Introduction

Noted that coronavirus attacks the respiratory system of children, which was initially identified in the 1960s. The severe acute respiratory infection significantly increased illness and death, discovered in 20031. Acute respiratory syndrome was a major outcome of coronavirus which was designated by the International Committee on Virus Taxonomy2. In3 a novel coronavirus-related respiratory sickness began in Wuhan, China in December 2019, and very fast it spread to other regions of China and other countries of the world. From 1 to 15 January, recorded some cases of COVID-19 and also noted the unreported cases and reproduction number \({R}_{0}\). As of 5 April 2020, there had been 3819 cases of coronavirus, in including 106 death cases, since 30 January, the first case of 2019-nCoV was reported in India. The researchers have examined a Bats–Hosts–Reservoir–People transmission fractional-order COVID-19 model in this work to simulate possible transmission while accounting for individual responses and government control measures4. The symptoms of COVID-2019 vary according to the severity but to an extent, symptoms are the same as those caused by other family members of SARS like the common cold, respiratory disease, etc5. In6 employs artificial intelligence to establish the frame work for data driven forecast of 2019-CoV and mathematics to predict some confinement strategies. In7 Mathematics is blessed with a unique field that relates physical phenomena to calculations, named 'Modeling'. Health agencies collect information which is related to the death rate and recovered persons, as well as the number of persons who tested, symptomatic, or asymptomatic due to COVID-2019, and publish it daily. For accuracy of data compilation, the modeling process gives appropriate value. In8 the world economy has been severely impacted by the infectious and fatal coronavirus or COVID-19. This fear will probably reverberate across the world of education. There are several debates surrounding e-learning. Some of the arguments surrounding online pedagogy like life-long learning, accessibility, etc. It's said that Internet education is widely available and may even reach outlying and rural locations. During the COVID-19 pandemic remaining at home is no longer a safe option for a large number of mothers and children worldwide. During the lockdown, there is an increase in the number of reports of assaulting kids and domestic violence. Raising awareness of domestic abuse is so essential9. The research which relied on a large number of instructors and students from institutions in the Arab world, attempts to highlight the challenges to attaining quality in distance learning during COVID-19. The different ways that COVID-19-related institutional suspensions caused students to continue their education at home10. Even though several studies have been conducted, the study on the COVID-19 pandemic's effect on teaching and learning worldwide concludes that more research is necessary to determine appropriate pedagogy and platforms for various class levels in higher secondary, middle, and primary education in developing nations11. The coronavirus crisis put the educational system under strain and compelled teachers to switch entirely to online instructions. Modern teaching technologies have expanded the scope of education, and its practical orientation, intensified students, and increased their cognitive activity as a result of remote learning of foreign languages during the COVID-19 shutdown12. The actual threat posed by COVID-19 is still unknown, but due to the large number of cases and the virus's quick spread, it has brought the virus into the public eye and forced organizations to emergency plans and take preventative measures13. Many models released their work during COVID-19 in an effort to deliver the best data with highest level of precision. The SIR model was selected in order to obtain the highest possible ratio of COVID-19 patient; nevertheless, this model also required time to reach the pandemic level. Researchers investigate in a variety of methods in an effort to develop a cure for the virus and guarantee public safety as quick as feasible. There are dynamic symptoms of COVID-2019 which include mild, respiratory disorders, aches, fever, sore throat, etc.14. In 2020, Hellewell et al.15 worked on the isolation technique for COVID-2019 and were successful in this case. The National Center for disease control of Georgia recommended that the education process be discontinued nationwide in colleges and schools on March 02, 2020, citing the COVID-19 outbreak as the reason for the suspension of the current system16. According to its human-to-human transmission, SARS-CoV-2 is becoming to more widespread and posing a serious threat to public health. It is still unknown who SARS-CoV-2 intermediate hosts. It is essential to identify the potential intermediate host of SARS-CoV-2 to stop17. Numerous interconnected attitudinal elements impact student satisfaction in distant online learning. One such element, particularly for students who propose their education in a second language is self-efficacy in academic L2 use18. As a nosocomial illness that was acquired in the community, MERS-CoV-2 is transmitted from person to person. The study determined which patients were at high risk and required specialized cases to enhance their prognosis19. In southern China, the severe acute respiratory syndrome (SARS) first appeared in 2002–2003. Its etiological agent, the SARS coronavirus, has yet to be identified. We show here that bat species are natural hosts of coronavirus closely related to the ones causing the SARS pandemic20. In21, the authors studied the dynamics of a fractional-order delayed model of COVID-19 with vaccination efficacy. In22, the author gave a study to delay differential equations and applications to biology. In23,24,25, the authors studied the dynamics of a stochastic epidemic model with time delays for COVID‐19 including different mathematical aspects.

Moving away from a mathematical model that may explain the spread of infections such as COVID-19, we will focus on the current study. Concerning the existence of several distinct groups of people and their complex conversations, the mathematical model is a deterministic system. We study the generic expansion of this system in the form of a stochastic differential equation to introduce a delay element. We present the extension in its most generic version, considering states and probabilities. One of the more probabilistic, realistic, and nature-inspired modeling methods is stochastic modeling. Put another way, stochastic delayed modeling is the true means of understanding the dynamics of infectious diseases and the effect they have on physical processes. Here a stochastic model was used to represent the coronavirus in the human population. Our main goal is to provide an analysis of the stochastic coronavirus model that preserves dynamical structure. This motivates us to investigate the stochastic COVID-19 model's non-standard computational analysis due to the unavailability of analytical solutions. The model is an SIR system, that divides the human population into several smaller groups such as S for susceptible (who are not affected by the virus), I for infected (who are affected by the virus), and R for recovered (who recovered their virus due to hospitalization or vaccination).

The paper is organized into six sections. "Model formulation", model formulation and its feasible properties. "Reproduction number", stochastic model with delay formulation (stochastic delay differential equations (SDDEs) and their properties like positivity, boundedness, extinction, and persistence of disease. "Euler Maruyama method", for the implementation of standard and non-standard finite difference methods, and "Results" and "Conclusion", for results and conclusion.

