Stability Analysis of a Mathematical Model for Infection Diseases with Stochastic Perturbations

Abstract

A well-known model of infectious diseases, described by a nonlinear system of delay differential equations, is investigated under the influence of stochastic perturbations. Using the general method of Lyapunov functional construction combined with the linear matrix inequality (LMI) approach, we derive sufficient conditions for the stability of the equilibria of the considered system. Numerical simulations illustrating the system’s behavior under stochastic perturbations are provided to support the thoretical findings. The proposed method for stability analysis is broadly applicable to other systems of nonlinear stochastic differential equations across various fields.

1. Introduction

The study of mathematical model of infectious diseases has a long history (see, for instance, [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45]) and remains highly relevant today, as it enables the derivation of results that closely reflect real-world processes.

In this article, we examine the spread of an infectious disease from a clinical rather than an epidemiological sense. To provide a mathematical explanation of the disease’s dynamics, it is necessary to first understand its behavior in a therapeutic context. Based on the two primary approaches to the human immune response (humoral and cellular), two corresponding types of mathematical models are developed to describe these mechanisms.

The mechanism of infectious diseases is based on the interaction between antigens (virus) and antibodies, which are represented in the model by the functions $V\left(t\right)$ and $F\left(t\right)$, respectively. Antibodies are understood as components of the immune system (such as immunoglobulins and cell receptors) that neutralize antigens. The population of cells respinsible for carrying and producing antibodies (i.e., immunocompetent cells and immunoglobulin-producing cells) is denoted by $C\left(t\right)$. The function $m\left(t\right)$ represents the extent of organ damage caused by the infection.

First, an antibody binds to an antigen, forming antibody–antigen complexes. In proportion to the quantity of these complexes, plasma cells are produced in the organism after a delay of $\tau$ leading to the mass production of antibodies. The process is described in the first term of the second equation in the system.

If we aim to predict the individual course and outcome of the disease, the basic mathematical model proves inadequate. This is because the model contains unknown parameters, which require a greater number of conditions to generate a reliable solution. Even with daily measurements, which is often impractical, it would still take days to collect sufficient data. By that time, the progression and outcome of the disease would likely be evident without the need for a mathematical model. To address this challenge, the model is individualized through the use of Wiener process-based uncertainty.

The proposed mathematical model of infectious diseases is described by the following system of delay differential equations [39,44]

$$\begin{array}{c}\dot{V}\left(t\right)=\beta V\left(t\right)-\gamma F\left(t\right)V\left(t\right),\\ \dot{C}\left(t\right)=\xi \left(m\right)\alpha F(t-\tau )V(t-\tau )-{\mu }_c(C\left(t\right)-C^{*}),\\ \dot{F}\left(t\right)=\rho C\left(t\right)-\eta \gamma F\left(t\right)V\left(t\right)-{\mu }_{f}F\left(t\right),\\ \dot{m}\left(t\right)=\sigma V\left(t\right)-{\mu }_{m}m\left(t\right),\end{array}$$

(1)

where $\beta ,\gamma ,\alpha ,\rho ,\eta ,\sigma ,{\mu }_c,{\mu }_f,{\mu }_m,C^{*}$ are positive parameters; $\xi \left(m\right)$ is a body health function defined bellow; $\tau$ represents the immune reaction delay time; and the initial conditions are given by

$$C\left(0\right)=C_0,m\left(0\right)=m_0,V\left(\theta \right)=V_0\left(\theta \right),F\left(\theta \right)=F_0\left(\theta \right),\theta \in [-\tau ,0].$$

(2)

The first equation of system (1) describes the dynamics of the virus with the following parameters:

-

$\beta$—coefficient describing the antigen activity;

-

$\gamma$—the antigen neutralizing factor.

The term describing the rate of change in the plasma cell concentration is represented in the second equation of system (1), where:

-

$\xi \left(m\right)$—takes into account the destruction of a normal functioning immune system, m is a feature of the organ;

-

$\alpha$—coefficient taking into account the probability of an antigen–antibody meeting;

-

${\mu }_c$—coefficient of reduction of plasma cells due to aging;

-

$C^{*}$—the plasma cell rate concentration of healthy body.

The dynamics of the antibodies are described by the third equation of the system (1), where:

-

$\rho$—production rate of antibodies by one plasma cell;

-

$\eta$—number of antibodies to neutralize one antigen;

-

$\gamma$—the antigen neutralizing factor;

-

${\mu }_f$—coefficient inversely proportional to the decay time of the antibodies.

