Genetic Algebras Associated with the SIR Model

Abstract

The present study proposes a novel connection between the susceptible–infected–recovered $\left(SIR\right)$ model and genetic algebras through the construction of quadratic stochastic processes (QSPs). To determine the underlying dynamics, explicit formulations of the QSP are generated and their long-term behavior is examined under different cases. Moreover, we evaluate the attributes of the related limiting genetic algebras, focusing on fundamental characteristics like Rota–Baxter operators, automorphisms, and derivations.

1. Introduction

Since the susceptible–infected–recovered $\left(SIR\right)$ model was introduced in the 19th century, there has been a significant increase in the study of mathematical modeling for epidemiological phenomena. In 1927, the SIR model as an important tool to understand and predict the spread of infectious diseases was introduced [1]. A number of models came into existence to anticipate different dimensions of the outbreaks like Spanish flu, human immunodeficiency virus/acquired immunodeficiency virus, Ebola, and COVID-19.

According to the $SIR$ model, which is classified as a compartmental model, the population is divided into three groups of individuals: susceptible (S), infected (I), and those who have either deceased (removed) or become resistant (recovered) (R). Even though the traditional $SIR$ model assumes a stable population and homogeneous mixing, significant modifications have been incorporated over the last century to account for the evolutionary progression of an enormous variety of epidemic processes. The SIRS model, SEIR model, MSIR model, network-based SIR model, age-structured SIR models, spatial SIR models, stochastic SIR models, seasonal SIR models, and SIR with vital dynamics are a few such versions. As a result, the $SIR$ model and its variants are used in numerous areas of research [2,3,4,5,6]. Furthermore, estimation of the parameters of the $SIR$ model have also gained lot of attention over the last decade. Optimization algorithms [7], second-order finite differences [8], an approximate analytic solution [9], and the iterative method [10] were some of the techniques proposed to enhance the accuracy of these parameters.

In the present work, we propose to study the quadratic stochastic operators inspired by the parameters of the $SIR$ model. We stress that while previous studies have explored QSOs in a generic context, the connection between QSOs and epidemiological models like SIR remains underexplored. The present paper bridges this gap by constructing QSOs from SIR model parameters and analyzing their long-term behavior and associated genetic algebras. Bernstein [11] first proposed the quadratic stochastic operator (QSO) as a mapping connected to a cubic stochastic matrix. Even though Bernstein first used these operators to show the time evolution of various species in population genetics, they are now recognized as an important analytical tool for the analysis of dynamic characteristics and simulations in a wide range of disciplines, including biology, physics, economics, and mathematics [12,13,14,15]. This resulted in the wide range of research being conducted with respect to different aspects such as models of mathematical genetics.

Let the set $E=\{1,2,\cdots ,m\}$ denote m number of species (or traits) of the population under consideration with the probability distribution at an initial state presumed as ${\mathbf{x}}^{\left(0\right)}$$=\left(x_1^{\left(0\right)},\cdots ,x_m^{\left(0\right)}\right)$. $P_{ij,k}$ denotes a probability that members from the ith and jth species interbreed to produce an offspring belonging to kth species. Subsequently, the distribution of probability of the species in the first generation ${\mathbf{x}}^{\left(1\right)}=\left(x_1^{\left(1\right)},\cdots ,x_m^{\left(1\right)}\right)$ can be established as a total probability, i.e.,

$$x_k^{\left(1\right)}=\sum _{i,j\in E}{P}_{ij,k}{x}_i^{\left(0\right)}{x}_j^{\left(0\right)},k\in \{1,\dots ,m\}.$$

(1)

Thus, the quadratic stochastic operator $\left(QSO\right)$, also known as the evolution operator, is the mapping ${\mathbf{x}}^{\left(0\right)}\to {\mathbf{x}}^{\left(1\right)}$. The quadratic stochastic operator, being the most basic nonlinear mapping, specifies a distribution of the system’s subsequent generation if the current distribution is known. In this way, the quadratic stochastic operator serves as a major resource for population genetics research on evolution [16,17]. Even on a small dimensional simplex, the main challenge is to analyze the behavior of the trajectories of quadratic stochastic operators [18,19]. The properties and limit behavior of quadratic stochastic operator trajectories were studied, along with the population genetics implications of these findings. Yet, a set of all QSOs would not be covered by these classes taken together. As a result, numerous classes of QSOs have not yet been thoroughly investigated. The crucial task persists to investigate the behavior of the trajectories of quadratic stochastic operators.

The dynamical structures generated by the SIR model in discrete time are taken into account to explore the behavior of QSO trajectory [20,21]. We aim to expand the QSO to genetic algebras generated by the SIR model. Etherington’s series of pioneering publications introduced the formal understanding of abstract algebra to the field of genetics study [22]. The study of genetic algebras and their connection to quadratic stochastic operators have been a topic of significant interest in the field of mathematical biology [23,24,25,26]. Through the establishment of links between the algebraic properties of the genetic algebras and dynamics of the SIR model, we want to provide a new perspective on the mathematical framework of the underlying epidemiological process.

The construction of paper is as follows: Section 2 provides preliminaries for quadratic stochastic operator and processes. The discrete version of the SIR model is also provided. In Section 3 we give the construction of the quadratic stochastic process associated with the SIR model. The explicit values of $P_{ij,k}^{\left(t\right)}$ are calculated. Section 4 deals with the limiting behavior of the constructed quadratic stochastic process. In Section 5, we explore the properties of the resulting limiting genetic algebra such as derivations and automorphisms. It turns out that if the model has unique equilibrium point, then the corresponding limiting algebra has a trivial derivation. The non-trivial derivation appears while the model has two equilibrium points. Finally, Section 6 is devoted to the analysis Rota–Baxter operators defined on the limiting genetic algebras. The study of Rota–Baxter operators on genetic algebras was initiated in [27].

2. Preliminaries

The necessary notations are covered in this section. Let $E=\{1,\dots ,m\}$ be as above. The standard basis in ${\mathbb{R}}^m$ are represented by ${\left\{{\mathbf{e}}_i\right\}}_{i\in E}$ where ${\mathbf{e}}_i=({\delta }_{i1},\dots ,{\delta }_{im})$, and $\delta$ is known as Kronecker’s Delta. The simplex taken into consideration throughout this paper is given as

$$S^{m-1}=\left\{x=(x_1,\cdots ,x_m)\in {\mathbb{R}}^m:\sum _{i\in E}{x}_i=1,x_i\ge 0,i\in E\right\}$$

(2)

A quadratic stochastic operator $\left(QSO\right)$ is a correspondence of the simplex $S^{m-1}$ into itself of the subsequent form

$$V:x_k^{\prime }=\sum _{i,j\in E}{P}_{ij,k}{x}_i{x}_j,k=\overline{1,m},$$

(3)

where $V\left(x\right)=x^{\prime }=(x_1^{\prime },\cdots ,x_m^{\prime })$. The coefficient of heredity is symbolized by $P_{ij,k}$, which fulfills the subsequent conditions

$$P_{ij,k}\ge 0,P_{ij,k}=P_{ji,k},\sum _{k\in E}{P}_{ij,k}=1.$$

(4)

The binary rule laid out for the arbitrary vectors $\mathbf{x},\mathbf{y}\in {\mathbb{R}}^m$ for the $QSO$ V is expressed by

$${\left(\mathbf{x}{\circ }_V\mathbf{y}\right)}_k=\sum _{i,j=1}^m{P}_{ij,k}{x}_i{y}_j$$

(5)

where $\mathbf{x}=(x_1,x_2,\dots ,x_m),\mathbf{y}=(y_1,y_2,\dots ,y_m)\in {\mathbb{R}}^m$. In terms of canonical basis the above multiplication can be articulated as follows

$${\mathbf{e}}_i{\circ }_V{\mathbf{e}}_j=\sum _{i,j\in E}{P}_{ij,k}{\mathbf{e}}_k$$

(6)

Thus, the evolution operator (3) generates an algebra $({\mathbb{R}}^m,{\circ }_V)$ called the genetic algebra. These algebras were researched thoroughly and some algebraic properties were examined [24,25,28]. These algebras are commutative, however there is ambiguity in their associative nature.