Model formulation

In this section, we talk about the dynamical properties of COVID-19 which remained common in the past few years (See Fig. 1). Let us consider, \(N\left(t\right)\) represents as total population and \(N\left(t\right),t\ge 0\) is as defined \(N:\left[0,\infty \right)\to \mathcal{R}.\) Similarly, \(S, I, R:\left[0,\infty \right)\to \mathcal{R}\) represented the relation of population components, which are non-negative differential functions. Where \(S(t)\) denotes the susceptible humans, \(I(t)\) denotes the infected persons, \(and R(t)\) represents recovered persons.

Figure 1
Figure 1
Full size image

Flow dynamics of coronavirus26.

The delay differential equations of the given model as shown:

$${{S}^{\prime}(t)}={\text{a}}-cS(t-\tau )I(t-\tau )(1+\gamma I(t-\tau )){e}^{-\mu \tau }-\mu S(t)+\alpha R(t),\forall t\ge 0,$$
(1)
$${I}^{\prime}\left(t\right)=cI\left(t-\tau \right)S\left(t-\tau \right)\left(1+\gamma I\left(t-\tau \right)\right){e}^{-\mu \tau }-\left(\beta +\mu +\delta -b\right)I\left(t\right),\forall t\ge 0,$$
(2)
$${R}^{\prime}\left(t\right)={\beta {\rm I}}\left(\text{t}\right)-\left(\alpha +\mu \right)R\left(t\right),\forall t\ge 0,$$
(3)

\(N\left(t\right)=S\left(t\right)+I\left(t\right)+R(t)\) which is satisfied at time \(t\ge 0 and \tau <t.\)

$$S\left(0\right)\ge 0,I\left(0\right)\ge 0,T\left(0\right)\ge 0,R\left(0\right)\ge 0.$$
(4)

Existence and uniqueness of the model

To explore the fundamental properties of the model, firstly we verify that the system (1–3) has bounded, unique, and global positive solutions. In system (1–3), variables display the negative free solution \(\forall\) times \(t\ge 0\) with non-negative initial conditions.

Theorem 1

The system (1–3) is bounded at any time t, if \(\underset{t\to \infty }{\text{lim}}SupN(t)\le \frac{a}{\mu }\).

Proof

The population function is \(N(t). So,\)

$$N(t) = S(t) + I(t) + R(t).$$

Therefore,

$$\frac{dN}{dt}= \frac{dS}{dt}+ \frac{dI}{dt}+\frac{dR}{dt}$$
$$= a-\left(\delta -b\right)I\left(t\right)- \mu N=0$$
$$N\left(t\right)= N\left(0\right){e}^{-\mu t}+\frac{a}{\mu },$$
(5)

\(\underset{t\to \infty }{\text{lim}}supN(t)\le \frac{a}{\mu }\), as desired.

Theorem 2

(Uniqueness) For any time t, the system (1–3) shows the uniqueness and existence of the solutions.

Proof

Firstly, we will explain the term norm.

\({\Vert \lambda \Vert }_{\infty }={sup}_{t\epsilon {D}_{\lambda }}\left|\lambda (t)\right|,\) and also considered the Banach Space. To determine the results of the given problems, we need to verify the growth and Lipschitz condition. Let us consider three positive constants \({M}_{1}, {M}_{2}\) and \({M}_{3}<\infty\) such that \({S}_{\infty }<{M}_{1}\) , \({I}_{\infty }<{M}_{2}\) and \({R}_{\infty }<{M}_{3}\)

We have,

$$\left\{\begin{array}{c}{S}^{\prime}={f}_{1}(S,I,R,t)\\ {I}^{\prime}={f}_{2}\left(S,I,R,t\right) \\ {R}^{\prime}={f}_{3}(S,I,R,t)\end{array}\right., \forall t\ge 0$$
(6)

We verify that,

$${\left|{f}_{i}(S,t)\right|}^{2}<{k}_{i}\left({\left|{S}_{i}\right|}^{2}+1\right), where \,i=\text{1,2},3$$
(7)
$${\left|{f}_{i}\left({S}^{1},t\right)-{f}_{i}\left({S}^{2},t\right)\right|}^{2}<\overline{{k }_{i}}{\left|{S}^{1}-{S}^{2}\right|}^{2}.$$
(8)

For proof, we consider the function \({f}_{1}\left(S, I,R,t\right)\), and the following assumptions are hold.

$${\left|{f}_{1}\right|}^{2}= {\left|a-cSI\left(1+\gamma I\right){e}^{-\mu \tau }-\mu S+\alpha R\right|}^{2}.$$
(9)
$${\left|{f}_{1}\right|}^{2} \le {\left|a\right|}^{2}+3{\left|cSI\left(1+\gamma I\right){e}^{-\mu \tau }\right|}^{2}+3{\left|\mu S+\alpha R\right|}^{2}.$$
$${\left|{f}_{1}\right|}^{2} \le ({\left|a\right|}^{2}+{sup}_{t\epsilon {D}_{\lambda }}{\left|cSI\left(1+\gamma I\right){e}^{-\mu \tau }\right|}^{2}+{sup}_{t\epsilon {D}_{\lambda }}{\left|\mu S+\alpha R\right|}^{2}).$$
$${\left|{f}_{1}\right|}^{2} \le ({\left|a\right|}^{2}+\left|c\right|SI{e}^{-\mu \tau }(1+\gamma {\Vert S\Vert }_{\infty })+{\left|\mu \right|}^{2}{\Vert S\Vert }_{\infty }+{\left|\alpha \right|}^{2}{\Vert R\Vert }_{\infty }).$$
$${\left|{f}_{1}\right|}^{2} \le \left({\left|a\right|}^{2}+{\left|\mu \right|}^{2}{\Vert S\Vert }_{\infty }+{\left|\alpha \right|}^{2}{R}_{\infty }\right)\left(1+\frac{\left|c\right|{\Vert S\Vert }_{\infty }{\Vert I\Vert }_{\infty }{e}^{-\mu \tau }\left(1+\gamma {\Vert I\Vert }_{\infty }\right)}{{\left|a\right|}^{2}+{\left|\mu \right|}^{2}{\Vert S\Vert }_{\infty }+{\left|\alpha \right|}^{2}{\Vert R\Vert }_{\infty }}\right).$$
$$\text{where }\left(\frac{\left|c\right|{\Vert S\Vert }_{\infty }{\Vert I\Vert }_{\infty }{e}^{-\mu \tau }\left(1+\gamma {\Vert I\Vert }_{\infty }\right)}{{\left|a\right|}^{2}+{\left|\mu \right|}^{2}{\Vert S\Vert }_{\infty }+{\left|\alpha \right|}^{2}{\Vert R\Vert }_{\infty }}\right)<1.$$
$$\text{So}, {\left|{f}_{1}(S,I,R,t)\right|}^{2} <{k}_{1}\left(1+{\left|S\right|}^{2}\right)$$
(10)