Finally, the target organ damage rate block is described by the final equation of the system (1), with the following parameters:

-

$\sigma$—constant related to a particular disease;

-

${\mu }_m$—coefficient describing the generation rate of the target organ.

$$\xi \left(m\right)=\left\{\begin{array}{cc}1,& 0\le m\le m^{*},\\ {\displaystyle \frac{1-m}{1-m^{*}}},& m^{*}<m\le 1,\end{array}\right.$$

(3)

Using to the function where $m^{*}$ is the threshold value for the relative damage to an organ. We describe the system (1) using two cases

$$\begin{array}{c}\dot{V}\left(t\right)=\beta V\left(t\right)-\gamma F\left(t\right)V\left(t\right),\\ \dot{C}\left(t\right)=\alpha F(t-\tau )V(t-\tau )-{\mu }_c(C\left(t\right)-C^{*}),\\ \dot{F}\left(t\right)=\rho C\left(t\right)-\eta \gamma F\left(t\right)V\left(t\right)-{\mu }_{f}F\left(t\right),\\ \dot{m}\left(t\right)=\sigma V\left(t\right)-{\mu }_{m}m\left(t\right),\end{array}$$

(4)

and

$$\begin{array}{c}\dot{V}\left(t\right)=\beta V\left(t\right)-\gamma F\left(t\right)V\left(t\right),\\ \dot{C}\left(t\right)={\displaystyle \frac{1-m\left(t\right)}{1-m^{*}}}\alpha F(t-\tau )V(t-\tau )-{\mu }_c(C\left(t\right)-C^{*}),\\ \dot{F}\left(t\right)=\rho C\left(t\right)-\eta \gamma F\left(t\right)V\left(t\right)-{\mu }_{f}F\left(t\right),\\ \dot{m}\left(t\right)=\sigma V\left(t\right)-{\mu }_{m}m\left(t\right).\end{array}$$

(5)

Following the work of [44], we transform the system (1) to a dimensionless form. We insert

$$V\left(t\right)=v\left(t\right)V_m,C\left(t\right)=s\left(t\right)C^{*},F\left(t\right)=f\left(t\right)F^{*},$$

where $F^{*}$ are antibodies of a healthy organism and $V_m$ is the maximum antigen’s value, which denotes the following

$$\begin{gathered} a_1=\beta ,a_2=\gamma F^{*},a_3=\alpha V_m{\displaystyle \frac{F^{*}}{C^{*}}},a_4={\mu }_f={\displaystyle \frac{\rho C^{*}}{F^{*}}}, \\ {a}_5={\mu }_c,a_6=\sigma V_m,a_7={\mu }_m,a_8=\eta \gamma V_m, \end{gathered}$$

and we obtain a dimensionless system of the following form:

$$\begin{array}{c}\dot{v}\left(t\right)=a_{1}v\left(t\right)-a_{2}f\left(t\right)v\left(t\right),\\ \dot{s}\left(t\right)=a_3\xi \left(m\right)f(t-\tau )v(t-\tau )-a_5(s\left(t\right)-1),\\ \dot{f}\left(t\right)=a_4(s\left(t\right)-f\left(t\right))-a_{8}f\left(t\right)v\left(t\right),\\ \dot{m}\left(t\right)=a_{6}v\left(t\right)-a_{7}m\left(t\right).\end{array}$$

(6)

Remark 1. 

The basic system (6) has different types of qualitative solutions. Based on laboratory data on pneumonia, Marchuk G.I. presented different forms of the disease in in [45] by varying the values of the parameters in system (6), which serve as key indicators characterizing the disease:

-

for the subclinical form of the disease:

$$a_1=8,a_2=10,a_3=\mathrm{10,000},a_4=0.17,a_5=0.5,a_6=10,a_7=0.12,a_8=8;$$

(7)

-

for the acute form of the disease:

$$a_1=2,a_2=0.8,a_3=\mathrm{10,000},a_4=0.17,a_5=0.5,a_6=10,a_7=0.12,a_8=8;$$

(8)

-

for the chronic form of the disease:

$$a_1=1,a_2=0.8,a_3=1000,a_4=0.17,a_5=0.5,a_6=10,a_7=0.12,a_8=8;$$

(9)

-

for the fatal outcome form of the disease:

$$a_1=1.54,a_2=0.77,a_3=880,a_4=0.15,a_5=0.5,a_6=12,a_7=0.12,a_8=8.$$

(10)

2. Equilibria

The equilibria of the system (6) are described using the following system of algebraic equations

$$\begin{array}{c}(a_1-a_{2}f)v=0,\\ a_3\xi \left(m\right)fv=a_5(s-1),\\ a_4(s-f)=a_{8}fv,\\ a_{6}v=a_{7}m.\end{array}$$

(11)

2.1. Equilibria $E_1$ and $E_2$

It is easy to see that regardless of the values of $a_1,\dots ,a_8$, one of the solutions for the system (11) is the equilibrium $E_1(v_1,s_1,f_1,m_1)=(0,1,1,0)$. Equilibrium $E_1$ describes the healthy state of an organism.

Using conditions

$$v\ne 0,\xi \left(m\right)=1,a_1\ne a_2,a_3{a}_4\ne a_5{a}_8$$

we can obtain the second equilibrium $E_2(v_2,s_2,f_2,m_2)$. From the first Equation (11), it follows that $f_2={\displaystyle \frac{a_1}{a_2}}$. From the second and third Equation (11), we have $a_3{a}_4(s-f_2)=a_5{a}_8(s-1)$. As a result, we obtain:

$$s_2={\displaystyle \frac{f_2{a}_3{a}_4-a_5{a}_8}{a_3{a}_4-a_5{a}_8}}.$$

(12)

From the second and final Equation (11), we obtain

$$v_2={\displaystyle \frac{a_2{a}_5}{a_1{a}_3}}(s_2-1)={\displaystyle \frac{a_4{a}_5(a_1-a_2)}{a_1(a_3{a}_4-a_5{a}_8)}},m_2={\displaystyle \frac{a_6}{a_7}}{v}_2={\displaystyle \frac{a_4{a}_5{a}_6(a_1-a_2)}{a_1{a}_7(a_3{a}_4-a_5{a}_8)}}.$$

(13)

Remark 2. 

