}.
\end{equation}
The representations (\ref{svAb22}) and (\ref{svAb24}) imply 
that as the right eigenvector (without normalization to unity)  
can take the form
\begin{equation*} 
u_{(i,i)}=u_{i},\ u_{(i,j)}=u_{j}\mathbf{N}_{(i,j)(j,j)}^{\top}.
\end{equation*}
where $u_{k}$ is an eigenvector from (\ref{svAbl}). The representation (\ref{Abl10}) is proved.

\section{Comments on applications and generalization } \label{s6}


The problem under consideration in a general case is 
traditionally solved by methods of statistical simulation. To 
test the operation of programs, it is convenient to use 
analytical results. The latter are also useful for creating 
simulation algorithms. Particular cases of the proposed model 
were investigated in \cite{19LPT} -- \cite{21BLPT}. A number of 
specific models and some results of numerical experiments are 
presented there. The behavior of the process on arcs in real 
problems excludes its description in terms of processes simpler 
than the Sevast'yanov branching processes. Here we used a more 
general the Crump-Mode-Jagers model.

The value of the Perron root for the the offspring mean matrix 
significantly affects the global behavior of processes. 
Supercritical and subcritical processes are well studied and 
quite simply arranged. With critical processes and processes 
close to them in terms of the value of the Perron root, there 
is a special interest. They are boundary when epidemics appear 
and have nontrivial properties for various parameters.
When simulating complex processes, random experiments are 
repeated many times and conclusions are drawn based on 
statistical estimates. In the study of critical processes,  
experimenters encountered strange behavior on a number of 
trajectories: with a traditionally small number of virions, 
sharp surges in their numbers occurred. This effect is easily 
explained by Yaglom's type limit Theorem for critical 
branching process (large deviation Theorem). If process beginning from one particle of any 
fixed type and up to time $t$ it is not degenerate (this happens 
with probability of the order 1/t), then the vector of particles 
number multiplied to $t^{-1}$ convergence to some random vector, 
as $t\to\infty$. The precise formula of the limiting random 
vector distribution essentially depends from the eigenvectors 
for the initial model. It means that the number of particles at 
this moment is random, but this number has the order $t$.
This phenomenon must be taken into account in practice for 
repetitive diseases.

The main results of this article are naturally generalized to a 
more general model. Instead of two global types of particles $A$ 
and $B$, one can take any finite number of types $A^{(l)}$, 
$1\leq l\leq k$, $1\leq i\leq k$. At the vertex of the graph, 
their transformations can be specified by some common branching 
process for particles $A^{(l)}_{i,i}$, $1\leq l\leq k$, with the 
possibility of particle escape $ A^{(l)}_{i,i}$ from vertex 
$N_{i}$ to arc $N_{i,j}$, $(i,j)\in \mathcal{S}$, 
{\Large(}transformation of $A^{(l)}_{i,i}$ into $A^{(l)}_{i,j}$ 
{\Large)} with the  probabilities $g_{i,j}$ defined above, 
where without dying they pass the entire arc
and at the moment of moving to $N_{j}$ they die and spawn a 
random number of offspring $A^{(l)}_{j,j}$, $1\leq l\leq k$, as 
in the Crump-Mode-Jagers process.
Further, the matrices $\mathbf{A}$ and $\mathbf{A}_{1}$ are 
defined by relations similar to (\ref{A01})--(\ref{2i,j}) in 
terms of new matrices of the size $k\times k$. 
The statement of the theorem should be written in terms of  
these matrices of the size $k\times k$ like  
(\ref{Abl9}) and (\ref{Abl10}).


\bigskip
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