Similarly, for function \({f}_{2}\left(S,I,R,t\right)\), we get

$${\left|{f}_{2}\right|}^{2}= {\left|cSI{e}^{-\mu \tau }\left(1+\gamma I\right)-\left(\beta +\mu +\delta -b\right)I\right|}^{2}.$$
(11)
$${\left|{f}_{2}\right|}^{2} \le {\left|cSI{e}^{-\mu \tau }\left(1+\gamma I\right)\right|}^{2}+{\left|\left(\beta +\mu +\delta -b\right)I\right|}^{2}.$$
$${\left|{f}_{2}\right|}^{2} \le ({sup}_{t\epsilon {D}_{\lambda }}{\left|cSI{e}^{-\mu \tau }\left(1+\gamma I\right)\right|}^{2}+{{sup}_{t\epsilon {D}_{\lambda }}\left|\left(\beta +\mu +\delta -b\right)I\right|}^{2})$$
$${\left|{f}_{2}\right|}^{2} \le (\left|c\right|{\Vert S\Vert }_{\infty }{\Vert I\Vert }_{\infty }{e}^{-\mu \tau }\left(1+\gamma {\Vert I\Vert }_{\infty }\right)+{\left|\left(\beta +\mu +\delta -b\right)\right|}^{2}{\Vert I\Vert }_{\infty })$$
$${\left|{f}_{2}\right|}^{2} \le \left({\left|\left(\beta +\mu +\delta -b\right)\right|}^{2}{\Vert I\Vert }_{\infty }\right)\left(1+\frac{\left|c\right|{\Vert S\Vert }_{\infty }{\Vert I\Vert }_{\infty }{e}^{-\mu \tau }\left(1+\gamma {\Vert I\Vert }_{\infty }\right)}{{\left|\left(\beta +\mu +\delta -b\right)\right|}^{2}{\Vert I\Vert }_{\infty }}\right).$$
$$\text{where }\left(\frac{\left|c\right|{\Vert S\Vert }_{\infty }{\Vert I\Vert }_{\infty }{e}^{-\mu \tau }\left(1+\gamma {\Vert I\Vert }_{\infty }\right)}{{\left|\left(\beta +\mu +\delta -b\right)\right|}^{2}{\Vert I\Vert }_{\infty }}\right)<1.$$
$${\left|{f}_{2}\left(S,I,R,t\right)\right|}^{2} <{k}_{2}\left(1+{\left|I\right|}^{2}\right).$$
(12)

For \({f}_{3},\) we have

$${\left|{f}_{3}\right|}^{2}={\left|\beta I-\left(\alpha +\mu \right)R\right|}^{2}.$$
(13)
$${\left|{f}_{3}\right|}^{2}\le {\left|\beta I\right|}^{2}+{\left|\left(\alpha +\mu \right)R\right|}^{2}.$$
$${\left|{f}_{3}\right|}^{2}\le \left({sup}_{t\epsilon {D}_{\lambda }}{\left|\beta I\right|}^{2}+{sup}_{t\epsilon {D}_{\lambda }}{\left|\left(\alpha +\mu \right)R\right|}^{2}\right).$$
$${\left|{f}_{3}\right|}^{2}\le \left({\left|\beta \right|}^{2}{\Vert I\Vert }_{\infty }+{\left|\alpha +\mu \right|}^{2}{\Vert R\Vert }_{\infty }\right).$$
$${\left|{f}_{3}\right|}^{2}\le \left({\left|\alpha +\mu \right|}^{2}{\Vert R\Vert }_{\infty }\right)(1+\frac{{\left|\beta \right|}^{2}{\Vert I\Vert }_{\infty }}{{\left|\alpha +\mu \right|}^{2}{\Vert R\Vert }_{\infty }})$$
$$\text{Where }(\frac{{\left|\beta \right|}^{2}{\Vert I\Vert }_{\infty }}{{\left|\alpha +\mu \right|}^{2}{\Vert R\Vert }_{\infty }})<1.$$
$$\text{So}, {\left|{f}_{3}\right|}^{2}< {k}_{3}\left({1+\left|R\right|}^{2}\right).$$
(14)

Therefore, the condition of linear growth is verified if

\(max\left\{\frac{\left|c\right|SI{e}^{-\mu \tau }\left(1+\gamma {\Vert I\Vert }_{\infty }\right)}{{\left|a\right|}^{2}+{\left|\mu \right|}^{2}{\Vert S\Vert }_{\infty }+{\left|\alpha \right|}^{2}{\Vert R\Vert }_{\infty }}, \frac{\left|c\right|SI{e}^{-\mu \tau }\left(1+\gamma {\Vert I\Vert }_{\infty }\right)}{{\left|\left(\beta +\mu +\delta -b\right)\right|}^{2}{\Vert I\Vert }_{\infty }},\frac{{\left|\beta \right|}^{2}{\Vert I\Vert }_{\infty }}{{\left|\alpha +\mu \right|}^{2}{\Vert R\Vert }_{\infty }} \right\} <1,\) is desired.