Note that positive $v_2$ and $m_2$, it must be

$$a_1>a_2,a_3{a}_4>a_5{a}_8$$

(14)

or

$$a_1<a_2,a_3{a}_4<a_5{a}_8.$$

(15)

In the case of (14), we have

$$f_2{a}_3{a}_4>a_3{a}_4>a_5{a}_8$$

and in the case of (15), we have

$$f_2{a}_3{a}_4<a_3{a}_4<a_5{a}_8.$$

Via (12), both cases are $s_2>0$.

Remark 3. 

Note that via (3) for $\xi \left(m\right)=1$, the obtained $m_2$ must satisfy the condition $m_2\in [0,m^{*}]$.

Remark 4. 

Note that if $a_1=a_2$, then $f_2=1$, via (12) $s_2=1$, via (13) $v_2=0$ and $m_2=0$, i.e., equilibrium $E_2$ coincides with $E_1$.

Example 1. 

(1) Let $m^{*}=0.1$ and $a_1,\dots ,a_8$, as given in (8). Then, condition (14) holds, and equilibrium $E_2$ exists

$$v_2=0.000037,s_2=2.5035,f_2=2.5,m_2=0.0025.$$

(2) Let be $m^{*}=0.1$ and $a_1,\dots ,a_8$ are given in (9). Then, condition (14) holds, and the following equilibrium $E_2$ exists

$$v_2=0.000128,s_2=1.256,f_2=1.25,m_2=0.0085.$$

(3) Let $m^{*}=0.1$ and $a_1,\dots ,a_8$, as given in (10). Then, condition (14) holds, and equilibrium $E_2$ exists

$$v_2=0.0004394,s_2=2.03125,f_2=2,m_2=0.02929.$$

2.2. Equilibria $E_3$ and $E_4$

Using $\xi \left(m\right)={\displaystyle \frac{1-m}{1-m^{*}}}$ and $f={\displaystyle \frac{a_1}{a_2}}$, from the second, third, and final Equation (11), we obtain

$$a_3{\displaystyle \frac{1-m}{1-m^{*}}}{\displaystyle \frac{a_1}{a_2}}{\displaystyle \frac{a_7}{a_6}}m=a_5\left({\displaystyle \frac{a_1}{a_2}}+{\displaystyle \frac{a_8{a}_1{a}_7}{a_4{a}_2{a}_6}}m-1\right)$$

or

$$(1-m)m{P}_1=P_2+P_{3}m,$$

where

$$P_1={\displaystyle \frac{a_1{a}_3{a}_7}{(1-m^{*})a_2{a}_6}},P_2=a_5\left({\displaystyle \frac{a_1}{a_2}}-1\right),P_3={\displaystyle \frac{a_1{a}_5{a}_7{a}_8}{a_2{a}_4{a}_6}},$$

or

$$m^2-2bm+c=0,$$

where

$$b={\displaystyle \frac{1}{2}}\left(1-{\displaystyle \frac{P_3}{P_1}}\right)={\displaystyle \frac{1}{2}}\left(1-{\displaystyle \frac{a_5{a}_8}{a_3{a}_4}}(1-m^{*})\right),c={\displaystyle \frac{P_2}{P_1}}=\left(1-{\displaystyle \frac{a_2}{a_1}}\right){\displaystyle \frac{a_5{a}_6}{a_3{a}_7}}(1-m^{*}).$$

(16)

As a result, we obtain two equilibria, $E_3=(v_3,s_3,f_3,m_3)$ and $E_4=(v_4,s_4,f_4,m_4)$ with

$$m_3=b+\sqrt{b^2-c},m_4=b-\sqrt{b^2-c},$$

(17)

and

$$v_j={\displaystyle \frac{a_7}{a_6}}{m}_j,f_j={\displaystyle \frac{a_1}{a_2}},s_j=\left(1+{\displaystyle \frac{a_7{a}_8}{a_4{a}_6}}{m}_j\right){\displaystyle \frac{a_1}{a_2}},j=3,4.$$

(18)

Remark 5. 

Note that via (3), the obtained $m_j$ must satisfy the condition $m_j\in (m^{*},1]$, $j=3,4$ Further, via (16) and (17):

(1) 

For existence and positivity of $m_3$ and $m_4$, it is required that $b>0$, $b^2\ge c>0$, i.e., $a_1>a_2$.

(2) 

If $b>0$, then $P_3<P_1$, or equivalently, $a_5{a}_8(1-m^{*})<a_3{a}_4$.

(3) 

For $m_3>m^{*}$, it must hold that $m^{*}<2b=1-P_3/P_1$ or $P_3/P_1<1-m^{*}$, or equivalently $a_5{a}_8<a_3{a}_4$.

(4) 

If $m_3$ is defined, then $m_3<1$.

Lemma 1. 

