Stochastic COVID-19 SEIQ epidemic model with time-delay

Abstract

In this work, we consider an epidemic model for corona-virus (COVID-19) with random perturbations as well as time delay, composed of four different classes of susceptible population, the exposed population, the infectious population and the quarantine population. We investigate the proposed problem for the derivation of at least one and unique solution in the positive feasible region of non-local solution. For one stationary ergodic distribution, the necessary result of existence is developed by applying the Lyapunov function in the sense of delay-stochastic approach and the condition for the extinction of the disease is also established. Our obtained results show that the effect of Brownian motion and noise terms on the transmission of the epidemic is very high. If the noise is large the infection may decrease or vanish. For validation of our obtained scheme, the results for all the classes of the problem have been numerically simulated.

Introduction

Epidemiology deals with various epidemic models or problems for investigation of different outbreaks using the available data from medical sciences. The importance of this area may be seen by its interest flourishing from day to day. Therefore, many mathematical models were established in the past, like SI, SIR, H1N1, HBV, SIS model, SARS, SIER model, H5N1 etc. as may be seen in [1], [2]. These all problems were formulated mathematically to provide some realistic predictions and the society gains information about the diseases which is helpful for stable society and stable health [3], [4], [5], [6]. The needful and necessary issues are the stability and preventing of various diseases in the societies of human population. Due to this each and every biological infection or disease is converted to mathematical model as soon as possible and the field of mathematical epidemiology were established for such formulation. After The first attempt of Mckendrick and Kermack [7], [8] the said models were highly analyzed for controlling of different diseases. Using this gate way and basic concepts, different researchers analyzed the epidemic models of SEIS, SEIRS, SIRS, vaccinated models and delay-models by including different parameter for vaccination and delay-time [9], [10], [11], [12], [13], [14], [15], [16], [17], [18].

As in the case of covid-19, spreading of disease have a direct relation with the quarantine of human population. Commonly, we have two types of quarantine, one is susceptible quarantine and second is infected quarantine. In our work we take the infected quarantine which means that those people will be quarantined if they are infected. Using this idea Chen et al. [19] constructed the epidemic model of covid-19 as follows

$$\begin{array}{ccc}\hfill \frac{dS}{dt}& \hfill =\hfill & A-\beta SI(t-\tau )-\mu S\left(t\right)+cI\left(t\right),\hfill \\ \hfill \frac{dE}{dt}& \hfill =\hfill & \beta S\left(t\right)I(t-\tau )-(\mu +\epsilon +{\sigma }_1+{\gamma }_1)E\left(t\right),\hfill \\ \hfill \frac{dI}{dt}& \hfill =\hfill & \epsilon E\left(t\right)-(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)I\left(t\right),\hfill \\ \hfill \frac{dQ}{dt}& \hfill =\hfill & {\sigma }_{1}E\left(t\right)+{\sigma }_{2}I\left(t\right)-(\mu +\alpha +{\gamma }_3)Q.\hfill \end{array}$$ (1)

here $S\left(t\right)$ is healthy class, $E\left(t\right)$ is expose to infection class, $I\left(t\right)$ is infectious class and $Q\left(t\right)$ is the quarantine class at the any time $\left(t\right)$ respectively. The explanation of all the parameters used in the proposed model are given in Table 1.

Table 1.

Parameters and there meaning in this paper.

Parameter	Description
A	The constant rate of birth or recruitment for people
$\beta$	The rate of infection for healthy people
$\mu$	Natural rate of mortality for all people
$c$	The recovered rate from infectious class to healthy people
$\epsilon$	Rate of infection from expose class
$\alpha$	Rate of epidemic mortality for infectious and quarantine people
${\sigma }_1$	Rate of quarantine from expose people
${\sigma }_2$	Rate of quarantine from infectious people
${\gamma }_1$	Rate of recovered people from exposed people
${\gamma }_2$	Rate of recovered people from infectious people
${\gamma }_3$	Rate of recovered people from quarantine classes

In the mathematical modeling of a biological phenomenon the stochastic differential equation models are more suitable than the deterministic one, because it can provide an additional degree of realism in comparison to their deterministic counterparts. Stochastic models produce more valuable output as compared to the deterministic ones because running a stochastic model several times, we can build up a distribution of the predicted outcomes, e.g., the number of infected classes at time $t$. On the other hand, a deterministic model will just give a single predicted value [20], [21], [22].

Various infectious models are unable to provide complete information about the concerned disease because of less time or lake of observation about the disease. Therefore, as in [23], these models may be more informative if we investigate it after all the symptoms appeared in human bodies or wait up to incubation duration. This incubation period is called time-delay which will be very helpful for more realistic results [24], [25]. The analysis of delay models are not easy as compared to the others problems which have no time delay. Time delays are considered as a natural elements of the dynamic process of economics, biology, epidemiology, ecology, mechanics and physiology. In the recent time some scholars worked on the delay models. Wu and Bai in [26] deals with non-linear incidence stationary waves for healthy, infectious and recovered (SIR) disease model. Based on temporary immunity, Liu et al. in [27], deals with asymptotic characteristics time delay stochastic SIR disease model. They also deals in [28] with non-local attraction and presence of the time delay scaling-free networking SIRS disease problem. Therefore our consideration will also deals with inclusion of delay-time representing the duration of incubation for the full symptoms appearance in Chen et al. [19] model. We will perturbed our problem by an external factor of environmental noise or brownian motion and by changing the given parameters.

We added the latent delay into system 1, by keeping the above assumption to deals with time delay problem as follows;

$$\begin{array}{ccc}\hfill dS& \hfill =\hfill & [A-\beta SI(t-\tau )-\mu S\left(t\right)+cI\left(t\right)]dt+v_{1}Sd{B}_1\left(t\right),\hfill \\ \hfill dE& \hfill =\hfill & [\beta S\left(t\right)I(t-\tau )-(\mu +\epsilon +{\sigma }_1+{\gamma }_1)E\left(t\right)]dt+v_{2}Ed{B}_2\left(t\right),\hfill \\ \hfill dI& \hfill =\hfill & [\epsilon E\left(t\right)-(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)I\left(t\right)]dt+v_{3}Id{B}_3\left(t\right),\hfill \\ \hfill dQ& \hfill =\hfill & [{\sigma }_{1}E\left(t\right)+{\sigma }_{2}I\left(t\right)-(\mu +\alpha +{\gamma }_3)Q]dt+v_{4}Qd{B}_4\left(t\right).\hfill \end{array}$$ (2)

Here $B_i\left(t\right);i=1,2,3,4$ are the free Brownian or noise motions. $v_i^2;i=1,2,3,4$ and $v_i>0$ are the intensities of the environmental external white noises, having the initial approximation as follows:

$$S\left(\theta \right)={\varphi }_1\left(\theta \right),E\left(\theta \right)={\varphi }_2\left(\theta \right),$$$$I\left(\theta \right)={\varphi }_3\left(\theta \right),Q\left(\theta \right)={\varphi }_4\left(\theta \right),-\tau \le \theta \le 0,$$$${\varphi }_i\left(\theta \right)\in C,i=1,2,3,4.$$ (3)

Here $C$ is the class of existing integral of Lebesgue operator through $[--\tau ,0]$ to ${\mathbb{R}}_{+}^4$.