Theorem 3

(Positivity) For given data in Eq. (4) (\(S\left(0\right), I\left(0\right), R(0))\epsilon {R}_{+}^{3}\), then the solution of the system (1–3) is positive for all time \(t>0\).

Proof

Let us consider a given system is as follows:

$${S}^{{{\prime}}}=\text{a}-cSI\left(1+\gamma \right){e}^{-\mu \tau }-\mu S+\alpha R.$$
$${S}^{{{\prime}}}\ge -\left[cI\left(1+\gamma I\right){e}^{-\mu \tau }+\mu \right]S.$$
$$\Rightarrow S\left(t\right)=S\left(0\right)\times {e}^{-\left[cI\left(1+\gamma I\right){e}^{-\mu \tau }+\mu \right]t}\ge 0.$$

Similarly, \(I\left(t\right)\ge I\left(0\right)\times {e}^{\left[\beta +\mu +\delta -b\right]t}\ge 0,R\left(t\right)\ge R\left(0\right)\times {e}^{-\left[\alpha +\mu \right]t}\ge 0\), as desired.

Equilibrium points

For calculating the equilibrium points of the system, we consider that the state variables are constant. Let put the right side of the system (1–3) equal to zero. The system (1–3) is in equilibrium condition as shown below:

  1. (i)

    Corona virus free equilibrium = \({E}_{1}=\left({S}^{1}, {I}^{1}, {R}^{1}\right)=(\frac{a}{\mu },\text{0,0})\)

  2. (ii)

    Corona virus existing equilibrium = \({E}_{2}=\left({S}^{*}, {I}^{*}, {R}^{*}\right)\),

Where:\({S}^{*}=\frac{a+\alpha {R}^{*}}{1+(c{I}^{*}\left(1+\gamma {I}^{*}\right)-\mu )}\), \({I}^{*}=(\frac{\beta +\mu +\delta -b}{{cS}^{*}\gamma {e}^{-\mu \tau }}-c{S}^{*})\), \({R}^{*}=\frac{\beta {I}^{*}}{\alpha +\mu }\).

Reproduction number

For finding the reproduction number, we use the next-generation method which is presented in27. This method gives an idea about transmission matrices represented as \(F\) and transition matrices represented as \(V\) respectively. In this way, we ignored the susceptible part and we took the remaining two parts infectious denoted by \(I\left(t\right)\), and recovered denoted by \(R\left(t\right).\) To obtain the result we use corona-free equilibrium points. Then we get,

$$F=\left[\begin{array}{cc}\frac{ca}{\mu }{e}^{-\mu \tau }& 0\\ 0& 0\end{array}\right]\text{ and }V=\left[\begin{array}{cc}\beta +\mu +\delta -b& 0\\ 0& \alpha +\mu \end{array}\right]$$

Then find \(F{V}^{-1}\).

$$F{V}^{-1}=\left[\begin{array}{cc}\frac{ca}{\mu (\beta +\mu +\delta -b)}{e}^{-\mu \tau }& 0\\ 0& 0\end{array}\right]$$

The spectral radius of \(\rho (F{V}^{-1})\) is denoted by \({R}_{0}=\frac{ac}{\mu (\beta +\mu +\delta -b)}{e}^{-\mu \tau }\).

Stability analysis

In this part, we check the local and global stability of the system (1–3), by using the equilibrium points.

Theorem 4

The system (1–3) at \({C}_{1}=\left({S}^{1},{I}^{1},{R}^{1}\right)=\left(\frac{a}{\mu }, \text{0,0}\right)\) is locally asymptotically stable if \({R}_{0}<1\).

Proof

By using the Jacobian Matrix of the system (1–3), as follows:

$$J=\left[\begin{array}{ccc}\frac{\partial F}{\partial S}& \frac{\partial F}{\partial I}& \frac{\partial F}{\partial R}\\ \frac{\partial G}{\partial S}& \frac{\partial G}{\partial I}& \frac{\partial G}{\partial R}\\ \frac{\partial H}{\partial S}& \frac{\partial H}{\partial I}& \frac{\partial H}{\partial R}\end{array}\right]$$
$$\text{J }= \left[\begin{array}{ccc}-cI\left(1+\gamma I\right){e}^{-\mu \tau }-\mu & cS\gamma (1+\gamma I){e}^{-\mu \tau }& \alpha \\ cI\left(1+\gamma I\right){e}^{-\mu \tau }& cS\gamma (1+\gamma I){e}^{-\mu \tau }-\left(\beta +\mu +\delta -b\right)& 0\\ 0& \beta & -\left(\alpha +\mu \right)\end{array}\right]$$
$$\text{J}({C}_{1}) = \left[\begin{array}{ccc}-\mu & c\frac{a}{\mu }\gamma {e}^{-\mu \tau }& \alpha \\ 0& c\frac{a}{\mu }\gamma {e}^{-\mu \tau }-\left(\beta +\mu +\delta -b\right)& 0\\ 0& \beta & -\left(\alpha +\mu \right)\end{array}\right]$$

Consider \(\left|J-\lambda I\right|=0.\)

So,

$$J=\left|\begin{array}{ccc}-\mu -\lambda & c\frac{a}{\mu }\gamma {e}^{-\mu \tau }& \alpha \\ 0& c\frac{a}{\mu }\gamma {e}^{-\mu \tau }-\left(\beta +\mu +\delta -b\right)-\lambda & 0\\ 0& \beta & -\left(\alpha +\mu \right)-\lambda \end{array}\right|=0$$
$${\lambda }_{1}=-\mu <0, {\lambda }_{2}= c\frac{a}{\mu }\gamma {e}^{-\mu \tau }-\left(\beta +\mu +\delta -b\right)<0.$$
$${\lambda }_{3}= -\left(\alpha +\mu \right)<0.$$

It shows that \({C}_{1}\) is asymptotically stable, \({R}_{0}<1\).