It is to be noted that a matrix ${\left(H_{ij}\right)}_{i,j\in E}$ is called stochastic if

$$H_{ij}\ge 0,\forall i,j\in E,\sum _{j\in E}{H}_{ij}=1$$

(7)

The collection of stochastic matrices $\{{\left(H_{ij}^{[s,t]}\right)}_{i,j\in E}:s,t\in {\mathbb{R}}_{+},0\le s\le t\}$ is known as the Markov process if the ensuing factors are established:

$$H_{ij}^{[s,t]}=\sum _{\ell =1}^n{H}_{i\ell }^{[t,t^{\prime }]}{H}_{\ell j}^{[t^{\prime },t]},s<t^{\prime }<t.$$

(8)

The resultant equation is called the Kolmogorov–Chapman equation.

Now, let us consider a family of functions

$$\{P_{ij,k}^{[s,t]}:i,j,k\in E,s,t\in {\mathbb{R}}_{+},t-s\ge 1\}$$

(9)

with an initial state ${\mathbf{x}}^{\left(0\right)}={\left(x_k^{\left(0\right)}\right)}_{k\in E}\in S^{m-1}$.

This arrangement is known to be a quadratic stochastic process (QSP) [17,18] if for fixed $s,t\in {\mathbb{R}}_{+}$ it meets the requirements listed below:

i.

$P_{ij,k}^{[s,t]}=P_{ji,k}^{[s,t]}$ for any $i,j,k\in E$;

ii.

$P_{ij,k}^{[s,t]}\ge 0$ and ${\displaystyle \sum _{k\in E}}{P}_{ij,k}^{[s,t]}=1$ for any $i,j,k\in E$;

iii.

There are two adaptations of the Kolmogorov–Chapman equation’s analogue: for the initial point ${\mathbf{x}}^{\left(0\right)}\in S^{m-1}$ and $s<r<t$ such that $t-r\ge 1$, $r-s\ge 1$ one has

(iii_A)

$$P_{ij,k}^{[s,t]}=\sum _{m,\ell }{P}_{ij,m}^{[s,r]}{P}_{m\ell ,k}^{[r,t]}{x}_{\ell }^{\left(r\right)},$$

where $x_k^{\left(r\right)}$ is outlined as follows:

$$x_k^{\left(r\right)}=\sum _{i,j}{P}_{ij,k}^{[0,r]}{x}_i^{\left(0\right)}{x}_j^{\left(0\right)};$$

(iii_B)

$$P_{ij,k}^{[s,t]}=\sum _{m,\ell ,g,h}{P}_{im,\ell }^{[s,r]}{P}_{jg,h}^{[s,r]}{P}_{\ell h,k}^{[r,t]}{x}_m^{\left(s\right)}{x}_g^{\left(s\right)}.$$

We indicate the definite process by $(E,P_{ij,k}^{[s,t]},{\mathbf{x}}^{\left(0\right)})$. The QSP is claimed to be of type (A) or (B) if it satisfies the foundational equations either $\left(ii{i}_A\right)$ or $\left(ii{i}_B\right)$, respectively.

The involvement between elements i and j at time s resulted in the element k at time t and is conveyed by the probability functions $P_{ij,k}^{[s,t]}$. We assume that the maximum time needed for interactions in physical, chemical, and biological phenomena is equal to 1 because these processes take a specific amount of time to occur ([16,29]). Hence, the likelihood $P_{ij,k}^{[s,t]}$ is defined for $t-s\ge 1.$

We know that the basic version of a SIR model is based on a set of coupled ordinary differential equations given by

$$\begin{array}{cc}\hfill {\displaystyle \frac{ds}{dt}}& =-\beta is\hfill \\ \hfill {\displaystyle \frac{di}{dt}}& =\beta is-\gamma i\hfill \\ \hfill {\displaystyle \frac{dr}{dt}}& =\gamma i\hfill \end{array}$$

(10)

where $\beta$ and $\gamma$ are rates of transmission. We analyze this model in discrete time, but it is established by a quadratic stochastic operator. The above set of Equation (10) in a discrete setting can be written as

$$\left\{\begin{array}{c}{s}_{n+1}=s_n-\beta s_n{i}_n\hfill \\ i_{n+1}=i_n-\gamma i_n+\beta s_n{i}_n\hfill \\ r_{n+1}=r_n+\gamma i_n\hfill \end{array}\right.$$

where n is a non-negative integer number.

Consider an operator $V:{\mathbb{R}}_{+}^3\to {\mathbb{R}}_{+}^3,\mathbf{x}=\left(x_1,x_2,x_3\right)\to {\mathbf{x}}^{\prime }=V\left(\mathbf{x}\right)=$$\left(x_1^{\prime },x_2^{\prime },x_3^{\prime }\right)$ defined by

$$V:\left\{\begin{array}{c}{x}_1^{\prime }=x_1\left(1-\beta x_2\right)\hfill \\ x_2^{\prime }=x_2\left(1-\gamma +\beta x_1\right)\hfill \\ x_3^{\prime }=x_3+\gamma x_2\hfill \end{array}\right.$$

(11)

where $x_1^{\prime }=s_{n+1},x_2^{\prime }=i_{n+1},x_3^{\prime }=r_{n+1},x_1=s_n,x_2=i_n,x_3=r_n$ for a natural n.

Using $x_1+x_2+x_3=1$ one can rewrite (11) in the form

$$V:\left\{\begin{array}{c}{x}_1^{\prime }=x_1^2+(1-\beta )x_1{x}_2+x_1{x}_3\hfill \\ x_2^{\prime }=(1-\gamma )x_2^2+(1-\gamma +\beta )x_1{x}_2+(1-\gamma )x_2{x}_3\hfill \\ x_3^{\prime }=\gamma x_2^2+x_3^2+\gamma x_1{x}_2+x_1{x}_3+(1+\gamma )x_2{x}_3.\hfill \end{array}\right.$$

(12)

In this setting, the last one can be written as follows

$$\left\{\begin{array}{c}{x}_1^{\prime }=p_{11,1}{x}_1^2+p_{22,1}{x}_2^2+p_{33,1}{x}_3^2+2p_{12,1}{x}_1{x}_2+2p_{13,1}{x}_1{x}_3+2p_{23,1}{x}_2{x}_3,\hfill \\ x_2^{\prime }=p_{11,2}{x}_1^2+p_{22,2}{x}_2^2+p_{33,2}{x}_3^2+2p_{12,2}{x}_1{x}_2+2p_{13,2}{x}_1{x}_3+2p_{23,2}{x}_2{x}_3,\hfill \\ x_3^{\prime }=p_{11,3}{x}_1^2+p_{22,3}{x}_2^2+p_{33,3}{x}_3^2+2p_{12,3}{x}_1{x}_2+2p_{13,3}{x}_1{x}_3+2p_{23,3}{x}_2{x}_3.\hfill \end{array}\right.$$

(13)

where (from (12)) one has

$$\begin{array}{ccccc}\hfill p_{11,1}=1,& & \hfill 2p_{12,1}=1-\beta ,& & \hfill 2p_{13,1}=1,\\ \hfill p_{11,2}=0,& & \hfill 2p_{12,2}=1-\gamma +\beta ,& & \hfill 2p_{13,2}=0,\\ \hfill p_{11,3}=0,& & \hfill 2p_{12,3}=\gamma ,& & \hfill 2p_{13,3}=1,\\ \hfill p_{22,1}=0,& & \hfill 2p_{23,1}=0,& & \hfill p_{33,1}=0,\\ \hfill p_{22,2}=1-\gamma ,& & \hfill 2p_{23,2}=1-\gamma ,& & \hfill p_{33,2}=0,\\ \hfill p_{22,3}=\gamma ,& & \hfill 2p_{23,3}=1+\gamma ,& & \hfill p_{33,3}=1\end{array}$$

For the operator given by (13) to be stochastic, one needs some constraint on the parameters, which is stated in the next proposition.