The purpose of our article is to analyze the dynamical properties of the root of the system before disease occurrence and checking of at least one repeating solutions greater than zero of the considered stochastic COVID-19 epidemic model with time delay.

The paper is organized by the following sections. In the second Section ‘Qualitative Analysis of positive solution’ the solution of (2) is derived to be positive and having upper and lower bounds in a feasible region, which is not changing. Also, maintenance of the considered problem and necessary results for reducing the infection are studied. The valid results for the dynamical behaviors of the stationary distribution are achieved in Section ‘Existence of ergodic stationary distribution’. In Section ‘Qualitative Analysis of positive solution’, the results for extinction of stochastic model is provided. Based on this, in Section ‘Extinction’, we draw our obtained scheme by numerical simulation for stochastic-stability. The last Section ‘Numerical simulations for stochastic stability, is the inclusion of some remarks related to conclusion and future work.

Qualitative analysis of positive solution

For investigation of dynamical behavior of SDE (2), we have to prove the problem (2) has one non-local solution in the feasible region. This can be achieved that if the coefficients of system (2) are fulfilling the growth and Lipschitzian conditions then their will exist one positive solution. So this can be very easy and we omit it. Next for positive and non-local solution we have to go ahead and use the techniques of Lyapunov operator [29], [30], [31].

Theorem 1

Problem (2) has one root greater then zero $(S\left(t\right),E\left(t\right),I\left(t\right),Q\left(t\right))$ on $t\ge -\tau$ , and the solution lie in ${\mathbb{R}}_{+}^4$ for the subjected initial approximation (3) with occurrence chance of one.

Proof

For given initial approximation $(S\left(0\right),E\left(0\right),I\left(0\right),Q\left(0\right))$, the coefficients of problem (2) fulfilling the local Lipschitzian result, so model (2) has one local root $(S\left(t\right),E\left(t\right),I\left(t\right),Q\left(t\right))$ on $t\in [--\tau ,{\tau }_e)$ a.s., here ${\tau }_e$ shows the time taken for infection [29], [30], [31].

Our objective is to prove that the solution is non-local i.e. ${\tau }_e=\infty$ a.s. Take $n_0\ge 1$ be largely; $S\left(\theta \right),E\left(\theta \right),I\left(\theta \right)$, and $Q\left(\theta \right)(\theta \in [--\theta ,0])$ $\in$ $[\frac{1}{n_0},n_0]$. $\forall n\ge n_0,n\in \mathbb{N}$, give the definition the time break as

$${\tau }_n=inf\{t\in [-\tau ,{\tau }_e):min(S\left(t\right),E\left(0\right),I\left(t\right),Q\left(t\right))\le \frac{1}{n}ormax(S\left(t\right),E\left(t\right),I\left(t\right),Q\left(t\right))\ge n\},$$

consider $min\varphi =\infty$ $(\varphi$ is void set). As, ${\tau }_n$ is growing as $n\to \infty$. Consider ${\tau }_{\infty }={lim}_{n\to \infty }{\tau }_n$, then ${\tau }_{\infty }\le {\tau }_e$ a.s. Hence, we have to prove that ${\tau }_{\infty }=\infty$ a.s, then ${\tau }_e=\infty$ a.s. and $(S\left(t\right),E\left(t\right),I\left(t\right),Q\left(t\right))\in {\mathbb{R}}_{+}^4$ a.s. $\forall t\ge --\tau$. If $\iota \in (0,1)$ and $\tilde{T}>0$; $P\{{\tau }_{\infty }\le \tilde{T}\}>\iota$. Then $\exists$ an integer $n_1\ge n_0$;

$$P\{\tilde{T}\ge {\tau }_n\}\ge \iota \forall n\ge n_1.$$ (4)

We take a $C^2-$ operator $V:{\mathbb{R}}_{+}^4\to {\mathbb{R}}_{+}$ as given:

$$V(S,E,I,Q)=(S-a-a\frac{lnS}{a})+(E-1-lnE)+(I-1-lnI)+(Q-1-lnQ)+{\int }_t^{t+\tau }a\beta I(s-\tau )ds,$$

here $a>0$ which will be evaluated. Applying Ito’s, we get

$$dV=LVdt+v_1(S-a)d{B}_1\left(t\right)+v_2(E-1)d{B}_2\left(t\right)+v_3(I-1)d{B}_3\left(t\right)+v_4(Q-1)d{B}_4\left(t\right).$$

where

$$LV=(1-\frac{a}{S})(A-\beta SI(t-\tau )-\mu S\left(t\right)+cI\left(t\right))+(1-\frac{1}{E})(\beta S\left(t\right)I(t-\tau )-(\mu +\epsilon +{\sigma }_1+{\gamma }_1)E\left(t\right))+(1-\frac{1}{I})(\epsilon E\left(t\right)-(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)I\left(t\right))+(1-\frac{1}{Q})({\sigma }_{1}E\left(t\right)+{\sigma }_{2}I\left(t\right)-(\mu +\alpha +{\gamma }_3)Q\left(t\right))+\frac{a{v}_1^2+v_2^2+v_3^2+v_4^2}{2}+a\beta I\left(t\right)-a\beta I(t-\tau )\le A+a\mu +3\mu +\epsilon +{\sigma }_1+{\gamma }_1+2\alpha +c+{\sigma }_2+{\gamma }_2+{\gamma }_3+[a\beta -(\mu +\alpha +{\gamma }_2)]I+\frac{a{v}_1^2+v_2^2+v_3^2+v_4^2}{2}$$

Let $a\beta -(\mu +\alpha +{\gamma }_2)=0$, then $a=\frac{\mu +\alpha +{\gamma }_2}{\beta }$, hence

$$LV\le A+\mu (a+3)+\epsilon +{\sigma }_1+{\gamma }_1+2\alpha +c+{\sigma }_2+{\gamma }_2+{\gamma }_3+\frac{a{v}_1^2+v_2^2+v_3^2+v_4^2}{2}≔M,$$ (5)

here $M>0$ which is free of $S\left(t\right),E\left(t\right),I\left(t\right)$, and $Q\left(t\right)$. Hence,

$$dV(S,E,I,Q)=Mdt+v_1(S-a)d{B}_1\left(t\right)+v_2(E-1)d{B}_2\left(t\right)+v_3(I-1)d{B}_3\left(t\right)+v_4(Q-1)d{B}_4\left(t\right).$$ (6)