Theorem 5

The system (1–3) at \({C}_{2}=\left({S}^{*},{I}^{*},{R}^{*}\right)\) is locally asymptotically stable if \({R}_{0}>1.\)

Proof

By using the Jacobian Matrix of system (1–3) at \({C}_{2}\), we obtained

$$J\left({C}_{2}\right)=\left[\begin{array}{ccc}-c{I}^{*}\left(1+\gamma {I}^{*}\right){e}^{-\mu \tau }-\mu & c{S}^{*}\gamma (1+\gamma {I}^{*}){e}^{-\mu \tau }& \alpha \\ c{I}^{*}\left(1+\gamma {I}^{*}\right){e}^{-\mu \tau }& c{S}^{*}\gamma (1+\gamma {I}^{*}){e}^{-\mu \tau }-\left(\beta +\mu +\delta -b\right)& 0\\ 0& \beta & -\left(\alpha +\mu \right)\end{array}\right]$$

Consider \(\left|J({C}_{2})-\lambda I\right|=0.\)

$$\left|\begin{array}{ccc}-c{I}^{*}\left(1+\gamma {I}^{*}\right){e}^{-\mu \tau }-\mu -\lambda & c{S}^{*}\gamma (1+\gamma {I}^{*}){e}^{-\mu \tau }& \alpha \\ c{I}^{*}\left(1+\gamma {I}^{*}\right){e}^{-\mu \tau }& c{S}^{*}\gamma (1+\gamma {I}^{*}){e}^{-\mu \tau }-\left(\beta +\mu +\delta -b\right)-\lambda & 0\\ 0& \beta & -\left(\alpha +\mu \right)-\lambda \end{array}\right|=0.$$
$${\lambda }^{3}+{\lambda }^{2}{A}_{2}+\lambda {A}_{1}+{A}_{0}=0$$

where \({A}_{2}=\) \(c{I}^{*}\left(1+\gamma {I}^{*}\right){e}^{-\mu \tau }-\mu +c{S}^{*}\gamma \left(1+\gamma {I}^{*}\right){e}^{-\mu \tau }-A-\alpha -\mu\),

$$\begin{aligned}{A}_{1} & = [{c}^{2}{I}^{*}{S}^{*}\gamma \left(1+\gamma {I}^{*}\right){e}^{-\mu \tau }-Ac{I}^{*}\left(1+\gamma {I}^{*}\right){e}^{-\mu \tau }-\alpha c{I}^{*}\left(1+\gamma {I}^{*}\right){e}^{-\mu \tau }-\mu c{I}^{*}\left(1+\gamma {I}^{*}\right){e}^{-\mu \tau }\\ &\quad+\mu c{S}^{*}\gamma \left(1+\gamma {I}^{*}\right){e}^{-\mu \tau }-\mu A-\alpha A-{\mu }^{2}+\alpha c{S}^{*}\gamma \left(1+\gamma {I}^{*}\right){e}^{-\mu \tau }+\mu c{S}^{*}\gamma \left(1+\gamma {I}^{*}\right){e}^{-\mu \tau }\\ &\quad-\alpha A-\mu A]\end{aligned}$$
$$\begin{aligned}{A}_{0} & = [\alpha {c}^{2}{I}^{*}{S}^{*}\gamma \left(1+\gamma {I}^{*}\right){e}^{-\mu \tau }+\mu {c}^{2}{I}^{*}{S}^{*}\gamma \left(1+\gamma {I}^{*}\right){e}^{-\mu \tau }-\alpha Ac{I}^{*}\left(1+\gamma {I}^{*}\right){e}^{-\mu \tau }-\mu Ac{I}^{*}\left(1+\gamma {I}^{*}\right){e}^{-\mu \tau } \\ & \quad - \mu \alpha c{S}^{*}\gamma \left(1+\gamma {I}^{*}\right){e}^{-\mu \tau }+{\mu }^{2}c{S}^{*}\gamma \left(1+\gamma {I}^{*}\right){e}^{-\mu \tau }-\mu \alpha A-{\mu }^{2}A+\alpha {c}^{2}{I}^{*}{S}^{*}\gamma \left(1+\gamma {I}^{*}\right){e}^{-\mu \tau }\\ & \quad +\mu {c}^{2}{I}^{*}{S}^{*}\gamma \left(1+\gamma {I}^{*}\right){e}^{-\mu \tau } +\alpha \beta c{I}^{*}\left(1+\gamma {I}^{*}\right){e}^{-\mu \tau }.\end{aligned}$$

By Routh Hurwitz's criterion when \({R}_{0}>1, here \,{A}_{2},{A}_{1, }and {A}_{0}\) are positive. So, \({C}_{2}\) is locally asymptotically stable.

Theorem 6

The system (1–3) at \({C}_{1}=\left({S}^{1},{I}^{1},{R}^{1}\right)=\left(\frac{a}{\mu },\text{0,0}\right)\) is globally asymptotically stable if \({R}_{0}<1.\)

Proof

To prove this theorem, we let the Lyapunov function \(U:\Omega \to R\) defined as

$$U\left(I\right)=\text{ln}\left(\frac{I}{{I}_{0}}\right),$$
$$\frac{dU\left(I\right)}{dt}=\frac{{I}^{{{\prime}}}}{I}\times \frac{1}{{I}^{{{\prime}}}}\frac{dI}{dt},$$
$$\frac{dU}{dt}\le \left(\beta +\mu +\delta -b\right)\left[\frac{ca{e}^{-\mu \tau }}{\mu \left(\beta +\mu +\delta -b\right)}-1\right].$$
$$\frac{dU}{dt}\le \left(\beta +\mu +\delta -b\right)\left({R}_{0}-1\right).$$
$$\frac{dU}{dt}\le 0 , \text{if }{R}_{0} <1.$$

Hence, the system is globally asymptotically stable.

Theorem 7

The system (1–3) is globally asymptotically stable at \({C}_{2}=({S}^{*},{I}^{*},{R}^{*})\), if \({R}_{0}>1\).