For $m_4\in (m^{*},1]$ it must hold that:

$$a_1>a_2,a_5{a}_8(1-m^{*})<a_3{a}_4,\left(1-{\displaystyle \frac{a_2}{a_1}}\right){\displaystyle \frac{a_6}{a_7{m}^{*}}}+{\displaystyle \frac{a_8}{a_4}}>{\displaystyle \frac{a_3}{a_5}}.$$

(19)

Proof. 

Using (16) and (17), the existence and positivity of $m_4$ require that $b>0$ and $b^2\ge c>0$. According to (16), for $c>0$, it must hold that $a_1>a_2$; and for $b>0$, the second inequality in (19) must be satisfied. From (17) and (16), we have also have $m_4<b<1/2<1$. From $m_4=b-\sqrt{b^2-c}>m^{*}$, it follows that $c+{\left(m^{*}\right)}^2>2b{m}^{*}$ or

$$\left(1-{\displaystyle \frac{a_2}{a_1}}\right){\displaystyle \frac{a_5{a}_6}{a_3{a}_7}}(1-m^{*})+{\left(m^{*}\right)}^2>\left(1-{\displaystyle \frac{a_5{a}_8}{a_3{a}_4}}(1-m^{*})\right)m^{*}$$

or

$$\left(1-{\displaystyle \frac{a_2}{a_1}}\right){\displaystyle \frac{a_5{a}_6}{a_3{a}_7}}(1-m^{*})>(1-m^{*})m^{*}-{\displaystyle \frac{a_5{a}_8}{a_3{a}_4}}(1-m^{*})m^{*},$$

from which the last inequality (19) follows. The proof is completed. □

Corollary 1. 

For the parameters in (7)–(10), equilibrium $E_4$ does not exist.

Proof. 

The condition $a_1>a_2$ does not hold for (7), and the last inequality (19) does not hold for (8)–(10); in particular, for (8)

$$\left(1-{\displaystyle \frac{a_2}{a_1}}\right){\displaystyle \frac{a_6}{a_7{m}^{*}}}+{\displaystyle \frac{a_8}{a_4}}=2047.059<{\displaystyle \frac{a_3}{a_5}}=\mathrm{20,000};$$

for (9)

$$\left(1-{\displaystyle \frac{a_2}{a_1}}\right){\displaystyle \frac{a_6}{a_7{m}^{*}}}+{\displaystyle \frac{a_8}{a_4}}=743.725<{\displaystyle \frac{a_3}{a_5}}=2000;$$

for (10)

$$\left(1-{\displaystyle \frac{a_2}{a_1}}\right){\displaystyle \frac{a_6}{a_7{m}^{*}}}+{\displaystyle \frac{a_8}{a_4}}=553.333<{\displaystyle \frac{a_3}{a_5}}=1760.$$

The proof is completed. □

3. Stochastic Perturbations, Centralization, and Linearization

Let $\{\mathsf{\Omega },\mathfrak{F},\mathbf{P}\}$ be a complete probability space; ${\left\{{\mathfrak{F}}_t\right\}}_{t\ge 0}$ be a nondecreasing family of sub-$\sigma$-algebras of $\mathfrak{F}$, i.e., ${\mathfrak{F}}_{\theta }\subset {\mathfrak{F}}_t$ for $\theta <t$; let $\mathbf{P}\{·\}$ be the probability of an event enclosed in the braces; let $\mathbf{E}$ be the mathematical expectation with respect to the probability $\mathbf{P}$; and let $H_2$ be the space of ${\mathfrak{F}}_0$-adapted stochastic processes $\phi \left(\theta \right)$, $\theta \le 0$, ${\parallel \phi \parallel }_0={sup}_{\theta \le 0}\left|\phi \left(\theta \right)\right|$, ${\parallel \phi \parallel }_1^2={sup}_{\theta \le 0}\mathbf{E}{\left|\phi \left(\theta \right)\right|}^2$.

Let us suppose that the system (6) in the case of $\xi \left(m\right)=1$ experiences white noise stochastic perturbations, which are directly proportional to the deviation of the system state from one of the equilibria $(\overline{v},\overline{s},\overline{f},\overline{m})$. So, we obtain the system of Ito’s stochastic differential equations [31,46]

$$\begin{array}{c}dv\left(t\right)=(a_{1}v\left(t\right)-a_{2}f\left(t\right)v\left(t\right))dt+{\sigma }_1(v\left(t\right)-\overline{v})d{w}_1\left(t\right),\\ ds\left(t\right)=(a_{3}f(t-\tau )v(t-\tau )-a_5(s\left(t\right)-1))dt+{\sigma }_2(s\left(t\right)-\overline{s})d{w}_2\left(t\right),\\ df\left(t\right)=(a_4(s\left(t\right)-f\left(t\right))-a_{8}f\left(t\right)v\left(t\right))dt+{\sigma }_3(f\left(t\right)-\overline{f})d{w}_3\left(t\right),\\ dm\left(t\right)=(a_{6}v\left(t\right)-a_{7}m\left(t\right))dt+{\sigma }_4(m\left(t\right)-\overline{m})d{w}_4\left(t\right),\end{array}$$

(20)

where ${\sigma }_1,\dots ,{\sigma }_4$ are constants and $w_1\left(t\right),\dots ,w_4\left(t\right)$ are mutually independent ${\mathfrak{F}}_t$-measurable Wiener processes on the probability space $\{\mathsf{\Omega },\mathfrak{F},\mathbf{P}\}$.