Upon integration (6) from 0 to ${\tau }_n\wedge \tilde{T}=min\{{\tau }_n,\tilde{T}\}$ and then consider the expected value $E$ on both sides, we have

$$EV\left(S({\tau }_n\wedge \tilde{T}),E({\tau }_n\wedge \tilde{T}),I({\tau }_n\wedge \tilde{T}),Q({\tau }_n\wedge \tilde{T})\right)\le EV\left(S\left(0\right),E\left(0\right),I\left(0\right),Q\left(0\right)\right)+M\tilde{T}.$$ (7)

Let ${\Omega }_n=\{{\tau }_n\le \tilde{T}\}$, for $n\ge n_1$ and in view of (4), we obtain $P\left({\Omega }_n\right)\ge \epsilon$ such that, for every $\omega \in {\Omega }_n$, there is at least one of $S({\tau }_n,\omega ),E({\tau }_n,\omega ),I({\tau }_n,\omega )$, or $Q({\tau }_n,\omega )$ equating either n or $\frac{1}{n}$, as

$$V\left(S({\tau }_n\wedge \tilde{T}),E({\tau }_n\wedge \tilde{T}),I({\tau }_n\wedge \tilde{T}),Q({\tau }_n\wedge \tilde{T})\right)\ge (n-1-lnn)\wedge \left(\frac{1}{n}-1-ln\frac{1}{n}\right).$$ (8)

According to (7), we get

$$EV(S\left(0\right),E\left(0\right),I\left(0\right),Q\left(0\right))+M\tilde{T}\ge E[1_{{\Omega }_n\left(\omega \right)}VS({\tau }_n,\omega ),E({\tau }_n,\omega ),I({\tau }_n,\omega ),Q({\tau }_n,\omega )]\ge \epsilon (n-1-lnn)\wedge \left(\frac{1}{n}-1-ln\frac{1}{n}\right),$$ (9)

where $1_{{\Omega }_n}$ shows the indication operator of ${\Omega }_n$. Taking $n\to \infty$ gives

$$\infty >EV(S\left(0\right),E\left(0\right),I\left(0\right),Q\left(0\right))+M\tilde{T}=\infty ,$$ (10)

which contradicts the supposition. So we conclude that ${\tau }_{\infty }=\infty$ a.s., hence the required result is proved. □

Existence of ergodic stationary distribution

Now in this section, we make a proper Lyapunov operator in sense of stochastic to deals with the existence of a one ergodic stationary division of the positive root to model (2) [32]. Firstly, take $X\left(t\right)$ is a regular time-homogeneous “Markov process” in ${\mathbb{R}}^d$ shown by the SDE

$$dX\left(t\right)=\sum _{r=1}^d{g}_r(t,X\left(t\right))d{B}_r\left(t\right)+f(X(t-\tau ),X\left(t\right),t)dt.$$ (11)

The diffusion matrix is

$$\Lambda \left(x\right)=\left({\lambda }_{ij}\left(x\right)\right),{\lambda }_{ij}\left(x\right)=\sum _{r=1}^d{g}_r^i\left(x\right)g_r^j\left(x\right).$$

Lemma 1

The process of Markov X(t) has one ergodic stationary division $\pi (.)$

$\exists$ a bounded pre-domain $U\subset {\mathbb{R}}^d$ with continuous boundary $\Gamma$ , and

(i): $\exists$ $K>0$ ; ${\sum }_{i,j=1}^d{\lambda }_{ij}\left(x\right){\zeta }_i{\zeta }_j\ge K{\left|\zeta \right|}^2,x\in U,\xi \in {\mathbb{R}}^d$ .

(ii): $\exists$ a non-negative $C^2$ operator $\tilde{V}$ ; $L\tilde{V}$ is less than zero for any ${\mathbb{R}}^d\setminus U$ .

Take the basic reproduction number of the model in the stochastic approach as follows:

$$R_0^s=\frac{A\beta \epsilon }{\hat{\mu }\hat{\epsilon }\hat{\alpha }\hat{\gamma }}$$ (12)

where $\hat{\mu }=\mu +\frac{v_1^2}{2},\hat{\epsilon }=\mu +\epsilon +{\sigma }_1+{\gamma }_1+\frac{v_2^2}{2},\hat{\alpha }=\mu +\alpha +c+{\sigma }_2+{\gamma }_2+\frac{v_3^2}{2}$

and $\hat{\gamma }=\mu +\alpha +{\gamma }_3+\frac{v_4^2}{2}$ .

Theorem 2

Assume that $R_0^s>1$ and $\mu -\frac{v_1^2\vee v_2^2\vee v_3^2\vee v_4^2}{2}>0$ , then for $(S\left(0\right),E\left(0\right),I\left(0\right),Q\left(0\right))\in {\mathbb{R}}_{+}^4$ , model (2) has one ergodic stationary division $\pi (.)$ .

Proof

Firstly, we have to validated the conditions (i) and (ii) of Lemma 1. To derive result (i), the diffusion matrix is:

$$\Lambda =\left(\begin{array}{cccc}\hfill v_1^2{S}^2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill v_2^2{E}^2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill v_3^2{I}^2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill v_4^2{Q}^2\hfill \\ \hfill \hfill \end{array}\right).$$

The matrix $\Lambda$ well be positive-definite on any compact subset of ${\mathbb{R}}_{+}^4$, and result (i) of Lemma 1 is fulfilling.