Proposition 1

([20]). The operator (12) maps $S^2$ to itself if and only if

$$0\le \gamma \le 1,\gamma -1\le \beta \le 1$$

(14)

Proof. 

Since each $P_{ij,k}\ge 0$, we have

$$\begin{array}{ccccc}\hfill 1-\gamma \ge 0& & \hfill \gamma \ge 0& & \hfill {\displaystyle \frac{1-\beta }{2}}\ge 0\\ \hfill {\displaystyle \frac{1-\gamma }{2}}\ge 0& & \hfill {\displaystyle \frac{1+\gamma }{2}}\ge 0& & \hfill {\displaystyle \frac{1-\gamma +\beta }{2}}\ge 0\end{array}$$

but

$$0\le \gamma \le 1$$

Also,

$$1-\gamma +\beta \ge 0\Rightarrow \gamma -1\le \beta$$

Hence,

$$\beta \le 1\Rightarrow \gamma -1\le \beta \le 1$$

□

We point out that the obtained conditions imply that the transmission rate $\gamma$ should be in the range $0\le \gamma \le 1$, while the other one $\beta$ needs to obey the condition $\gamma -1\le \beta \le 1$. Under these assumptions, the operator (13) is a well-defined QSO. The constructed QSO is called an associated QSO with SIR model [20]. In the next section, we are going to construct a QSP generated by this QSO.

3. Construction of Quadratic Stochastic Process

Let $(E,P_{ij,k}^{[s,t]},{\mathbf{x}}^{\left(0\right)})$ be a quadratic stochastic process; then one defines

$$\begin{array}{ccc}\hfill & & H_{ij}^{[s,t]}=\sum _{\ell }{P}_{i\ell ,j}^{[s,t]}{x}_l^{\left(s\right)},\hfill \end{array}$$

(15)

where $i,j\in E$.

It is to be noted that for each pair $s,t\in {\mathbb{R}}_{+}$ the matrix $H_{ij}^{[s,t]}$ is stochastic. It was proved that $H_{ij}^{[s,t]}$ is a Markov process ([30] Theorem 2.3). In the present paper, we are going to construct a homogeneous quadratic stochastic process by means of a QSO associated with the SIR model.

Let $\left\{P_{ij,k}\right\}$ be a QSO associated with the SIR model. We first define the stochastic matrix as

$$H_{ik}=\sum _{j=1}^d{P}_{ij,k}{x}_j.$$

(16)

In what follows, we take $\mathbf{x}=(1,0,0)$. Then,

$$H=\left[\begin{array}{ccc}1& 0& 0\\ {\displaystyle \frac{1-\beta }{2}}& {\displaystyle \frac{1-\gamma +\beta }{2}}& {\displaystyle \frac{\gamma }{2}}\\ {\displaystyle \frac{1}{2}}& 0& {\displaystyle \frac{1}{2}}\end{array}\right]$$

By the diagonalization theorem, we know $H=PD{P}^{-1}$, where

$$D=\left[\begin{array}{ccc}1& 0& 0\\ 0& {\displaystyle \frac{1}{2}}& 0\\ 0& 0& {\displaystyle \frac{1-\gamma +\beta }{2}}\end{array}\right]$$

and

$$P=\left[\begin{array}{ccc}1& 0& 0\\ 1& {\displaystyle \frac{\gamma }{\gamma -\beta }}& 1\\ 1& 1& 0\end{array}\right]$$

Therefore, $H^t=P{D}^t{P}^{-1}$ yields

$$H^t=\left[\begin{array}{ccc}1& 0& 0\\ 1-{\displaystyle \frac{\gamma -\beta {(1-\gamma +\beta )}^t}{2^t(\gamma -\beta )}}& {\left({\displaystyle \frac{1-\gamma +\beta }{2}}\right)}^t& {\displaystyle \frac{\gamma (1-{(1-\gamma +\beta )}^t)}{2^t(\gamma -\beta )}}\\ {\displaystyle \frac{2^t-1}{2^t}}& 0& {\displaystyle \frac{1}{2^t}}\end{array}\right]$$

(17)

Using this, we construct QSP associated with $H^t$ by

$$P_{ij,k}^{\left(t\right)}=\sum _{m=1}^3{P}_{ij,m}{H}_{m,k}^{(t-1)}.$$

(18)

Thus, we have

$$\begin{array}{c}\hfill P_{11,1}^{\left(t\right)}=1,P_{11,2}^{\left(t\right)}=0,P_{11,3}^{\left(t\right)}=0,P_{13,1}^{\left(t\right)}=1-{\displaystyle \frac{1}{2^t}},P_{13,2}^{\left(t\right)}=0,P_{13,3}^{\left(t\right)}={\displaystyle \frac{1}{2^t}}\end{array}$$

(19)

$$\begin{array}{c}\hfill P_{33,1}^{\left(t\right)}={\displaystyle \frac{2^{t-1}-1}{2^{t-1}}},P_{33,2}^{\left(t\right)}=0,P_{33,3}^{\left(t\right)}={\displaystyle \frac{1}{2^{t-1}}}\end{array}$$

(20)

$$\begin{array}{c}\hfill P_{22,1}^{\left(t\right)}=(1-\gamma )\left(1-{\displaystyle \frac{\gamma -{(1-\gamma +\beta )}^t}{2^t(\gamma -\beta )}}\right)+{\displaystyle \frac{\gamma (2^{t-1}-1)}{2^{t-1}}}\end{array}$$

(21)

$$\begin{array}{c}\hfill P_{22,2}^{\left(t\right)}={\displaystyle \frac{(1-\gamma ){(1-\gamma +\beta )}^{t-1}}{2^{t-1}}}\end{array}$$

(22)

$$\begin{array}{c}\hfill P_{22,3}^{\left(t\right)}={\displaystyle \frac{(1-\gamma )\gamma (1-{(1-\gamma +\beta )}^{t-1})}{2^{t-1}(\gamma -\beta )}}+{\displaystyle \frac{\gamma }{2^{t-1}}}\end{array}$$

(23)

$$\begin{array}{c}\hfill P_{12,1}^{\left(t\right)}={\displaystyle \frac{1-\beta }{2}}+{\displaystyle \frac{1-\gamma +\beta }{2}}\left(1-{\displaystyle \frac{\gamma -\beta {(1-\gamma +\beta )}^{t-1}}{2^{t-1}(\gamma -\beta )}}\right)+{\displaystyle \frac{\gamma (2^{t-1}-1)}{2^t}}\end{array}$$

(24)

$$\begin{array}{c}\hfill P_{12,2}^{\left(t\right)}={\left({\displaystyle \frac{1-\gamma +\beta }{2}}\right)}^t,P_{23,2}^{\left(t\right)}={\displaystyle \frac{(1-\gamma ){(1-\gamma +\beta )}^{t-1}}{2^t}}\end{array}$$

(25)

$$\begin{array}{c}\hfill P_{12,3}^{\left(t\right)}={\displaystyle \frac{1-\gamma +\beta }{2}}\left({\displaystyle \frac{\gamma (1-{(1-\gamma +\beta )}^{t-1})}{2^{t-1}(\gamma -\beta )}}\right)+{\displaystyle \frac{\gamma }{2^t}}\end{array}$$

(26)

$$\begin{array}{c}\hfill P_{23,1}^{\left(t\right)}=\left({\displaystyle \frac{1-\gamma }{2}}\right)\left(1-{\displaystyle \frac{\gamma -\beta {(1-\gamma +\beta )}^{t-1}}{2^{t-1}(\gamma -\beta )}}\right)+\left({\displaystyle \frac{1+\gamma }{2}}\right)\left({\displaystyle \frac{2^{t-1}-1}{2^{t-1}}}\right)\end{array}$$

(27)

$$\begin{array}{c}\hfill P_{23,3}^{\left(t\right)}={\displaystyle \frac{1-\gamma }{2}}\left({\displaystyle \frac{\gamma (1-{(1-\gamma +\beta )}^{t-1})}{2^t(\gamma -\beta )}}\right)+{\displaystyle \frac{1+\gamma }{2^t}}.\end{array}$$

(28)

4. Limiting Behavior of Quadratic Stochastic Processes

The processes will associate with the flow of genetic algebras. Studies have been carried out over certain types of flow of genetic algebras [31,32]. In this part of the article, we will investigate limiting quadratic stochastic operator generated by the SIR model. Furthermore, we want to investigate limiting genetic algebras.