Remark 6. 

It is clear that the equilibrium $(\overline{v},\overline{s},\overline{f},\overline{m})$ of the system for deterministic differential Equation (6) is also the solution for the system of stochastic differential Equation (20). Note that the first stochastic perturbations for (20) were used in [13] and later in man other works (see, for instance, [31] and the references therein).

By adding new variables into (20)

$$x_1\left(t\right)=v\left(t\right)-\overline{v},x_2\left(t\right)=s\left(t\right)-\overline{s},x_3\left(t\right)=f\left(t\right)-\overline{f},x_4\left(t\right)=m\left(t\right)-\overline{m},$$

and using (11), we obtain a nonlinear system of stochastic delay differential equations with the zero solution

$$\begin{array}{c}d{x}_1\left(t\right)=[-(a_2\overline{f}-a_1)x_1\left(t\right)-a_2\overline{v}{x}_3\left(t\right)-a_2{x}_1\left(t\right)x_3\left(t\right)]dt+{\sigma }_1{x}_1\left(t\right)d{w}_1\left(t\right),\\ d{x}_2\left(t\right)=[a_3(x_1(t-\tau )x_3(t-\tau )+\overline{f}{x}_1(t-\tau )+\overline{v}{x}_3(t-\tau ))-a_5{x}_2\left(t\right)]dt+{\sigma }_2{x}_2\left(t\right)d{w}_2\left(t\right),\\ d{x}_3\left(t\right)=[a_4(x_2\left(t\right)-x_3\left(t\right))-a_8{x}_1\left(t\right)(x_3\left(t\right)+\overline{f})-a_8\overline{v}{x}_3\left(t\right)]dt+{\sigma }_3{x}_3\left(t\right)d{w}_3\left(t\right),\\ d{x}_4\left(t\right)=[a_6{x}_1\left(t\right)-a_7{x}_4\left(t\right)]dt+{\sigma }_4{x}_4\left(t\right)d{w}_4\left(t\right).\end{array}$$

(21)

Rejecting the nonlinear terms in (21), we obtain the linear part of the system (21):

$$\begin{array}{c}d{y}_1\left(t\right)=[-(a_2\overline{f}-a_1)y_1\left(t\right)-a_2\overline{v}{y}_3\left(t\right)]dt+{\sigma }_1{y}_1\left(t\right)d{w}_1\left(t\right),\\ d{y}_2\left(t\right)=[a_3(\overline{f}{y}_1(t-\tau )+\overline{v}{y}_3(t-\tau ))-a_5{y}_2\left(t\right)]dt+{\sigma }_2{y}_2\left(t\right)d{w}_2\left(t\right),\\ d{y}_3\left(t\right)=[-a_8\overline{f}{y}_1\left(t\right)+a_4{y}_2\left(t\right)-(a_4+a_8\overline{v})y_3\left(t\right)]dt+{\sigma }_3{y}_3\left(t\right)d{w}_3\left(t\right),\\ d{y}_4\left(t\right)=[a_6{y}_1\left(t\right)-a_7{y}_4\left(t\right)]dt+{\sigma }_4{y}_4\left(t\right)d{w}_4\left(t\right).\end{array}$$

(22)

Matrix Form

Note that the linear part (22) of the systems (21) can be represented in the matrix form

$$dy\left(t\right)=[Ay\left(t\right)+By(t-\tau )]dt+\sum _{i=1}^4{C}_{i}y\left(t\right)d{w}_i\left(t\right),$$

(23)

where $y\left(t\right)={(y_1\left(t\right),y_2\left(t\right),y_3\left(t\right),y_4\left(t\right))}^{\prime }$ (here and everywhere below, ^′ indicates transposition),

$$A=\left[\begin{array}{cccc}-(a_2\overline{f}-a_1)& 0& -a_2\overline{v}& 0\\ 0& -a_5& 0& 0\\ -a_8\overline{f}& a_4& -(a_4+a_8\overline{v})& 0\\ a_6& 0& 0& -a_7\end{array}\right],B=\left[\begin{array}{cccc}0& 0& 0& 0\\ a_3\overline{f}& 0& a_3\overline{v}& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],$$

(24)

the matrix $C_i\mathrm{has}\mathrm{all}\mathrm{zero}\mathrm{elements}\mathrm{besides}{c}_{ii}={\sigma }_i,i=1,\dots ,4.$

In particular, for equilibrium $E_1=(0,1,1,0)$, matrices A and B are

$$A_1=\left[\begin{array}{cccc}-(a_2-a_1)& 0& 0& 0\\ 0& -a_5& 0& 0\\ -a_8& a_4& -a_4& 0\\ a_6& 0& 0& -a_7\end{array}\right],B_1=\left[\begin{array}{cccc}0& 0& 0& 0\\ a_3& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],$$

(25)

and, similarly, for equilibrium $E_2$

$$A_2=\left[\begin{array}{cccc}0& 0& -a_2{v}_2& 0\\ 0& -a_5& 0& 0\\ -a_8{f}_2& a_4& -(a_4+a_8{v}_2)& 0\\ a_6& 0& 0& -a_7\end{array}\right],B_2=\left[\begin{array}{cccc}0& 0& 0& 0\\ a_3{f}_2& 0& a_3{v}_2& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],$$

(26)

where $f_2={\displaystyle \frac{a_1}{a_2}}$ and $v_2$ are defined in (13).