Proof

By Lyapunov function, \(X:\Omega \to R\) defined as:

$$X=\left(S-{S}^{*}-{S}^{*}log\frac{S}{{S}^{*}}\right)+\left(I-{I}^{*}-{I}^{*}log\frac{I}{{I}^{*}}\right)+\left(R-{R}^{*}-{R}^{*}log\frac{R}{{R}^{*}}\right).$$
$$\frac{dX}{dt}=\left(-\frac{a}{S{S}^{*}}{\left(S-{S}^{*}\right)}^{2}-\frac{\alpha R}{S{S}^{*}}{\left(S-{S}^{*}\right)}^{2}\right)-(\frac{\beta I}{R{R}^{*}}{\left(R-{R}^{*}\right)}^{2})\le 0.$$

So, \(\frac{dx}{dt}\le 0,\) Hence, \({C}_{2}\), is globally asymptotically stable.

Transition probabilities

We employed the terminology \(U\left(t\right)={[S\left(t\right),I\left(t\right),R\left(t\right)]}^{T}\) to give the stochastic extension of this model. The quantity of possibilities for an occurrence is shown in Table 1 as shown below:

Table 1 Representation of changes in parameters.
Full size table

Next, we calculated the drift and diffusion of Eq. (1–3) as shown below:

$${\rm Expectation}={\text{E}}^{*}\left[\Delta \text{U}\right]=\sum_{\text{i}=1}^{7}{\textrm{P}}_{\text{i}}{\textrm{T}}_{\text{i}}=\left[\begin{array}{c}\text{a}-cSI\left(1+\gamma I\right){e}^{-\mu \tau }-\mu S+\alpha R\\ cIS\left(1+\gamma I\right){e}^{-\mu \tau }-(\beta +\mu +\delta -b)I\\ {\beta{\rm I}}-\left(\alpha +\mu \right)R\end{array}\right]\Delta t.$$
$$\text{Variance}={\text{E}}^{*}\left[\Delta \text{U}{\Delta \text{U}}^{\text{T}}\right]=\sum_{\text{i}=1}^{7}{\text{P}}_{\text{i}}{[\text{T}}_{\text{i}}{{][\text{T}}_{\text{i}}]}^{\text{T}}.$$

Drift \(=\text{H}\left(\text{U}\left(\text{t}\right),\text{t}\right)=\frac{{\text{E}}^{*}\left[\Delta \text{U}\right]}{\Delta \text{t}}\), Diffusion \(=\text{K}\left(\text{U}(\text{t}),\text{t}\right)=\sqrt{\frac{{\text{E}}^{*}[\Delta \text{U}{\Delta \text{U}}^{\text{T}}]}{\Delta \text{t}}}\), so

$$\text{dU}\left(\text{t}\right)=\text{H}\left(\text{U}(\text{t}),\text{t}\right)\text{dt}+\text{K}\left(\text{U}\left(\text{t}\right),\text{t}\right)\text{dB}\left(\text{t}\right).$$
(15)

The above Eq. (15) is called the stochastic delay differential equation. Where \(B(t)\) is Brownian motion.

Euler Maruyama method

The standard numerical methodology will be discussed in the proceeding section for approximation of stochastic model solutions. This is agreed by us that \({I}_{n}=\{\text{0,1},\text{2,3},\dots n\}\). Consider \(N\in {\mathbb{N}}\), as well as the consistent division of the temporal interval \([0,T]\) with uniform partition equal to \(\tau =\frac{T}{N}\), and their respective nodes are presented as \(0={t}_{0}<{t}_{1}<{t}_{2}<\dots {t}_{N}=T\).

For each \(n\in {I}_{N}\). Further, this will be agreed by us \({U}^{n}=U({t}_{n})\), however \(n\in {I}_{N}\) and \(U(t)={(S,I,R)}^{t}\). \(\Delta {W}_{n}=W\left({t}_{n}+1\right)-W({t}_{n})\).

Stochastic delayed model

The phrase "time delay" is used in transmission terms to denote the duration of viral infection's incubation or the interval between infection and the start of symptoms. The addition of time delay in the model could result in many periodic solutions for varying time delay \(\tau\) values. Considered that \(cdt=cdt+\sigma dB(t)\) with parametric perturbation technique for the Eqs. (1–3) as follows28,

$$\left\{\begin{array}{c}dS\left(\text{t}\right)=\left[\text{a}-\text{cS}\left(\text{t}\right)\text{I}\left(\text{t}\right)\left(1+{\gamma {\rm I}}\left(\text{t}\right)\right){\text{e}}^{-{\mu \tau }}-{\mu {\rm S}}\left(\text{t}\right)+{\alpha {\rm R}}\left(\text{t}\right)\right]dt-\sigma c\left[\text{S}\left(\text{t}\right)\text{I}\left(\text{t}\right)\left(1+{\gamma {\rm I}}\left(\text{t}\right)\right)\right]dB(t), \\ dI\left(\text{t}\right)=[cI(t)S(t)\left(1+{\gamma {\rm I}}\left(\text{t}\right)\right){\text{e}}^{-{\mu \tau }}-(\beta +\mu +\delta -b)I(t)]dt+\sigma cS\left(\text{t}\right)I\left(\text{t}\right)\left(1+{\gamma {\rm I}}\left(\text{t}\right)\right)dB(t),\\ dR\left(\text{t}\right)=\left[{\beta {\rm I}}\left(\text{t}\right)-\left({\alpha }+\upmu \right)\text{R}\left(\text{t}\right)\right]dt, \end{array}\right.$$
(16)

In the above equations, \(\tau\) is the delay effect, \(\upsigma\) is the randomness, \(\forall \text{ t}\ge 0\) and \(\text{B}(\text{t})\) is Brownian motion. These equations are non-integrable due to Brownian motion.

Positivity and boundedness

Let's introduce a probability space which is represented as \((\Omega , F, P)\) and \({\{{F}_{t}\}}_{t\epsilon R}\) represented as a filtration. For satisfying the right continuous conditions and increasing29, \({F}_{0}\) contains P-null sets. Symbolically represented as \(U\left(t\right)=(S\left(t\right), I\left(t\right), R\left(t\right))\) and norm as \(\left|U(t)\right|=\sqrt{{S}^{2}\left(t\right)+{I}^{2}\left(t\right)+{R}^{2}(t)}\).