We first calculate the limiting behavior of $H_{m,k}^{(t-1)}$.

Case 1: If $1-\gamma +\beta <2$, then $ast\to \infty$, we have

$$H^{\infty }=\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 1& 0& 0\end{array}\right]$$

Following (18), we calculate $P_{ij,k}^{(\infty )}$ as follows:

$$\begin{array}{ccccc}\hfill p_{11,1}^{\infty }=1,& & \hfill p_{12,1}^{\infty }=1,& & \hfill p_{13,1}^{\infty }=1,\\ \hfill p_{11,2}^{\infty }=0,& & \hfill p_{12,2}^{\infty }=0,& & \hfill p_{13,2}^{\infty }=0,\\ \hfill p_{11,3}^{\infty }=0,& & \hfill p_{12,3}^{\infty }=0,& & \hfill p_{13,3}^{\infty }=0,\\ \hfill p_{22,1}^{\infty }=1,& & \hfill p_{23,1}^{\infty }=1,& & \hfill p_{33,1}^{\infty }=1,\\ \hfill p_{22,2}^{\infty }=0,& & \hfill p_{23,2}^{\infty }=0,& & \hfill p_{33,2}^{\infty }=0,\\ \hfill p_{22,3}^{\infty }=0,& & \hfill p_{23,3}^{\infty }=0,& & \hfill p_{33,3}^{\infty }=0\end{array}$$

Using (5), the table of multiplication associated to $P_{ij,k}^{\infty }$ is given by

Table 1. Table of Multiplication: case 1.

	${\mathbf{e}}_1$	${\mathbf{e}}_2$	${\mathbf{e}}_3$
${\mathbf{e}}_1$	${\mathbf{e}}_1$	${\mathbf{e}}_1$	${\mathbf{e}}_1$
${\mathbf{e}}_2$	${\mathbf{e}}_1$	${\mathbf{e}}_1$	${\mathbf{e}}_1$
${\mathbf{e}}_3$	${\mathbf{e}}_1$	${\mathbf{e}}_1$	${\mathbf{e}}_1$

.

where

$${\mathbf{e}}_1=(1,0,0),{\mathbf{e}}_2=(0,1,0),{\mathbf{e}}_3=(0,0,1)$$

We point out that the condition $1-\gamma +\beta <2$ yields that the heredity coefficients in this regime are not zero. In this regime, we obtain equilibrium point ${\mathbf{e}}_1$.

Case 2: If $1-\gamma +\beta =2$, then $ast\to \infty$, we have

$$H^{\infty }=\left[\begin{array}{ccc}1& 0& 0\\ {\displaystyle \frac{\gamma }{\gamma -\beta }}& 1& {\displaystyle \frac{\gamma }{\beta -\gamma }}\\ 1& 0& 0\end{array}\right]$$

Hence, one calculates $P_{ij,k}^{(\infty )}$ as follows:

$$\begin{array}{ccccc}\hfill p_{11,1}^{\infty }=1,& & \hfill p_{12,1}^{\infty }={\displaystyle \frac{(\gamma -\beta )(1-\beta +\gamma )+2\gamma }{2(\gamma -\beta )}},& & \hfill p_{13,1}^{\infty }=1,\\ \hfill p_{11,2}^{\infty }=0,& & \hfill p_{12,2}^{\infty }=1,& & \hfill p_{13,2}^{\infty }=0,\\ \hfill p_{11,3}^{\infty }=0,& & \hfill p_{12,3}^{\infty }={\displaystyle \frac{\gamma }{\beta -\gamma }},& & \hfill p_{13,3}^{\infty }=0,\\ \hfill p_{22,1}^{\infty }=1+{\displaystyle \frac{(1-\gamma )\beta }{(\gamma -\beta )}},& & \hfill p_{23,1}^{\infty }={\displaystyle \frac{\gamma (1-\gamma )}{2(\beta -\gamma )}}+{\displaystyle \frac{1+\gamma }{2}},& & \hfill p_{33,1}^{\infty }=1,\\ \hfill p_{22,2}^{\infty }=1-\gamma ,& & \hfill p_{23,2}^{\infty }={\displaystyle \frac{1-\gamma }{2}},& & \hfill p_{33,2}^{\infty }=0,\\ \hfill p_{22,3}^{\infty }={\displaystyle \frac{\gamma (1-\gamma )}{(\beta -\gamma )}},& & \hfill p_{23,3}^{\infty }=\left({\displaystyle \frac{1-\gamma }{2}}\right)\left({\displaystyle \frac{\gamma }{\beta -\gamma }}\right),& & \hfill p_{33,3}^{\infty }=0\end{array}$$

From Proposition (1), we have $\gamma =0,\beta =1$, which reduces the above to the following

$$\begin{array}{ccccc}\hfill p_{11,1}^{\infty }=1,& & \hfill p_{12,1}^{\infty }=0,& & \hfill p_{13,1}^{\infty }=1,\\ \hfill p_{11,2}^{\infty }=0,& & \hfill p_{12,2}^{\infty }=1,& & \hfill p_{13,2}^{\infty }=0,\\ \hfill p_{11,3}^{\infty }=0,& & \hfill p_{12,3}^{\infty }=0,& & \hfill p_{13,3}^{\infty }=0,\\ \hfill p_{22,1}^{\infty }=0,& & \hfill p_{23,1}^{\infty }={\displaystyle \frac{1}{2}},& & \hfill p_{33,1}^{\infty }=1,\\ \hfill p_{22,2}^{\infty }=1,& & \hfill p_{23,2}^{\infty }={\displaystyle \frac{1}{2}},& & \hfill p_{33,2}^{\infty }=0,\\ \hfill p_{22,3}^{\infty }=0,& & \hfill p_{23,3}^{\infty }=0,& & \hfill p_{33,3}^{\infty }=0\end{array}$$

Using (5), the table of multiplication associated with $P_{ij,k}^{\infty }$ is given by

Table 2. Table of Multiplication: case 2.

	${\mathbf{e}}_1$	${\mathbf{e}}_2$	${\mathbf{e}}_3$
${\mathbf{e}}_1$	${\mathbf{e}}_1$	${\mathbf{e}}_2$	${\mathbf{e}}_1$
${\mathbf{e}}_2$	${\mathbf{e}}_2$	${\mathbf{e}}_2$	${\displaystyle \frac{1}{2}}{e}_1+{\displaystyle \frac{1}{2}}{e}_2$
${\mathbf{e}}_3$	${\mathbf{e}}_1$	${\displaystyle \frac{1}{2}}{\mathbf{e}}_1+{\displaystyle \frac{1}{2}}{\mathbf{e}}_2$	${\mathbf{e}}_1$

.

In the considered setting, we have $\gamma =0$. This means that the heredity coefficient from $\{1,2\}$ to 3 is always 0, which is naturally expected. The condition $\beta =1$ implies $p_{12,2}=1$, while $p_{12,1}=0$. In this regime, we find two equilibrium points ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$.