Further, we derive condition (ii). Take $C^2$-operator $V:{\mathbb{R}}_{+}^4\to \mathbb{R}$ as given:

$$V(S,E,I,Q)=N\left(-lnS{c}_1-lnE{c}_2-c_{3}lnI-lnQ+\beta {\int }_t^{t+\tau }I(-\tau +s)ds\right)$$$$-lnS+\beta {\int }_t^{t+\tau }I(-\tau +s)ds-lnE-lnQ+\frac{1}{\rho +1}{(S+E+I+Q)}^{\rho +1}$$$$=N{V}_1+V_2+V_3+V_4+V_5,$$ (13)

where $c_1=\frac{A\beta \epsilon }{{\hat{u}}^2\hat{\epsilon }\hat{\alpha }},c_2=\frac{A\beta \epsilon }{\hat{u}{\hat{\epsilon }}^2\hat{\alpha }}and{c}_3=\frac{A\beta \epsilon }{\hat{u}\hat{\epsilon }{\hat{\alpha }}^2}$. Note that $V(S,E,I,Q)$ is not only defines on each point, but also goes to $+\infty$ as $(S,E,I,Q)$ goes to the limit of ${\mathbb{R}}_{+}^4$ and $\Vert (S,E,I,Q)\Vert \to \infty$. Hence, we have a small point $(S\left(0\right),E\left(0\right),I\left(0\right),Q\left(0\right))$ in the domain of ${\mathbb{R}}_{+}^4$. We also take a $C^2-$ operator $\tilde{V}:{\mathbb{R}}_{+}^4\to {\mathbb{R}}_{+}$ as given:

$$\tilde{V}(S,E,I,Q)=N\left(-lnS{c}_1-lnE{c}_2-c_{3}lnI-lnQ+\beta {\int }_t^{\tau +t}I(-\tau +s)ds\right)-lnS+\beta {\int }_t^{\tau +t}I(-\tau +s)ds-lnE-lnQ+\frac{1}{\rho +1}{(S+E+I+Q)}^{\rho +1}-V(S\left(0\right),E\left(0\right),I\left(0\right),Q\left(0\right))≔N{V}_1+V_2+V_3+V_4+V_5-V(S\left(0\right),E\left(0\right),I\left(0\right),Q\left(0\right)),$$ (14)

here $(S,E,I,Q)\in (\frac{1}{n},n)\times (\frac{1}{n},n)\times (\frac{1}{n},n)\times (\frac{1}{n},n)$ and $n>1$ is a so larger integer,

$V_1=-c_{1}lnS-c_{2}lnE-c_{3}lnI-lnQ+\beta {\int }_t^{t+\tau }I(s-\tau )ds$,

$V_2=-lnS+\beta {\int }_t^{t+\tau }I(s-\tau )ds$,

$V_3=-lnE,v_4=-lnQ$,

$V_5=\frac{1}{\rho +1}{(S+E+I+Q)}^{\rho +1}$,   $\rho >1$, fulfilling $\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee V_3^2\vee v_4^2)>0$, and $N>0$ is a so larger value fulfilling condition $-N\delta +R\le -2$, here $\delta =\frac{A\beta \epsilon }{\hat{\mu }\hat{\epsilon }\hat{\alpha }}-(\mu +\alpha +{\gamma }_3+\frac{v_4^2}{2})>0$,

$$R=\underset{(S,E,I,Q)\in {\mathbb{R}}_{+}^4}{sup}(-\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]I^{\rho +1}+3\mu +\epsilon +{\sigma }_1+{\gamma }_1+\alpha +{\gamma }_3+B+\frac{v_1^2}{2}+\frac{v_2^2}{2}+\frac{v_3^2}{2})$$ (15)

and

$$B=\underset{(S,E,I,Q)\in {\mathbb{R}}_{+}^4}{sup}\{A{(S+E+I+Q)}^{\rho }-\frac{1}{2}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]\times {(S+E+I+Q)}^{\rho +1}\}<\infty .$$ (16)

Applying Itô’s formula to $V_1$, we have

$$L{V}_1=-\frac{c_{1}A}{S}+\mu c_1-\frac{cI}{S}+\frac{c_1{v}_1^2}{2}-\frac{c_2\beta SI}{E}+c_2(\mu +\epsilon +{\sigma }_1+{\gamma }_1)+\frac{c_2{v}_2^2}{2}-\frac{c_{3}E\epsilon }{I}+c_3(\mu +\alpha +c+{\gamma }_2)+\frac{c_3{v}_3^2}{2}-\frac{{\sigma }_{1}E}{Q}-\frac{{\sigma }_{2}I}{Q}+(\mu +\alpha +{\gamma }_3)+\frac{v_4^2}{2}+c_1\beta I\left(t\right)\le -3\sqrt[3]{A\beta \epsilon c_1{c}_2{c}_3}+c_1\left(\mu +\frac{v_1^2}{2}\right)+c_2\left(\mu +\epsilon +{\sigma }_1+{\gamma }_1+\frac{v_2^2}{2}\right)$$$$c_3\left(\mu +\alpha +c+{\gamma }_2+\frac{v_3^2}{2}\right)+\left(\mu +\alpha +{\gamma }_3+\frac{v_4^2}{2}\right)+c_1\beta I\left(t\right)\le \frac{A\beta \epsilon }{\hat{\mu }\hat{\epsilon }\hat{\alpha }}+\left(\mu +\alpha +{\gamma }_3+\frac{v_4^2}{2}\right)+c_1\beta I\left(t\right)=-\delta +c_1\beta I\left(t\right).$$ (17)

Similarly, we can get

$$L{V}_2=-\frac{A}{S}+\mu -\frac{cI}{S}+\beta I+\frac{v_1^2}{2}$$ (18)

$$L{V}_3=-\frac{\beta SI(t-\tau )}{E}+(\mu +\epsilon +{\sigma }_1+{\gamma }_1)+\frac{v_2^2}{2}$$ (19)

$$L{V}_4=-\frac{{\sigma }_{1}E}{Q}-\frac{{\sigma }_{2}I}{Q}+(\mu +\alpha +{\gamma }_3)+\frac{v_4^2}{2}$$ (20)

$$L{V}_5={(S+E+I+Q)}^{\rho }[A-\mu (S+E+I+Q)-{\gamma }_{1}E-(\alpha +{\gamma }_2)I-(\alpha +{\gamma }_3)Q]+\frac{\rho }{2}{(S+E+I+Q)}^{\rho -1}\times (v_1^2{S}^2\vee v_2^2{E}^2\vee v_3^2{I}^2\vee v_4^2{Q}^2)\le {(S+E+I+Q)}^{\rho }[A-\mu (S+E+I+Q)]+\frac{\rho }{2}{(S+E+I+Q)}^{\rho +1}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\le A{(S+E+I+Q)}^{\rho }-{(S+E+I+Q)}^{\rho +1}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]\le B-\frac{1}{2}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]{(S+E+I+Q)}^{\rho +1}\le B-\frac{1}{2}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right](S^{\rho +1}+E^{\rho +1}+I^{\rho +1}+Q^{\rho +1}),$$ (21)