4. Stability

Definition 1. 

The zero solution of the system (21) is considered to be stable in probability if for any ${\epsilon }_1>0$ and ${\epsilon }_2\in (0,1)$ there exists $\delta >0$, such that the solution $x\left(t\right)$ for the system (21) satisfies the condition $\mathbf{P}\{\underset{t\ge 0}{sup}\left|x\left(t\right)\right|>{\epsilon }_1\}<{\epsilon }_2$ for any initial function $\varphi \left(\theta \right)$, such that $\mathbf{P}\{\parallel \varphi {\parallel }_0<\delta \}=1$.

Definition 2. 

The zero solution of the system (22) with the initial condition $y\left(\theta \right)=\varphi \left(\theta \right)$, $\theta \le 0$, is called:

-

mean square stable if for each $\epsilon >0$ there exists a $\delta >0$ such that ${\mathbf{E}\left|y\left(t\right)\right|}^2<\epsilon$, $t\ge 0$, provided that ${\parallel \varphi \parallel }_1^2<\delta$;

-

asymptotically mean square stable if it is mean square stable and $\underset{t\to \infty }{lim}\mathbf{E}{\left|y\left(t\right)\right|}^2=0$ for each initial function ϕ.

Remark 7. 

The stability of the zero solution of system (21) is equivalent to the stability of the equilibrium point $(\overline{v},\overline{s},\overline{f},\overline{m})$ of system (20). It is important to note that the system of nonlinear stochastic differential Equation (21) shows a degree of nonlinearity greater than one. As shown in [31], in such cases, sufficient conditions for the asymptotic mean square stability of the zero solution of the linearized system (22) are also sufficient for the stability in probability of the zero solution of the full nonlinear system (21).

Below, $y\left(t\right)$ and $y_t$ denote the value of the solution at time t and a complete trajectory of the solution up to time t, respectively.

Consider a functional $W(t,\phi ):[0,\infty )\times H_2\to {\mathbf{R}}_{+}$ that can be represented in the form $W(t,\phi )=W(t,\phi (0),\phi (\theta \left)\right)$, $\theta <0$, and for $\phi =y_t$ put

$$\begin{gathered} W_{\phi }(t,y)=W(t,\phi )=W(t,y_t)=W(t,y,y(t+\theta )), \\ y=\phi \left(0\right)=y\left(t\right),\theta <0. \end{gathered}$$

(27)

Let D be the set of functionals, for which the function $W_{\phi }(t,y)$ defined by (27) has a continuous derivative with respect to t and two continuous derivatives with respect to y. The generator L of Equation (23) is defined by the functionals from D and has the following form [31,46]

$$\begin{gathered} LW(t,y_t)={\displaystyle \frac{\partial }{\partial t}}{W}_{\phi }(t,y\left(t\right))+\nabla W_{\phi }^{\prime }(t,y\left(t\right))(A_{k}y\left(t\right)+B_{k}y(t-h)) \\ +{\displaystyle \frac{1}{2}}\sum _{i=1}^4{y}^{\prime }\left(t\right)C_i^{\prime }{\nabla }^2{W}_{\phi }(t,y\left(t\right))C_{i}y\left(t\right). \end{gathered}$$

(28)

Theorem 1 

([31]). Let a functional $W(t,\phi )\in D$ and positive constants $c_1$, $c_2$, $c_3$ exist, such that the following conditions hold:

$$\mathbf{E}W(t,y_t)\ge c_1\mathbf{E}|y\left(t\right){|}^2,\mathbf{E}W(0,\varphi )\le c_2{\parallel \varphi \parallel }^2,\mathbf{E}LW(t,y_t)\le -c_3\mathbf{E}{\left|y\left(t\right)\right|}^2.$$

Then, the zero solution of the system (22) is asymptotically mean square stable.

Theorem 2. 

Let positive definite $4\times 4$-matrices P and R exist, such that the following linear matrix inequality holds:

$$\mathsf{\Phi }=\left[\begin{array}{cc}\mathsf{\Psi }& PB\\ B^{\prime }P& -R\end{array}\right]<0,\mathsf{\Psi }=A^{\prime }P+PA+\sum _{i=1}^4{C}_i^{\prime }P{C}_i+R,$$

(29)

where the matrices $A,B,C_1,\dots ,C_4$ are defined in (24). Then, the equilibrium $(\overline{v},\overline{s},\overline{f},\overline{m})$ of the system (20) has a stable probability.

Proof. 