5. Limiting Genetic Algebras

The following section will examine some of the features of genetic algebra provided by Table 1 and Table 2. These algebras, which are connected to quadratic stochastic processes defined by the $SIR$ model, are termed as $\mathit{limiting}\mathit{genetic}\mathit{algebras}$.

5.1. Derivations

The derivations of limiting genetic algebras pertaining to quadratic stochastic processes are discussed in this subsection. It is worthwhile to recollect that a derivation on algebra $(\mathcal{A},\circ )$ is defined as a linear mapping $D:\mathcal{A}\to \mathcal{A}$ such that

$$D(u\circ v)=D\left(u\right)\circ v+u\circ D\left(v\right)$$

for all $u,v\in \mathcal{A}$.

It is important to mention that $D\equiv 0$ is also a derivation, and such derivation is known as a trivial one. The set of all derivations of $\mathcal{A}$ is denoted $Der\left(\mathcal{A}\right)$. It is crucial to determine if the provided algebra possess a derivation that is non-trivial. The genetic interpretation of derivations was thoroughly examined in [12]. The mapping d is said to be a derivation if and only if

$$D({\mathbf{e}}_i\circ {\mathbf{e}}_j)=D\left({\mathbf{e}}_i\right)\circ {\mathbf{e}}_j+{\mathbf{e}}_i\circ D\left({\mathbf{e}}_j\right)$$

(29)

For comprehension of algebraic derivations, we validate the validity of (29). Suppose

$$D\left({\mathbf{e}}_i\right)=\sum _{j=1}^3{d}_{i,j}{\mathbf{e}}_j,i\in 1,2,3$$

(30)

for some matrix $\left(d_{ij}\right)$.

Case 1: Let $1-\gamma +\beta <2$, for $i=j=1$; we have

$$d\left({\mathbf{e}}_1\right)=2d\left({\mathbf{e}}_1\right)\circ {\mathbf{e}}_1$$

which yields

$$d_{11}{\mathbf{e}}_1+d_{21}{\mathbf{e}}_2+d_{31}{\mathbf{e}}_3=2(d_{11}{\mathbf{e}}_1+d_{21}{\mathbf{e}}_2+d_{31}{\mathbf{e}}_3)\circ {\mathbf{e}}_1$$

$$d_{11}{\mathbf{e}}_1+d_{21}{\mathbf{e}}_2+d_{31}{\mathbf{e}}_3=2(d_{11}{\mathbf{e}}_1+d_{21}{\mathbf{e}}_1+d_{31}{\mathbf{e}}_1)$$

$$d_{11}{\mathbf{e}}_1+d_{21}{\mathbf{e}}_2+d_{31}{\mathbf{e}}_3=2{\mathbf{e}}_1(d_{11}+d_{21}+d_{31})$$

Comparing coefficients of ${\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3$, we obtain

$$d_{21}=d_{31}=0$$

$$d_{11}=2d_{11}$$

$$d_{11}=0$$

Similarly, by considering other cases we deduce the subsequent set of equations.

$$d_{21}=d_{31}=d_{11}=0$$

$$d_{12}+d_{22}+d_{32}=0$$

$$d_{13}+d_{23}+d_{33}=0$$

$$d_{11}=2(d_{12}+d_{22}+d_{32})$$

$$d_{11}=d_{12}+d_{22}+d_{32}+d_{13}+d_{23}+d_{33}$$

$$d_{11}=2(d_{13}+d_{23}+d_{33})$$

Therefore,

$$d_{12}=-d_{22}-d_{32}$$

$$d_{13}=-d_{23}-d_{33}$$

Correspondingly, one finds

$$D=\left[\begin{array}{ccc}0& -\alpha -\beta & -\gamma -\delta \\ 0& \alpha & \gamma \\ 0& \beta & \delta \end{array}\right],\alpha ,\beta ,\gamma ,\delta \mathrm{is}\mathrm{arbitrary}.$$

(31)

Case 2: Let $1-\gamma +\beta =2$, for $i=2,j=2$; we have

$$d\left({\mathbf{e}}_2\right)=2d\left({\mathbf{e}}_2\right)\circ {\mathbf{e}}_2$$

which yields

$$d_{12}{\mathbf{e}}_1+d_{22}{\mathbf{e}}_2+d_{32}{\mathbf{e}}_3=2(d_{12}{\mathbf{e}}_1+d_{22}{\mathbf{e}}_2+d_{32}{\mathbf{e}}_3){\mathbf{e}}_2$$

$$d_{12}{\mathbf{e}}_1+d_{22}{\mathbf{e}}_2+d_{32}{\mathbf{e}}_3=2(d_{12}{\mathbf{e}}_2+d_{22}{\mathbf{e}}_2+d_{32}({\displaystyle \frac{1}{2}}{\mathbf{e}}_1+{\displaystyle \frac{1}{2}}{\mathbf{e}}_2))$$

Comparing coefficients of ${\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3$, we obtain

$$d_{32}=0$$

$$d_{22}{\mathbf{e}}_2=(2d_{22}+d_{32}){\mathbf{e}}_2$$

$$d_{22}=2d_{22}+d_{32}$$

$$d_{22}=2d_{22}$$

$$d_{22}=0$$

$$d_{12}{\mathbf{e}}_1=(2d_{12}+d_{32}){\mathbf{e}}_1$$

$$d_{12}=2d_{12}+d_{32}$$

$$d_{12}=2d_{12}$$

$$d_{12}=0$$

Similarly, by considering other cases we obtain the following,

$$d_{ij}=0\forall i,j=1,2,3$$

Correspondingly, one finds

$$D=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$$

(32)

Let us summarize the result.

Theorem 1.

Let A be the limiting genetic algebra, then the following assertions are valid:

(i) 

if $1-\gamma +\beta =2$, consequently the derivations reduce to a trivial case.

(ii) 

if $1-\gamma +\beta <2$, under this condition, the derivation is non-trivial, offered by

$$D=\left[\begin{array}{ccc}0& -\alpha -\beta & -\gamma -\delta \\ 0& \alpha & \gamma \\ 0& \beta & \delta \end{array}\right],\alpha ,\beta ,\gamma ,\delta \mathit{is}\mathit{arbitrary}.$$

5.2. Automorphisms

In this subsection, we evaluate the automorphisms of the limiting genetic algebras. We acknowledge that a automorphisms on algebra $(\mathcal{A},\circ )$ is defined as a one-to-one linear mapping $\alpha :\mathcal{A}\to \mathcal{A}$ such that

$$\alpha (u\circ v)=\alpha \left(u\right)\circ v+u\circ \alpha \left(v\right)$$

for all $u,v\in \mathcal{A}$. It is clear that $\alpha =I_d$ is a trivial automorphism. The collection of all automorphisms of $\mathcal{A}$ is represented by $Aut\left(\mathcal{A}\right)$. It is to be noted that $\alpha$ is an automorphism if and only if

$$\alpha ({\mathbf{e}}_i\circ {\mathbf{e}}_j)=\alpha \left({\mathbf{e}}_i\right)\circ \alpha \left({\mathbf{e}}_j\right)$$

Case 1: If $1-\gamma +\beta <2$, consequently we derive the corresponding set of equations

$${\alpha }_{31}=0,{\alpha }_{21}=0$$

$${\alpha }_{11}={\alpha }_{11}^2\Rightarrow {\alpha }_{11}=0,1$$

$${\alpha }_{11}={\alpha }_{11}({\alpha }_{12}+{\alpha }_{22}+{\alpha }_{32})$$

$${\alpha }_{11}={\alpha }_{11}({\alpha }_{13}+{\alpha }_{23}+{\alpha }_{33})$$

$${\alpha }_{11}={\alpha }_{11}{({\alpha }_{12}+{\alpha }_{22}+{\alpha }_{32})}^2$$

$${\alpha }_{11}={\alpha }_{11}({\alpha }_{12}+{\alpha }_{22}+{\alpha }_{32})({\alpha }_{13}+{\alpha }_{23}+{\alpha }_{33})$$

$${\alpha }_{11}={\alpha }_{11}({\alpha }_{13}+{\alpha }_{23}+{\alpha }_{33}^2)$$

(1) If ${\alpha }_{11}=0$

$$\Rightarrow {\alpha }_{21}={\alpha }_{31}=0$$

$$\Rightarrow ({\alpha }_{12}+{\alpha }_{22}+{\alpha }_{32})=0$$

$$\Rightarrow ({\alpha }_{13}+{\alpha }_{23}+{\alpha }_{33})=0$$

Therefore, we have

$$A=\left[\begin{array}{ccc}0& -\alpha -\beta & -\gamma -\sigma \\ 0& \alpha & \gamma \\ 0& \beta & \sigma \end{array}\right],\alpha ,\beta ,\gamma ,\sigma \mathrm{is}\mathrm{arbitrary}.$$

(33)

This is a singular matrix, hence not an automorphism.