$B$ is given in (16). From (17)–(21), we follows

$$L\tilde{V}\le -N\delta +N{c}_1\beta I-\frac{1}{2}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right](S^{\rho +1}+E^{\rho +1}+I^{\rho +1}+Q^{\rho +1})+3\mu +\epsilon +{\sigma }_1+{\gamma }_1+\alpha +{\gamma }_3-\frac{A}{S}-\frac{cI}{S}+\beta I-\frac{{\sigma }_{1}E}{Q}-\frac{{\sigma }_{2}I}{Q}+B+\frac{v_2^2}{2}+\frac{v_1^2}{2}+\frac{v_3^2}{2}\le -N\delta +N{c}_1\beta I-\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right](S^{\rho +1}+E^{\rho +1}+I^{\rho +1}+Q^{\rho +1})+3\mu +\epsilon +{\sigma }_1+{\gamma }_1+\alpha +{\gamma }_3-\frac{A}{S}-\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]I^{\rho +1}-\frac{cI}{S}+\beta I-\frac{{\sigma }_{1}E}{Q}-\frac{{\sigma }_{1}I}{Q}+B+\frac{v_1^2}{2}+\frac{v_2^2}{2}+\frac{v_4^2}{2}$$ (22)

For $\xi >0$, define a bounded closed set

$$D=\left\{(S,E,I,Q)\in {\mathbb{R}}_{+}^4:\xi \le S\le \frac{1}{\xi },\xi \le E\le \frac{1}{\xi },{\xi }^2\le I\le \frac{1}{{\xi }^2},{\xi }^3\le Q\le \frac{1}{{\xi }^3}\right\}.$$

In the set ${\mathbb{R}}_{+}^4\setminus D$, take obeying the conditions as follows:

$$-\frac{A}{\xi }+P\le -1$$ (23)

$$-{\sigma }_1\xi +P\le -1$$ (24)

$$-{\sigma }_1\xi +N{c}_1\beta {\xi }^2+R\le -1$$ (25)

$$-\frac{{\sigma }_2{\xi }^2}{{\xi }^3}+P\le -1$$ (26)

$$-\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]\frac{1}{{\xi }^{\rho +1}}+P\le -1$$ (27)

$$-\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]\frac{1}{{\xi }^{2(\rho +1)}}+P\le -1$$ (28)

$$-\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]\frac{1}{{\xi }^{3(\rho +1)}}+P\le -1$$ (29)

Where

$$P=\underset{(S,E,I,Q)\in {\mathbb{R}}_{+}^4}{sup}\{N{c}_1\beta I-\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]I^{\rho +1}+3\mu +\epsilon +{\sigma }_1+{\gamma }_1+\alpha +{\gamma }_3+B+\frac{v_2^2}{2}+\frac{v_1^2}{2}+\frac{v_4^2}{2}\}$$ (30)

We need to show that $L\tilde{V}\le -1$ for any $(S,E,I,Q)\in {\mathbb{R}}_{+}^4\setminus D$, and ${\mathbb{R}}_{+}^4\setminus D=\left[{\bigcup }_{i=1}^8{D}_i\right]$, where

$$D_1=\left\{(S,E,I,Q)\in {\mathbb{R}}_{+}^4;0<S<\xi \right\},$$$$D_2=\left\{(S,E,I,Q)\in {\mathbb{R}}_{+}^4;0<E<\xi \right\},$$$$D_3=\left\{(S,E,I,Q)\in {\mathbb{R}}_{+}^4;0<I<{\xi }^2,E\ge \xi \right\},$$$$D_4=\left\{(S,E,I,Q)\in {\mathbb{R}}_{+}^4;0<Q<{\xi }^3,I\ge {\xi }^2\right\},$$$$D_5=\left\{(S,E,I,Q)\in {\mathbb{R}}_{+}^4;S>\frac{1}{\xi }\right\},$$$$D_6=\left\{(S,E,I,Q)\in {\mathbb{R}}_{+}^4;E>\frac{1}{\xi }\right\},$$$$D_7=\left\{(S,E,I,Q)\in {\mathbb{R}}_{+}^4;I>\frac{1}{{\xi }^2}\right\},$$$$D_8=\left\{(S,E,I,Q)\in {\mathbb{R}}_{+}^4;Q>\frac{1}{{\xi }^3}\right\}.$$ (31)

Case 1. For any $(S,E,I,Q)\in D_1$, we obtain

$$L\tilde{V}\le -\frac{A}{S}+N{c}_1\beta I-\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]I^{\rho +1}+3\mu +\epsilon +{\sigma }_1+{\gamma }_1+\alpha +{\gamma }_3+B+\frac{v_2^2}{2}+\frac{v_1^2}{2}+\frac{v_4^2}{2}\le -\frac{A}{S}+P\le -\frac{A}{\xi }+P\le -1,$$ (32)

which is obtained from (23). Therefore, $L\tilde{V}\le --1$ for any $(S,E,I,Q)\in D_1$.

Case 2. For any $(S,E,I,Q)\in D_2$, we obtain

$$L\tilde{V}\le -\frac{{\sigma }_{1}E}{Q}+N{c}_1\beta I-\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]I^{\rho +1}+3\mu +\epsilon +{\sigma }_1+{\gamma }_1+\alpha +{\gamma }_3+B+\frac{v_2^2}{2}+\frac{v_1^2}{2}+\frac{v_4^2}{2}\le -\frac{{\sigma }_{1}E}{Q}+P\le -{\sigma }_1\xi +P\le -1,$$ (33)

which follows from (24). Therefore, $L\tilde{V}\le --1$ for any $(S,E,I,Q)\in D_2$.

Case 3. For any $(S,E,I,Q)\in D_3$, we obtain

$$L\tilde{V}\le N{c}_1\beta I-\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]I^{\rho +1}+3\mu +\epsilon +{\sigma }_1+{\gamma }_1+\alpha +{\gamma }_3-\frac{{\sigma }_{1}E}{Q}+B+\frac{v_2^2}{2}+\frac{v_1^2}{2}+\frac{v_4^2}{2}\le -\frac{{\sigma }_{1}E}{Q}++N{c}_1\beta I+R\le -{\sigma }_1\xi +N{c}_1\beta {\xi }^2+R\le -1,$$ (34)

which is obtained from (25). Therefore, $L\tilde{V}\le --1$ for any $(S,E,I,Q)\in D_3$.

Case 4. For any $(S,E,I,Q)\in D_4$, we obtain

$$L\tilde{V}\le -\frac{{\sigma }_{2}I}{Q}+N{c}_1\beta I-\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]I^{\rho +1}+3\mu +\epsilon +{\sigma }_1+{\gamma }_1+\alpha +{\gamma }_3+B+\frac{v_2^2}{2}+\frac{v_1^2}{2}+\frac{v_4^2}{2}\le -\frac{{\sigma }_{2}I}{Q}+P\le -\frac{{\sigma }_2{\xi }^2}{{\xi }^3}+P\le -1,$$ (35)

which is obtained from (26). Therefore, $L\tilde{V}\le --1$ for any $(S,E,I,Q)\in D_4$.