Theorem 8

For model (16) and any given initial value \(\left(S\left(0\right),I\left(0\right),R\left(0\right)\right)\in {R}_{+}^{3}\), there is a unique solution \(\left(S\left(t\right),I\left(t\right),R\left(t\right)\right)\), which is defined on \(t\in [0,\infty ]\) and the solution of the system will remain in \({\mathbb{R}}_{+}^{3}\) with probability one.

Proof

The initial value \(\left(S\left(0\right),I\left(0\right),R\left(0\right)\right)\epsilon {\mathbb{R}}_{+}^{3}\) of the model (16) satisfies the local Lipschitz condition, and has a unique local solution existing on \(t\in \left[0,{t}_{e}\right],{t}_{e}\) shows the explosion time. Next to show that the solution of the system is global, \({t}_{e}=\infty\).

Suppose \({d}_{0}\ge 1\) for initial values, they all lie within the interval [\(\frac{1}{{d}_{0}},{d}_{0}]\). For each integer, \(d\ge {d}_{0}\), this sequence is known as stopping time.

$${\tau }_{d}=\text{inf}\left\{t\in \left[0,{t}_{e}\right]:\text{min}\left(S\left(t\right),I\left(t\right),R\left(t\right)\right)\le \frac{1}{d} or\text{max}\left(S\left(t\right),I\left(t\right),R\left(t\right)\right)\ge d\right\}.$$
(17)

where inf ∅ = ∞ (∅ is an empty set) \(.\)

By the definition of stopping time, \({\tau }_{d}\) is increasing function as \(d\to \infty\).

$$\text{So}, {\tau }_{\infty }={lim}_{d\to \infty }{\tau }_{d}$$
(18)
$${\tau }_{\infty }\le {\tau }_{e}$$

If \({\tau }_{\infty }=\infty\), then \({\tau }_{e}=\infty\).

If this argument is false, then there exists a constant \(T>0\) and \(\varepsilon \epsilon (\text{0,1})\) such that,

$$P\{{\tau }_{\infty }\le T\}>\varepsilon .$$
(19)

There exists an integer \({d}_{1}\ge {d}_{0}\).

$$P\left\{{\tau }_{d}\le T\right\}>\varepsilon , \forall d\ge {d}_{1}.$$
(20)

For \(t\le {\tau }_{d}\), we get

$$d\left(S+I+R\right)=\left[a-cSI\left(1+\gamma I\right){e}^{-\mu \tau }-\mu S+\alpha R+cSI\left(1+\gamma I\right){e}^{-\mu \tau }-\left(\beta +\mu +\delta -b\right)I+\beta I-\left(\alpha +\mu \right)R\right].$$
$$=\left[a-\mu \left(S+I+R\right)-\delta I+bI\right]dt.$$
$$\le \left[a-\mu \left(S+I+R\right)\right]dt.$$

After calculation, we see that

$$S\left(t\right)+I\left(t\right)+R\left(t\right)\le \left\{\begin{array}{c}\begin{array}{cc}\frac{a}{\mu } & if \,S\left(0\right)+I\left(0\right)+R\left(0\right)\le \frac{a}{\mu }\end{array}\\ \begin{array}{cc}otherwise& S\left(0\right)+I\left(0\right)+R\left(0\right)\ge \frac{a}{\mu }\end{array}\end{array}\right.$$

Define a \({C}^{3}\)-function \(V:{R}_{+}^{3}\to {R}_{+}\) by

$$V\left(S,I,R\right)=\left(S-1-lnS\right)+\left(I-1-lnI\right)+\left(R-1-lnR\right).$$
(21)

By using Ito’s formula, we can obtain

$$dV\left(S,I,R\right)=\left(1-\frac{1}{S}\right)dS+\left(1-\frac{1}{I}\right)dI+\left(1-\frac{1}{R}\right)dR+\frac{{\sigma }^{2}}{2}dt.$$

Putting values of \(dS,dI,dR.\)

$$=\left[a+2\mu +\beta +\delta +b+\frac{{\sigma }^{2}}{2}\right]dt+\sigma c\left(S-I\right)\left(1+\gamma I\right){e}^{-\mu \tau }dB\left(t\right).$$

Let, \(M=a+2\mu +\beta +\delta +b+\frac{{\sigma }^{2}}{2}\); where M is a constant. Then above equation as

$$dV\left(S,I,R\right)\le Mdt+\sigma c\left(S-I\right)\left(1+\gamma I\right){e}^{-\mu \tau }dB\left(t\right).$$
(22)

Integrating above equation from limit \(0 to {\tau }_{d}\wedge \tau\). Now,

$${\int }_{0}^{{\tau }_{d}\wedge \tau }dV\left(S,I,R\right)\le {\int }_{0}^{{\tau }_{d}\wedge \tau }MdS+{\int }_{o}^{{\tau }_{d}\wedge \tau }\sigma c\left(S-I\right)\left(1+\gamma I\right){e}^{-\mu \tau }dB\left(t\right)$$
(23)

where \({\tau }_{d}\wedge \tau =\text{min}({\tau }_{d},T)\). Now, taking expectations.

$$EV\left(S\left({\tau }_{d}\wedge \tau \right),I\left({\tau }_{d}\wedge \tau \right),R\left({\tau }_{d}\wedge \tau \right)\right)\le V\left(S\left(0\right),I\left(0\right),R\left(0\right)\right)+MT.$$
(24)

Let, \({\Omega }_{d}=\{{\tau }_{d}\le \tau \}\)

$$P=\left\{{\Omega }_{d}\right\}>\varepsilon .$$

For every, \(u \epsilon {\Omega }_{d}\). For some \(i, {v}_{i}({\tau }_{d},u)\) equals either \(d or \frac{1}{d}\) for i = 1, 2, 3.