(2) If ${\alpha }_{11}=1$

$$\Rightarrow {\alpha }_{21}={\alpha }_{31}=0$$

$$\Rightarrow ({\alpha }_{12}+{\alpha }_{22}+{\alpha }_{32})=1$$

$$\Rightarrow ({\alpha }_{13}+{\alpha }_{23}+{\alpha }_{33})=1$$

Therefore, we have

$$A=\left[\begin{array}{ccc}1& 1-\alpha -\beta & 1-\gamma -\sigma \\ 0& \alpha & \gamma \\ 0& \beta & \sigma \end{array}\right],\alpha ,\beta ,\gamma ,\sigma \mathrm{is}\mathrm{arbitrary}.$$

(34)

Case 2:$1-\gamma +\beta =2$, which yields the succeeding set of equations

$${\alpha }_{31}=0$$

$${\alpha }_{21}=2{\alpha }_{11}{\alpha }_{21}+{\alpha }_{21}^2$$

$${\alpha }_{11}={\alpha }_{11}^2\Rightarrow {\alpha }_{11}=0,1$$

$${\alpha }_{32}=0$$

$${\alpha }_{22}={\alpha }_{11}{\alpha }_{22}+{\alpha }_{21}{\alpha }_{12}+{\alpha }_{21}{\alpha }_{22}$$

$${\alpha }_{12}={\alpha }_{11}{\alpha }_{12}$$

$${\alpha }_{21}={\alpha }_{11}{\alpha }_{23}+{\alpha }_{21}{\alpha }_{13}+{\alpha }_{21}{\alpha }_{23}+{\displaystyle \frac{1}{2}}{\alpha }_{21}{\alpha }_{33}$$

$${\alpha }_{11}={\alpha }_{11}{\alpha }_{13}+{\alpha }_{11}{\alpha }_{33}+{\displaystyle \frac{1}{2}}{\alpha }_{21}{\alpha }_{33}$$

$${\alpha }_{22}=2{\alpha }_{12}{\alpha }_{22}+{\alpha }_{22}^2$$

$${\alpha }_{12}={\alpha }_{12}^2\Rightarrow {\alpha }_{12}=0,1$$

$${\displaystyle \frac{1}{2}}{\alpha }_{21}+{\displaystyle \frac{1}{2}}{\alpha }_{22}={\alpha }_{12}{\alpha }_{23}+{\alpha }_{22}{\alpha }_{13}+{\alpha }_{22}{\alpha }_{23}+{\displaystyle \frac{1}{2}}{\alpha }_{22}{\alpha }_{33}$$

$${\displaystyle \frac{1}{2}}{\alpha }_{11}+{\displaystyle \frac{1}{2}}{\alpha }_{12}={\alpha }_{12}{\alpha }_{13}+{\alpha }_{12}{\alpha }_{33}+{\displaystyle \frac{1}{2}}{\alpha }_{22}{\alpha }_{33}$$

$${\alpha }_{21}=2{\alpha }_{13}{\alpha }_{23}+{\alpha }_{23}^2+{\alpha }_{23}{\alpha }_{33}$$

$${\alpha }_{11}={\alpha }_{13}^2+2{\alpha }_{13}{\alpha }_{33}+{\alpha }_{23}{\alpha }_{33}+{\alpha }_{33}^2$$

(1) If ${\alpha }_{11}=0$ and ${\alpha }_{12}=0$, then we have

$$\alpha =\left[\begin{array}{ccc}0& 0& \sqrt{-\gamma \sigma }-\sigma \\ [2\gamma (\sqrt{-\gamma \sigma }-\sigma )+{\gamma }^2+\gamma \sigma ]& \alpha & \gamma \\ 0& 0& \sigma \end{array}\right],\gamma ,\sigma \mathrm{is}\mathrm{arbitrary}.$$

(35)

This is a singular matrix, hence not an automorphism.

(2) If ${\alpha }_{11}=1$ and ${\alpha }_{12}=0$, then we have

$$\alpha =\left[\begin{array}{ccc}1& 0& 1-\sigma \\ 0& \alpha & 0\\ 0& 0& \sigma \end{array}\right],\alpha ,\sigma \mathrm{is}\mathrm{arbitrary}.$$

(36)

(3) If ${\alpha }_{11}=1$ and ${\alpha }_{12}=1$, then we have

$$\alpha =\left[\begin{array}{ccc}1& 1& 1-\sigma \\ 0& 0& 0\\ 0& 0& \sigma \end{array}\right],\sigma \mathrm{is}\mathrm{arbitrary}.$$

(37)

This is a singular matrix, hence not an automorphism.

Theorem 2.

Let A be the limiting genetic algebra, then the succeeding claims hold.

(i) 

If $1-\gamma +\beta <2$, then the non-trivial automorphism is given as

$$A=\left[\begin{array}{ccc}1& 1-\alpha -\beta & 1-\gamma -\sigma \\ 0& \alpha & \gamma \\ 0& \beta & \sigma \end{array}\right],\alpha ,\beta ,\gamma ,\sigma \mathit{is}\mathit{arbitrary}.$$

(ii) 

If $1-\gamma +\beta =2$, then there is a non-trivial automorphism given by

$$\alpha =\left[\begin{array}{ccc}1& 0& 1-\sigma \\ 0& \alpha & 0\\ 0& 0& \sigma \end{array}\right],\alpha ,\sigma \mathit{is}\mathit{arbitrary}.$$

We define the set of exponents of all derivations of $\mathcal{A}$ as

$$Exp\left(D\right)=\left\{e^d\right|d\in Der\left(\mathcal{A}\right)\}$$

It is well known $Exp\left(D\right)\subset Aut\left(\mathcal{A}\right)$. We are interested whether in this inclusion is strict.

Theorem 3.

The set $Exp\left(D\right)$ is a proper subset of $Aut\left(\mathcal{A}\right)$.

Proof. 