Case 5. For any $(S,E,I,Q)\in D_5$, we obtain

$$L\tilde{V}\le -\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]S^{\rho +1}+N{c}_1\beta I-\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]I^{\rho +1}+3\mu +\epsilon +{\sigma }_1+{\gamma }_1+\alpha +{\gamma }_3+B+\frac{v_2^2}{2}+\frac{v_1^2}{2}+\frac{v_4^2}{2}\le -\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]S^{\rho +1}+P\le -\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]\frac{1}{{\xi }^{\rho +1}}+P\le -1,$$ (36)

which is obtained from (27). Therefore, $L\tilde{V}\le --1$ for any $(S,E,I,Q)\in D_5$.

Case 6. For any $(S,E,I,Q)\in D_6$, we obtain

$$L\tilde{V}\le -\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]E^{\rho +1}+N{c}_1\beta I-\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]I^{\rho +1}+3\mu +\epsilon +{\sigma }_1+{\gamma }_1+\alpha +{\gamma }_3+B+\frac{v_2^2}{2}+\frac{v_1^2}{2}+\frac{v_4^2}{2}\le -\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]E^{\rho +1}+P\le -\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]\frac{1}{{\xi }^{\rho +1}}+P\le -1,$$ (37)

which is obtained from (27). Therefore, $L\tilde{V}\le --1$ for any $(S,E,I,Q)\in D_6$.

Case 7. For any $(S,E,I,Q)\in D_7$, we obtain

$$L\tilde{V}\le -\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]I^{\rho +1}+N{c}_1\beta I-\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]I^{\rho +1}+3\mu +\epsilon +{\sigma }_1+{\gamma }_1+\alpha +{\gamma }_3+B+\frac{v_2^2}{2}+\frac{v_1^2}{2}+\frac{v_4^2}{2}\le -\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]I^{\rho +1}+P\le -\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]\frac{1}{{\xi }^{2\rho +2}}+P\le -1,$$ (38)

which is obtained from (28). Therefore, $L\tilde{V}\le --1$ for any $(S,E,I,Q)\in D_7$.

Case 8. For any $(S,E,I,Q)\in D_8$, we obtain

$$L\tilde{V}\le -\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]Q^{\rho +1}+N{c}_1\beta I-\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]I^{\rho +1}+3\mu +\epsilon +{\sigma }_1+{\gamma }_1+\alpha +{\gamma }_3+B+\frac{v_2^2}{2}+\frac{v_1^2}{2}+\frac{v_4^2}{2}\le -\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]Q^{3\rho +3}+P\le -\frac{1}{4}\left[\mu -\frac{\rho }{2}(v_1^2\vee v_2^2\vee v_3^2\vee v_4^2)\right]\frac{1}{{\xi }^{3\rho +3}}+P\le -1,$$ (39)

this implies from (29). Hence, $L\tilde{V}\le --1$ at any $(S,E,I,Q)\in D_8$.

Consequently, result (ii) of Lemma 1 satisfied. So, we concluded that model (2) have one stationary distribution $\pi (.)$. □

Extinction

For the vanishing or reducing of the epidemic, just see the lemmas as follows.

Lemma 2

Let $M={\left\{M_t\right\}}_{t\ge 0}$ be a real valued define local martingale reduced at t = 0. Then

${lim}_{t\to \infty }{〈M,M〉}_t=\infty a.s.\Rightarrow {lim}_{t\to \infty }\frac{M_t}{{〈M,M〉}_t}=0a.s$ .,

and also

${lim}_{t\to \infty }sup\frac{{〈M,M〉}_t}{t}<\infty a.s.\Rightarrow {lim}_{t\to \infty }\frac{M_t}{t}=0,a.s.$ ,

here ${〈M,M〉}_t$ shows the quadratic variants of M.

Lemma 3

Take $(S\left(t\right),E\left(t\right),I\left(t\right),Q\left(t\right))$ be root of (2) with $(S\left(0\right),E\left(0\right),I\left(0\right),Q\left(0\right))\in {\mathbb{R}}_{+}^4$ , then

${lim}_{t\to \infty }\frac{S\left(t\right)}{t}=0,{lim}_{t\to \infty }\frac{E\left(t\right)}{t}=0,{lim}_{t\to \infty }\frac{I\left(t\right)}{t}=0,{lim}_{\to \infty }\frac{Q\left(t\right)}{t}=0,a.s$ .,

Furthermore, if $\mu >\frac{v_1^2\vee v_2^2\vee v_3^2\vee v_4^2}{2}$ , then

${lim}_{t\to \infty }\frac{{\int }_0^{t}S\left(s\right)d{B}_1\left(s\right)}{t}=0,{lim}_{t\to \infty }\frac{{\int }_0^{t}E\left(s\right)d{B}_2\left(s\right)}{t}=0,{lim}_{t\to \infty }\frac{{\int }_0^{t}I\left(s\right)d{B}_3\left(s\right)}{t}=0$ ,

${lim}_{t\to \infty }\frac{{\int }_0^{t}Q\left(s\right)d{B}_4\left(s\right)}{t}=0$ ,   a.s.

Theorem 3

If $R_0^s<1$ and $\mu >\frac{v_1^2\vee v_2^2\vee v_3^2\vee v_4^2}{2}$ , then root of (2) fulfilling the given as:

$$\underset{t\to \infty }{lim}sup\frac{1}{t}ln({\sigma }_2(E+I)+(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)Q)\le \beta -\frac{1}{2{\left({\sigma }_2\right)}^2}\{{\sigma }_2^2\frac{v_3^2}{2}\wedge {\sigma }_2^2(\mu +\alpha -{\sigma }_1(\alpha +\mu +c+\gamma )+\frac{v_2^2}{2})\wedge {(\alpha +\mu +c+{\sigma }_2+{\gamma }_2)}^2(\alpha +{\gamma }_3+\frac{v_4^2}{2})\}<0$$ (40)

and ${lim}_{t\to \infty }〈S〉=1$ a.s.