Using Remark 7, to prove the stability in probability of the equilibrium of system (20), it is enough to prove the asymptotic mean square stability of the zero solution of Equation (23). Using the general method of Lyapunov functional construction [31], let us construct the functional $W=W_1+W_2$, where $W_1\left(y\right)=y^{\prime }Py$ and $W_2$ are chosen below. Let L be the generator [31,46] of Equation (23). Then

$$\begin{gathered} L{W}_1\left(y\left(t\right)\right)=2y^{\prime }\left(t\right)P[Ay\left(t\right)+By(t-\tau )]+\sum _{i=1}^4{y}^{\prime }\left(t\right)C_i^{\prime }P{C}_{i}y\left(t\right) \\ =y^{\prime }\left(t\right)\left[PA+A^{\prime }P+\sum _{i=1}^4{C}_i^{\prime }P{C}_i\right]y\left(t\right)+2y^{\prime }\left(t\right)PBy(t-\tau ). \end{gathered}$$

(30)

Using the additional functional $W_2(t,y_t)={\int }_{t-\tau }^t{y}^{\prime }\left(s\right)Ry\left(s\right)ds$, we have

$$L{W}_2(t,y\left(t\right))=y^{\prime }\left(t\right)Ry\left(t\right)-y^{\prime }(t-\tau )Ry(t-\tau ).$$

(31)

As a result, from (30), (31) for the functional $W=W_1+W_2$, $\eta \left(t\right)={(y^{\prime }\left(t\right),y^{\prime }(t-\tau ))}^{\prime }$ and some $c>0$, we obtain

$$\begin{gathered} LW(t,y_t)=y^{\prime }\left(t\right)\left[PA+A^{\prime }P+\sum _{i=1}^4{C}_i^{\prime }P{C}_i\right]y\left(t\right)+2y^{\prime }\left(t\right)PBy(t-\tau ) \\ +y^{\prime }\left(t\right)Ry\left(t\right)-y^{\prime }(t-\tau )Ry(t-\tau ) \\ ={\eta }^{\prime }\left(t\right)\mathsf{\Phi }\eta \left(t\right)\le -c|\eta \left(t\right){|}^2\le -c{\left|y\left(t\right)\right|}^2. \end{gathered}$$

Using Theorem 1, we know that the zero solution of Equation (23) is asymptotically mean square stable. The proof is completed. □

4.1. Routh–Hurwitz Criterion

Definition 3 

([31]). The trace of the k-th order of an $n\times n$-matrix A with elements $a_{ij}$ is defined as follows

$$S_k=\sum _{1\le i_1<\dots <i_k\le n}\left|\begin{array}{ccc}{a}_{i_1{i}_1}& \dots & a_{i_1{i}_k}\\ \dots & \dots & \dots \\ a_{i_k{i}_1}& \dots & a_{i_k{i}_k}\end{array}\right|,k=1,\dots ,n.$$

(32)

Here, in particular, $S_1=Tr\left(A\right)$, $S_n=det\left(A\right)$.

The characteristic equation of the matrix $A_k$ in Equation (23) has the following form

$${\lambda }^4-S_1{\lambda }^3+S_2{\lambda }^2-S_3\lambda +S_4=0.$$

(33)

It is known [31] that for LMI (29), all roots $\lambda$ of the characteristic Equation (33) must have negative real parts.

Theorem 3 

([31]). (Routh–Hurwitz criterion) All roots λ of the characteristic Equation (33) have negative real parts, if and only if

$$\begin{gathered} {\mathsf{\Delta }}_1=-S_1>0,{\mathsf{\Delta }}_2=\left|\begin{array}{cc}-S_1& -S_3\\ 1& S_2\end{array}\right|>0, \\ {\mathsf{\Delta }}_3=\left|\begin{array}{ccc}-S_1& -S_3& 0\\ 1& S_2& S_4\\ 0& -S_1& -S_3\end{array}\right|>0,{\mathsf{\Delta }}_4=\left|\begin{array}{cccc}-S_1& -S_3& 0& 0\\ 1& S_2& S_4& 0\\ 0& -S_1& -S_3& 0\\ 0& 1& S_2& S_4\end{array}\right|>0. \end{gathered}$$

(34)

Corollary 2. 

From (34), it follows that the characteristics of Equation (33) have negative real parts if and only if

$$S_1<0,S_1{S}_2<S_3<0,0<S_1^2{S}_4<S_3(S_1{S}_2-S_3).$$

(35)

4.2. Equilibrium $E_1(0,1,1,0)$

Theorem 4. 

If

$${\sigma }_1^2<2(a_2-a_1),{\sigma }_2^2<2a_5,{\sigma }_3^2<2a_4,{\sigma }_4^2<2a_7$$

(36)

then, the equilibrium $E_1$ is stable in probability.

Proof. 

It is clear that the stability of equilibrium $E_1$ is equivalent to the stability of the zero solution of the system (21). The order of nonlinearity of the nonlinear system (21) is higher than one. Following Remark 7 for stability in probability of the equilibrium $E_1$, it is enough to prove the asymptotic mean square stability of the zero solution of the linear system (22).