If $1-\gamma +\beta =2$, then this statement is obvious. Therefore, we assume $1-\gamma +\beta <2$. By Theorems 1 and 2, we have

$$D=\left[\begin{array}{ccc}0& -\alpha -\beta & -\gamma -\delta \\ 0& \alpha & \gamma \\ 0& \beta & \delta \end{array}\right],\alpha ,\beta ,\gamma ,\delta \mathrm{is}\mathrm{arbitrary}.$$

$$\forall d\in D,Spec\left(e^d\right)=\{1,e^{{\lambda }_1},e^{{\lambda }_2}\}$$

and

$$\alpha =\left[\begin{array}{ccc}1& 1-\alpha -\beta & 1-\gamma -\sigma \\ 0& \alpha & \gamma \\ 0& \beta & \sigma \end{array}\right],\alpha ,\beta ,\gamma ,\delta \mathrm{is}\mathrm{arbitrary}.$$

$$\forall \alpha \in Aut\left(\mathcal{A}\right),Spec\left(\alpha \right)=\{1,{\lambda }_1,{\lambda }_2\}$$

Here ${\lambda }_1$ and ${\lambda }_2$ are the eigenvalues given by

$${\lambda }_1={\displaystyle \frac{(a+d)+\sqrt{a^2-2ad+4bc+d^2}}{2}}$$

and

$${\lambda }_2={\displaystyle \frac{(a+d)-\sqrt{a^2-2ad+4bc+d^2}}{2}}$$

Since $e^{{\lambda }_i}\ge 0but{\lambda }_i\in \mathbb{R}\forall i=1,2$ it implies that

$$Exp\left(D\right)\subset Aut\left(\mathcal{A}\right)$$

□

6. Rota–Baxter Operator

As a solution to a probability problem arising from the theory of associative algebras, G. Baxter originally proposed Rota–Baxter operators [33]. It was thereafter the subject of extensive research in numerous mathematical domains such as combinatorics, quantum field theory, lie algebra, integrable systems, and the flow of quantum genetic Lotka–Volterra algebras [32].

In the following section, we dive into Rota–Baxter operators defined on limiting genetic algebra. Recall that a linear operator $\mathcal{R}:\mathcal{A}\to \mathcal{A}$, called a Rota–Baxter operator if

$$\mathcal{R}\left(x\right)\mathcal{R}\left(y\right)=\mathcal{R}\left[\mathcal{R}\right(x\left)\right(y)+(x\left)\mathcal{R}\right(y)+\lambda (x\circ y\left)\right]$$

The pair ($\mathcal{A},R$) is called a Rota–Baxter algebra of weight $\lambda$. Its well-known weight $\lambda$ can be 0 or 1.

In order to simplify, we describe the Rota–Baxter operators on basis elements as

$$\mathcal{R}\left({\mathbf{e}}_i\right)\mathcal{R}\left({\mathbf{e}}_j\right)=\mathcal{R}[\mathcal{R}\left({\mathbf{e}}_i\right)\left({\mathbf{e}}_j\right)+\left({\mathbf{e}}_i\right)\mathcal{R}\left({\mathbf{e}}_j\right)+\lambda ({\mathbf{e}}_i\circ {\mathbf{e}}_j)]$$

where

$$\mathcal{R}\left({\mathbf{e}}_k\right)=\sum _{j=1}^3{r}_{kj}{\mathbf{e}}_{j}wherek=1,2,3.$$

Case 1: $1-\gamma +\beta <2$, we deduce the below system of equations

$${(r_{11}+r_{12}+r_{13})}^2=r_{11}[\lambda +2(r_{11}+r_{12}+r_{13})]$$

$$0=r_{12}[\lambda +2(r_{11}+r_{12}+r_{13})]$$

$$0=r_{13}[\lambda +2(r_{11}+r_{12}+r_{13})]$$

$${(r_{21}+r_{22}+r_{23})}^2=r_{11}[\lambda +2(r_{21}+r_{22}+r_{23})]$$

$$0=r_{12}[\lambda +2(r_{21}+r_{22}+r_{23})]$$

$$0=r_{13}[\lambda +2(r_{21}+r_{22}+r_{23})]$$

$${(r_{31}+r_{32}+r_{33})}^2=r_{11}[\lambda +2(r_{31}+r_{32}+r_{33})]$$

$$0=r_{12}[\lambda +2(r_{31}+r_{32}+r_{33})]$$

$$0=r_{13}[\lambda +2(r_{31}+r_{32}+r_{33})]$$

$$(r_{11}+r_{12}+r_{13})(r_{21}+r_{22}+r_{23})=r_{11}[\lambda +(r_{11}+r_{12}+r_{13})+(r_{21}+r_{22}+r_{23})]$$

$$0=r_{12}[\lambda +(r_{11}+r_{12}+r_{13})+(r_{21}+r_{22}+r_{23})]$$

$$0=r_{13}[\lambda +(r_{11}+r_{12}+r_{13})+(r_{21}+r_{22}+r_{23})]$$

$$(r_{11}+r_{12}+r_{13})(r_{31}+r_{32}+r_{33})=r_{11}[\lambda +(r_{11}+r_{12}+r_{13})+(r_{31}+r_{32}+r_{33})]$$

$$0=r_{12}[\lambda +(r_{11}+r_{12}+r_{13})+(r_{31}+r_{32}+r_{33})]$$

$$0=r_{13}[\lambda +(r_{11}+r_{12}+r_{13})+(r_{31}+r_{32}+r_{33})]$$

$$(r_{21}+r_{22}+r_{23})(r_{31}+r_{32}+r_{33})=r_{11}[\lambda +(r_{21}+r_{22}+r_{23})+(r_{31}+r_{32}+r_{33})]$$

$$0=r_{12}[\lambda +(r_{21}+r_{22}+r_{23})+(r_{31}+r_{32}+r_{33})]$$

$$0=r_{13}[\lambda +(r_{21}+r_{22}+r_{23})+(r_{31}+r_{32}+r_{33})]$$

Case: $\lambda =0$. Under this condition, we obtain the subsequent set of equations

$${(r_{11}+r_{12}+r_{13})}^2=2r_{11}(r_{11}+r_{12}+r_{13})$$

$$0=2r_{12}(r_{11}+r_{12}+r_{13})$$

$$0=2r_{13}(r_{11}+r_{12}+r_{13})$$

$${(r_{21}+r_{22}+r_{23})}^2=2r_{11}(r_{21}+r_{22}+r_{23})$$

$$0=2r_{12}(r_{21}+r_{22}+r_{23})$$

$$0=2r_{13}(r_{21}+r_{22}+r_{23})$$

$${(r_{31}+r_{32}+r_{33})}^2=2r_{11}(r_{31}+r_{32}+r_{33})$$

$$0=2r_{12}(r_{31}+r_{32}+r_{33})$$

$$0=2r_{13}(r_{31}+r_{32}+r_{33})$$

$$(r_{11}+r_{12}+r_{13})(r_{21}+r_{22}+r_{23})=r_{11}[(r_{11}+r_{12}+r_{13})+(r_{21}+r_{22}+r_{23})]$$

$$0=r_{12}[(r_{11}+r_{12}+r_{13})+(r_{21}+r_{22}+r_{23})]$$

$$0=r_{13}[(r_{11}+r_{12}+r_{13})+(r_{21}+r_{22}+r_{23})]$$

$$(r_{11}+r_{12}+r_{13})(r_{31}+r_{32}+r_{33})=r_{11}[(r_{11}+r_{12}+r_{13})+(r_{31}+r_{32}+r_{33})]$$

$$0=r_{12}[(r_{11}+r_{12}+r_{13})+(r_{31}+r_{32}+r_{33})]$$

$$0=r_{13}[(r_{11}+r_{12}+r_{13})+(r_{31}+r_{32}+r_{33})]$$

$$(r_{21}+r_{22}+r_{23})(r_{31}+r_{32}+r_{33})=r_{11}[(r_{21}+r_{22}+r_{23})+(r_{31}+r_{32}+r_{33})]$$

$$0=r_{12}[(r_{21}+r_{22}+r_{23})+(r_{31}+r_{32}+r_{33})]$$

$$0=r_{13}[(r_{21}+r_{22}+r_{23})+(r_{31}+r_{32}+r_{33})]$$

Let us examine a few instances:

Sub case 1: When $r_{11}=r_{12}=r_{13}=0$, then

$$\begin{array}{c}\hfill r_{21}+r_{22}+r_{23}\ne 0\\ \hfill r_{31}+r_{32}+r_{33}\ne 0\end{array}$$