Proof

Take $U\left(t\right)={\sigma }_2(E+I)+(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)Q$, applying Ito’s expression, we obtain

$$dlnU\left(t\right)=\{\frac{1}{{\sigma }_2(E+I)+(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)Q}\times [{\sigma }_2\beta SI(t-\tau )-{\sigma }_2(\mu +{\gamma }_1)E+{\sigma }_1(\mu +\alpha +c+{\gamma }_2)E-(\alpha +{\gamma }_3)(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)Q]-\frac{{\sigma }_2^2{v}_2^2{E}^2+{\sigma }_2^2{v}_3^2{I}^2+{(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)}^2{v}_4^2{Q}^2}{2{({\sigma }_2(E+I)+(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)Q)}^2}\}dt+\frac{{\sigma }_2{v}_{2}E}{{\sigma }_2(E+I)+(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)Q}d{B}_2+\frac{{\sigma }_2{v}_{3}I}{{\sigma }_2(E+I)+(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)Q}d{B}_3+\frac{(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)v_{4}Q}{{\sigma }_2(E+I)+(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)Q}d{B}_4\le \beta Sdt-\frac{1}{{\sigma }_2(E+I)+(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)Q}\times \{{\sigma }_2^2\frac{v_3^2}{2}{I}^2+{\sigma }_2^2(\mu +\alpha -{\sigma }_1(\mu +\alpha +c+\gamma )+\frac{v_2^2}{2})E^2+{(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)}^2(\alpha +{\gamma }_3+\frac{v_4^2}{2})Q^2\}dt+\frac{{\sigma }_2{v}_{2}E}{{\sigma }_2(E+I)+(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)Q}d{B}_2+\frac{{\sigma }_2{v}_{3}I}{{\sigma }_2(E+I)+(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)Q}d{B}_3+\frac{(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)v_{4}Q}{{\sigma }_2(E+I)+(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)Q}d{B}_4\le \beta Sdt+\frac{{\sigma }_2{v}_{2}E}{{\sigma }_2(E+I)+(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)Q}d{B}_2+\frac{{\sigma }_2{v}_{3}I}{{\sigma }_2(E+I)+(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)Q}d{B}_3-\frac{1}{2{\left({\sigma }_2\right)}^2}\{{\sigma }_2^2\frac{v_3^2}{2}\wedge {\sigma }_2^2(\mu +\alpha -{\sigma }_1(\mu +\alpha +c+\gamma )+\frac{v_2^2}{2})\wedge {(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)}^2(\alpha +{\gamma }_3+\frac{v_4^2}{2})\}dt+\frac{(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)v_{4}Q}{{\sigma }_2(E+I)+(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)Q}d{B}_4.$$ (41)

From model (2), we have

$$d(S+E+I+Q)$$$$=[A-\mu (S+E+I+Q)-{\gamma }_{1}E-(\alpha +{\gamma }_2)I-(\alpha +{\gamma }_3)Q]dt$$$$+v_{1}Sd{B}_1+v_{2}Ed{B}_2+v_{3}Id{B}_3+v_{4}Qd{B}_4.$$ (42)

Applying integral from $0$ to $t$, we gets

$$〈S+E+I+Q〉=\frac{A}{\mu }+{\psi }_1\left(t\right),$$ (43)

where

$${\psi }_1=\frac{1}{\mu }[\frac{1}{t}(S\left(0\right)+E\left(0\right)+I\left(0\right)+Q\left(0\right))-\frac{1}{t}(S\left(t\right)+E\left(t\right)+I\left(t\right)+Q\left(t\right))$$$$-{\gamma }_1\frac{{\int }_0^{t}E\left(t\right)}{t}-(\alpha +{\gamma }_2)\frac{{\int }_0^{t}I\left(t\right)}{t}-(\alpha +{\gamma }_3)\frac{{\int }_0^{t}Q\left(t\right)}{t}+\frac{v_1{\int }_0^{t}S\left(s\right)d{B}_1}{t}+\frac{v_2{\int }_0^{t}E\left(s\right)d{B}_2}{t}+\frac{v_3{\int }_0^{t}I\left(s\right)d{B}_3}{t}$$$$+\frac{v_4{\int }_0^{t}Q\left(s\right)d{B}_4}{t}].$$ (44)

Using Lemma 2, Lemma 3, we gets ${lim}_{t\to \infty }{\psi }_1\left(t\right)=0$a.s.

Consequently, (43) implies

$$\underset{t\to \infty }{lim}sup〈S+E+I+Q〉=\frac{A}{\mu }a.s.$$ (45)

Taking Integral of (41) we get

$$\frac{lnU\left(t\right)}{t}\le \beta -\frac{1}{2{\left({\sigma }_2\right)}^2}\{{\sigma }_2^2\frac{v_3^2}{2}\wedge {\sigma }_2^2(\mu +\alpha -{\sigma }_1(\mu +\alpha +c+\gamma )+\frac{v_2^2}{2})\wedge {(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)}^2(\alpha +{\gamma }_3+\frac{v_4^2}{2})\}+{\psi }_2,$$ (46)

where

$${\psi }_2\left(t\right)=\frac{lnU\left(0\right)}{t}+\frac{{\sigma }_2{v}_2}{t}{\int }_0^t\left(\frac{E\left(s\right)}{{\sigma }_2(E+I)+(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)Q}d{B}_2\right)+\frac{{\sigma }_2{v}_3}{t}{\int }_0^t\left(\frac{\left(s\right)}{{\sigma }_2(E+I)+(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)Q}d{B}_3\right)+\frac{(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)v_4}{t}{\int }_0^t\left(\frac{Q\left(s\right)}{{\sigma }_2(E+I)+(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)Q}d{B}_4\right).$$ (47)

Again from Lemma 2, Lemma 3, we approaches

$$\underset{t\to \infty }{lim}{\psi }_2\left(t\right)=0a.s.$$

Since $R_0^s<1$, so, by applying the above limit of (46), we get

$$\underset{t\to \infty }{lim}sup\frac{lnU\left(t\right)}{t}\le \beta -\frac{1}{2{\left({\sigma }_2\right)}^2}\{{\sigma }_2^2\frac{v_3^2}{2}\wedge {\sigma }_2^2(\mu +\alpha -{\sigma }_1(\mu +\alpha +c+\gamma )+\frac{v_2^2}{2})\wedge {(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)}^2(\alpha +{\gamma }_3+\frac{v_4^2}{2})\}<0,$$ (48)

This shows that ${lim}_{t\to \infty }E\left(t\right)=0,{lim}_{t\to \infty }I\left(t\right)=0,{lim}_{t\to \infty }Q\left(t\right)=0$ a.s., which leads to the disease $I$ may vanish with occurrence chance of one.