It is known [31] that the first inequality (36) is a necessary and sufficient condition for the asymptotic mean square stability of the zero solution of the first Equation (22). Using $\underset{t\to \infty }{lim}\mathbf{E}{\left|y_1\left(t\right)\right|}^2=0$ and the second and final inequalities (36), we obtain that $\mathbf{E}|y_2{\left(t\right)|}^2$ and $\mathbf{E}|y_4{\left(t\right)|}^2$ converge to zero too. Similarly, from $\underset{t\to \infty }{lim}\mathbf{E}{\left|y_1\left(t\right)\right|}^2=0$, $\underset{t\to \infty }{lim}\mathbf{E}{\left|y_2\left(t\right)\right|}^2=0$ and the third inequality (36), it follows that $\underset{t\to \infty }{lim}\mathbf{E}{\left|y_3\left(t\right)\right|}^2=0$. The proof is completed. □

Remark 8. 

Supposing that $a_2>a_1$, for matrix A (25), we obtain

$$\begin{array}{c}{S}_1=-(a_2-a_1+a_5+a_4+a_7),\\ S_2=(a_2-a_1)(a_5+a_4+a_7)+a_5(a_4+a_7)+a_4{a}_7,\\ S_3=-(a_2-a_1)(a_5{a}_4+a_5{a}_7+a_4{a}_7)-a_5{a}_4{a}_7,\\ S_4=(a_2-a_1)a_5{a}_4{a}_7.\end{array}$$

(37)

It is easy to see that all of the conditions (35) hold.

4.3. Equilibria $E_2=(v_2,s_2,f_2,m_2)$ and $E_3=(v_3,s_3,f_3,m_3)$

Remark 9. 

Note that for the matrix A (26), we have $S_4=-a_2{a}_5{a}_7{a}_8{v}_2{f}_2<0$, i.e., the conditions (35) for the equilibrium $E_2$ does not hold. It means that equilibrium $E_2$ is unstable even in the deterministic case: ${\sigma }_1={\sigma }_2={\sigma }_3={\sigma }_4=0$. For equilibrium $E_3$, the matrix A has the form as in (26), with $v_{2}and{f}_2$ replaced by $v_{3}and{f}_3$. So, equilibrium $E_3$ is also unstable.

5. Numerical Simulations

Let us consider some examples for the equilibria $E_1$, $E_2$, $E_3$, discussed in Figure 1, Figure 2 and Figure 3.

Example 2. 

Consider the system (20) with the parameters (7) and

$$\tau =1,{\sigma }_1=0.5,{\sigma }_2=0.3,{\sigma }_3=0.3,{\sigma }_4=0.3.$$

(38)

Using MATLAB (R2024a), it was shown that for equilibrium $E_1(0,1,1,0)$ with the parameters from (7) and (38), LMI (29) holds, i.e., this equilibrium is stable in probability. In Figure 1, 50 trajectories from the system (20) solution are shown with the following initial conditions

$$s\left(0\right)=1.5,m\left(0\right)=1.2,v\left(\theta \right)=0.017,f\left(\theta \right)=1.2,\theta \in [-\tau ,0].$$

The state of a healthy organism in the case $a_1<a_2$ (7) corresponds to the equilibrium

$$E_1(v_1,s_1,f_1,m_1)=(0,1,1,0).$$

Example 3. 

Consider equilibrium $E_2$ with the parameters from (8). In accordance with (14) and Remark 3, we obtain the following values for the equilibrium $E_2(v_2,s_2,f_2,m_2)$ = $(0.000037,2.5035,2.5,0.025)$. According to Remark 9, this equilibrium is unstable, even in the deterministic case. In Figure 2 50 trajectories of the system (20) solution are shown with $\tau =1$, ${\sigma }_1={\sigma }_2={\sigma }_3={\sigma }_4=0.1$ and the initial conditions

$$s\left(0\right)=2.5,m\left(0\right)=0.035,v\left(\theta \right)=0.0003,f\left(\theta \right)=2.55,\theta \in [-\tau ,0],$$

which are close enough to equilibrium $E_2$. It can be seen that all trajectories move away from the unstable equilibrium.

Example 4. 

Consider equilibrium $E_3$ with the parameters of (10). Using (17) and (18), we obtain

$$E_3(v_3,s_3,f_3,m_3)=(0.00946,2.03125,2,0.946).$$

According to Remark 9, this equilibrium is unstable. In Figure 3, 50 trajectories of the system (20) solution are shown with $\tau =1$, ${\sigma }_1={\sigma }_2={\sigma }_3={\sigma }_4=0.1$ and the following initial conditions

$$s\left(0\right)=2.03,m\left(0\right)=0.9,v\left(\theta \right)=0.009,f\left(\theta \right)=2,\theta \in [-\tau ,0],$$

which are close enough to equilibrium $E_3$. All trajectories move away from the unstable equilibrium.

Remark 10. 

Figure 1, Figure 2 and Figure 3 show that equilibrium $E_1$ is stable, and equilibria $E_2$ and $E_3$ are unstable. According to our results, from a medical point of view, it is shown that only a healthy body is in a stable state, while in other cases of equilibria, there is a high probability that the patient’s condition will change soon.

6. Conclusions

The known deterministic model of infectious diseases, proposed by G.I. Marchuk, is studied under stochastic perturbations and is described by a system of Ito’s stochastic differential equations with a delay. Conditions of stability or instability of equilibria for the considered system are investigated. We provide examples of numerical simulations for the system solutions and figures that illustrate the obtained results. The proposed research method can be used to investigate other nonlinear mathematical models in various applications.