Hence, we have

$$\mathcal{R}=\left[\begin{array}{ccc}0& 0& 0\\ \alpha & \beta & \gamma \\ \eta & \delta & \xi \end{array}\right],\alpha ,\beta ,\gamma ,\delta ,\xi ,\eta \mathrm{is}\mathrm{arbitrary}.$$

Sub case 2: When one of them is not zero, then

$$r_{11}^2+r_{12}^2+r_{13}^2\ne 0$$

We have

$$\begin{array}{c}\hfill r_{21}+r_{22}+r_{23}=0\\ \hfill r_{31}+r_{32}+r_{33}=0\end{array}$$

Hence, we have

$$\mathcal{R}=\left[\begin{array}{ccc}-\alpha -\beta & \alpha & \beta \\ -\gamma -\delta & \gamma & \delta \\ -\xi -\eta & \xi & \eta \end{array}\right],\alpha ,\beta ,\gamma ,\delta ,\xi ,\eta \mathrm{is}\mathrm{arbitrary}.$$

Sub case 3: When $r_{11}=0$ and $r_{12}\ne 0r_{13}\ne 0$. Hence, we have

$$\mathcal{R}=\left[\begin{array}{ccc}0& \alpha & \beta \\ -\gamma -\delta & \gamma & \delta \\ -\xi -\eta & \xi & \eta \end{array}\right],\alpha ,\beta ,\gamma ,\delta ,\xi ,\eta \mathrm{is}\mathrm{arbitrary}.$$

Case: $\lambda =1$. In this case, the system is reduced to the following

$${(r_{11}+r_{12}+r_{13})}^2=r_{11}[1+2(r_{11}+r_{12}+r_{13})]$$

$$0=r_{12}[1+2(r_{11}+r_{12}+r_{13})]$$

$$0=r_{13}[1+2(r_{11}+r_{12}+r_{13})]$$

$${(r_{21}+r_{22}+r_{23})}^2=r_{11}[1+2(r_{21}+r_{22}+r_{23})]$$

$$0=r_{12}[1+2(r_{21}+r_{22}+r_{23})]$$

$$0=r_{13}[1+2(r_{21}+r_{22}+r_{23})]$$

$${(r_{31}+r_{32}+r_{33})}^2=r_{11}[1+2(r_{31}+r_{32}+r_{33})]$$

$$0=r_{12}[1+2(r_{31}+r_{32}+r_{33})]$$

$$0=r_{13}[1+2(r_{31}+r_{32}+r_{33})]$$

$$(r_{11}+r_{12}+r_{13})(r_{21}+r_{22}+r_{23})=r_{11}[1+(r_{11}+r_{12}+r_{13})+(r_{21}+r_{22}+r_{23})]$$

$$0=r_{12}[1+(r_{11}+r_{12}+r_{13})+(r_{21}+r_{22}+r_{23})]$$

$$0=r_{13}[1+(r_{11}+r_{12}+r_{13})+(r_{21}+r_{22}+r_{23})]$$

$$(r_{11}+r_{12}+r_{13})(r_{31}+r_{32}+r_{33})=r_{11}[1+(r_{11}+r_{12}+r_{13})+(r_{31}+r_{32}+r_{33})]$$

$$0=r_{12}[1+(r_{11}+r_{12}+r_{13})+(r_{31}+r_{32}+r_{33})]$$

$$0=r_{13}[1+(r_{11}+r_{12}+r_{13})+(r_{31}+r_{32}+r_{33})]$$

$$(r_{21}+r_{22}+r_{23})(r_{31}+r_{32}+r_{33})=r_{11}[1+(r_{21}+r_{22}+r_{23})+(r_{31}+r_{32}+r_{33})]$$

$$0=r_{12}[1+(r_{21}+r_{22}+r_{23})+(r_{31}+r_{32}+r_{33})]$$

$$0=r_{13}[1+(r_{21}+r_{22}+r_{23})+(r_{31}+r_{32}+r_{33})]$$

Let us consider several cases:

Sub-case 1: When $r_{11}=0$

$$\begin{array}{c}\hfill r_{12}=0,r_{13}=0\\ \hfill r_{21}+r_{22}+r_{23}=0\\ \hfill r_{31}+r_{32}+r_{33}=0\end{array}$$

$$\mathcal{R}=\left[\begin{array}{ccc}0& 0& 0\\ -\alpha -\beta & \alpha & \beta \\ -\gamma -\delta & \gamma & \delta \end{array}\right],\alpha ,\beta ,\gamma ,\delta \mathrm{is}\mathrm{arbitrary}.$$

Sub-case 2: When $r_{11}=-1$

$$\begin{array}{c}\hfill r_{12}=0,r_{13}=0\\ \hfill r_{21}+r_{22}+r_{23}=-1\\ \hfill r_{31}+r_{32}+r_{33}=-1\end{array}$$

$$\mathcal{R}=\left[\begin{array}{ccc}-1& 0& 0\\ -1-\alpha -\beta & \alpha & \beta \\ -1-\gamma -\delta & \gamma & \delta \end{array}\right],\alpha ,\beta ,\gamma ,\delta \mathrm{is}\mathrm{arbitrary}.$$

Let us formulate the following theorem:

Theorem 4.

Let A be the limiting genetic algebra such that $1-\gamma +\beta <2$, then the following statements hold.

(1) 

Let $\lambda =0$, then Rota–Baxter operator is given by

(i) 

if $r_{11}=r_{12}=r_{13}=0$, then

$$\mathcal{R}=\left[\begin{array}{ccc}0& 0& 0\\ \alpha & \beta & \gamma \\ \eta & \delta & \xi \end{array}\right],\alpha ,\beta ,\gamma ,\delta ,\xi ,\eta \mathit{is}\mathit{arbitrary}.$$

(ii) 

if $r_{11}^2+r_{12}^2+r_{13}^2\ne 0$, then

$$\mathcal{R}=\left[\begin{array}{ccc}-\alpha -\beta & \alpha & \beta \\ -\gamma -\delta & \gamma & \delta \\ -\xi -\eta & \xi & \eta \end{array}\right],\alpha ,\beta ,\gamma ,\delta ,\xi ,\eta \mathit{is}\mathit{arbitrary}.$$

(iii) 

if $r_{11}=0$ and $r_{12}\ne 0r_{13}\ne 0$, then

$$\mathcal{R}=\left[\begin{array}{ccc}0& \alpha & \beta \\ -\gamma -\delta & \gamma & \delta \\ -\xi -\eta & \xi & \eta \end{array}\right],\alpha ,\beta ,\gamma ,\delta ,\xi ,\eta \mathit{is}\mathit{arbitrary}.$$

(2) 

Let $\lambda =1$, then Rota–Baxter operator takes the following form

(i) 

if $r_{11}=1$, then

$$\mathcal{R}=\left[\begin{array}{ccc}-1& 0& 0\\ -1-\alpha -\beta & \alpha & \beta \\ -1-\gamma -\delta & \gamma & \delta \end{array}\right],\alpha ,\beta ,\gamma ,\delta \mathit{is}\mathit{arbitrary}.$$

(ii) 

if $r_{11}=-1$, then

$$\mathcal{R}=\left[\begin{array}{ccc}-1& 0& 0\\ -1-\alpha -\beta & \alpha & \beta \\ -1-\gamma -\delta & \gamma & \delta \end{array}\right],\alpha ,\beta ,\gamma ,\delta \mathit{is}\mathit{arbitrary}.$$

7. Conclusions

In the current paper, we have proposed a construction of quadratic stochastic processes by means of parameters of the susceptible–infected–removed (SIR) model. Furthermore, the limiting behavior of the resulting quadratic stochastic processes were investigated. This led us to explore properties of the limiting genetic algebras like derivations and automorphisms. It is worthwhile to observe the set of exponents of all derivations of the algebra is a proper subset of the set of all automorphims of the algebra. Furthermore, the Rota–Baxter operator was studied using two cases of parameter $\lambda$.