This will be easy, by (45), (48), to derive that ${lim}_{t\to \infty }〈S〉=\frac{A}{\mu }a.s$. □

Numerical simulations for stochastic stability

Now, we provide numerical simulations for the illustration and validation of our obtained theoretical scheme. For this, we apply stochastic iterative techniques of fourth order Runge Kutta method and to obtain the following discretization-transformation of model (2),

$$S_{i+1}=S_i+\left[A-\beta S_i{I}_i(t-\tau )-\mu S_i\left(t\right)+c{I}_i\left(t\right)\right]△t$$$$+v_1{S}_i\sqrt{△t}{\zeta }_{1,i}+\frac{v_1^2}{2}{E}_i({\zeta }_{1,i}^2-1)△t$$$$E_{i+1}=E_i+\left[\beta S_i\left(t\right)I_i(t-\tau )-(\mu +\epsilon +{\sigma }_1+{\gamma }_1)E_i\left(t\right)\right]△t$$$$+v_2{E}_i\sqrt{△t}{\zeta }_{2,i}+\frac{v_2^2}{2}{E}_i({\zeta }_{2,i}^2-1)△t$$$$I_{i+1}=I_i+\left[\epsilon E_i\left(t\right)-(\mu +\alpha +c+{\sigma }_2+{\gamma }_2)I_i\left(t\right)\right]△t$$$$+v_3{I}_i\sqrt{△t}{\zeta }_{3,i}+\frac{v_3^2}{2}{Y}_i({\zeta }_{3,i}^2-1)△t$$$$Q_{i+1}=Q_i+\left[{\sigma }_1{E}_i\left(t\right)+{\sigma }_2{I}_i\left(t\right)-(\mu +\alpha +{\gamma }_3)Q_i\left(t\right)\right]△t$$$$+v_4{Q}_i\sqrt{△t}{\zeta }_{4,i}+\frac{v_4^2}{2}{Z}_i({\zeta }_{4,i}^2-1)△t$$ (49)

Where ${\zeta }_{k,i}(k=1,2,3,4)$, are four free Gaussian general variables having $N(0,1)$ and time-increment $\Delta t>0$.

To bring out numerical simulation for investigating the dynamics of stochastic stability and for optimality, we have to mention the values of the parameters given in the problem (2) and some parameters for optimal control.

Now we deals with the numerical approximation and biological feasibility of system (2) through numerical simulation. So we considered the parameters and the noises intensities values from Table 2 (Set A). For $t\in [0--200]$ units and the various initial classes size for each compartment individuals susceptible $S\left(t\right)$, exposed $E\left(t\right)$, infected $I\left(t\right)$, and quarantine $Q\left(t\right)$, are given in Table 2.

Table 2.

The values for Parameters given in model (2).

Parameters	Set A	References	Set B	References
$\Lambda$	0.9	Considered	5	Considered
${\gamma }_1$	0.02	Considered	0.02	Considered
${\gamma }_2$	0.03	Considered	0.02	Considered
${\gamma }_3$	0.02	Considered	0.02	Considered
$\beta$	0.05	Considered	5	Considered
$\mu$	0.02	Considered	0.05	Considered
${\sigma }_1$	0.05	Considered	0.5	Considered
${\sigma }_2$	0.5	Considered	0.5	Considered
$\alpha$	0.02	Considered	0.2	Considered
$\epsilon$	0.03	Considered	0.2	Considered
$\tau$	1	Considered	1	Considered
$c$	0.05	Considered	0.02	Considered
$v_1$	0.2	Considered	0.6	Considered
$v_2$	0.3	Considered	0.9	Considered
$v_3$	0.6	Considered	0.7	Considered
$v_4$	0.4	Considered	0.11	Considered
$S\left(0\right)$	10	Considered	10	Considered
$E\left(0\right)$	5	Considered	5	Considered
$I\left(0\right)$	2	Considered	2	Considered
$Q\left(0\right)$	10	Considered	10	Considered

Taking white noises into account and parameter value from Table 2 (Set A), then Theorem 3 deals with sufficient results for maintenance in the average and vanished, which may be proved from the stochastic approach theory. Hence, the result of Theorem 3 is fulfilled. It means that the epidemic in the system (2) finishes with one chance of occurrence. Now for the corresponding deterministic approach the epidemic free equilibrium point showing global asymptotic stability. The disease may die as can be seen in Fig. 1.

Fig. 1.

Trajectories of stochastic $SEIQ$ model (2) and its corresponding deterministic version.

Next, in Theorem 2 an ergodic techniques is used to prove that the stochastic system has a one stationary distribution. For stochastic system (2), we consider parameter values from Table 2 (Set B) and compute $R_0^s>1$, so, Theorem 2 is fulfilled. As seen in Fig. 2, the infection of problem (2) will lie in average which validate the results of Theorem 2, hence implies that problem (2) lie an ergodic stationary distribution. Observe $1000$ attempts at $t=200$, then compute average value, Theorem 2 implies that system (2) lie an ergodic stationary distribution as Fig. 3 explained this situation.

Fig. 2.

Trajectories of stochastic $SEIQ$ model (2) and its corresponding deterministic version.

Fig. 3.

Histogram for the stochastic $SEIQ$ model (2) and its corresponding deterministic version.

Example 1 Stochastic Disease-Free Dynamical Behavior —

We take the parameter values from Table 2 (Set A). So we compute the reproduction number $R_0^s<1$ and by Theorem 3 the root of system (2) may fulfilled

$$\underset{t\to \infty }{lim}sup\frac{\text{log}E\left(t\right)}{t}\le <0,a.s.$$

and

$$\underset{t\to \infty }{lim}sup\frac{\text{log}I\left(t\right)}{t}\le <0,a.s.$$

Hence this implies that the epidemic will vanish from the community as in Fig. 1 shows that the numerical-simulation verify our scheme.

Example 2 Stochastic Endemic Dynamical Behavior —

By same fashion we take the parameter values from Table 2 (Set B). We prove $R_0^s>1$, and by Theorem 2, the disease will lie or stable, and provide simulation to show our results in Fig. 2. Theorem 2 implies that system (2) has one stationary distribution which is verified by Fig. 2.

Conclusion

In the concluding remarks of this manuscript, we have established a SEIQ disease system having time-delay for the new-strain corona-virus COVID-19 in the sense of stochastic approach. The stochastically taken agents are treated in the system as white Gaussian noises because of external environment variances. We have derived some sufficient results for maintenance and reduction of disease in the average of the epidemic. The system has one stationary distribution which is ergodic for small intensities of white noises. Lastly, the numerical representation through various plotting is given for verification of our obtained results.

The researchers know that the stochastic SEIQ system is the try to know epidemiological properties of COVID-19. The system gives new scenes into epidemiological conditions when the environment noises (perturbations) and crossing-immunity are taken in the COVID-19 disease systems. The composition of white noises and time-delay, in the infectious systems, have more effect on the lying and vanishing of the infection and complicated the dynamical behavior of the system. The work of this paper may be analyzed by including controlling variables like vaccination and other treatment type actions.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.