Hence,

\(V(S\left({\tau }_{d},u\right),I\left({\tau }_{d},u\right),R\left({\tau }_{d},u\right))\) is less \(\text{min}\left\{d-1-lnd, \frac{1}{d}-1-ln\frac{1}{d}\right\},\) we obtain

$$V\left(S\left(0\right),I\left(0\right),R\left(0\right)\right)+MT\ge E\left[{I}_{{\Omega }_{d}u}V\left(S\left({\tau }_{d},u\right),I\left({\tau }_{d},u\right),R\left({\tau }_{d},u\right)\right)\right] \ge \varepsilon [(d-1-lnd)\wedge (\frac{1}{d}-1-ln\frac{1}{d} )]$$
(25)

Letting \({\varvec{d}}\to \boldsymbol{\infty },\) leads to the contradiction,

\(\infty =V\left(S\left(0\right),I\left(0\right),R\left(0\right)\right)+MT<\infty\) as desired.

Extinction and persistence

Consider \(B(t)\) represents the Brownian motion and \(I(t)\) be the Ito drift–diffusion process that satisfies the stochastic model as:

$$dI\left(t\right)=\mu \left(I\left(t\right),t\right)dt+\sigma \left(I\left(t\right),t\right)dB\left(t\right).$$

If \(k(I,t)\in {C}^{2}({\mathcal{R}}^{2},\mathcal{R})\) then \(k(I,t)\) is also Ito drift diffusion. So,

$$d\left(k\left(I,t\right)\right)=\frac{dk}{dt}\left(I\left(t\right),t\right)dt+\frac{dk}{dt}\left(I\left(t\right),t\right)dB\left(t\right)+\frac{1}{2}\frac{{d}^{2}k}{{dt}^{2}}\left(I\left(t\right),t\right)dB{\left(t\right)}^{2}.$$

Theorem 9

The model (16) has contained \((S(t), I(t), R(t))\) be the solution with the initial value \((S(0), I(0), R(0))\), then we have two conditions

\(\left(\text{i}\right)\quad If {\sigma }^{2}>\frac{ca{e}^{-\mu \tau }}{2\mu \left(\beta +\mu +\delta -b\right)}-\left(\beta +\mu +\delta -b\right)\) holds, then \({lim}_{t\to \infty }\frac{lnI(t)}{t}>0\).

\((ii) \quad If {\sigma }^{2}<\frac{ca{e}^{-\mu \tau }}{2\mu \left(\beta +\mu +\delta -b\right)}-(\left(\beta +\mu +\delta -b\right)\)holds, then \({lim}_{t\to \infty }sup\frac{lnI(t)}{t}\le 0\).

Proof

Consider \(the I\) component of SDDE’s as:

$$\text{dI}\left(\text{t}\right)=[\text{cI}(\text{t})\text{S}(\text{t})\left(1+{\gamma{\rm I}}\left(\text{t}\right)\right){\text{e}}^{-{\mu \tau }}-(\upbeta +\upmu +\updelta -\text{b})\text{I}(\text{t})]\text{dt}+{\sigma {\rm cS}}\left(\text{t}\right)\text{I}\left(\text{t}\right)\left(1+{\gamma {\rm I}}\left(\text{t}\right)\right)\text{dB}(\text{t})$$

Applying Ito’s formula, we get

$$dln\left(I\right)={k}^{{{\prime}}}\left(I\right)dI+\frac{1}{2}{k}^{{{\prime\prime}}}\left(I\right){I}^{2}{\sigma }^{2}dt.$$
$$dln\left(I\right)=\left(cS\left(1+\gamma I\right){e}^{-\mu \tau }-\left(\beta +\mu +\delta -b\right)-\frac{1}{2}{\sigma }^{2}\right)dt+\sigma cS\left(1+\gamma I\right){e}^{-\mu \tau }dB.$$

Integrate with respect to \(I\) on both sides from \(0 to t,\) we get

$$\text{ln}\left(I\right)=\text{ln}\left(0\right)+{\int }_{0}^{t}\left(cS\left(1+\gamma I\right){e}^{-\mu \tau }-\left(\beta +\mu +\delta -b\right)-\frac{1}{2}{\sigma }^{2}\right)dt+{\int }_{0}^{t}\sigma cS\left(1+\gamma I\right){e}^{-\mu \tau }dB.$$

\(If {\sigma }^{2}>\frac{ca{e}^{-\mu \tau }}{2\mu \left(\beta +\mu +\delta -b\right)}-\left(\beta +\mu +\delta -b\right)\), then

$$\frac{\text{ln}(I)}{t}>\left(\frac{ca{e}^{-\mu \tau }}{2\mu \left(\beta +\mu +\delta -b\right)}-\left(\beta +\mu +\delta -b\right)\right)+\frac{\text{M}\left(t\right)}{t}+\frac{\text{lnI}(0)}{t}$$
$${lim}_{t\to \infty }\frac{\text{ln}(I)}{t}>\left(\frac{ca{e}^{-\mu \tau }}{2\mu \left(\beta +\mu +\delta -b\right)}-\left(\beta +\mu +\delta -b\right)\right)>0$$

\(If {\sigma }^{2}<\frac{ca{e}^{-\mu \tau }}{2\mu \left(\beta +\mu +\delta -b\right)}-(\left(\beta +\mu +\delta -b\right)\) , then

$$\frac{\text{ln}(I)}{t}<\left(\beta +\mu +\delta -b\right)\left(\frac{ca{e}^{-\mu \tau }}{\mu {\left(\beta +\mu +\delta -b\right)}^{2}}-\frac{{\sigma }^{2}}{2\left(\beta +\mu +\delta -b\right)}-1\right)+\frac{\text{M}\left(t\right)}{t}+\frac{\text{lnI}(0)}{t}.$$
$${lim}_{t\to \infty }sup\frac{\text{ln}(I)}{t}<\left(\beta +\mu +\delta -b\right)\left({R}_{0}^{s}-1\right),$$

Therefore, \({R}_{0}^{s}<1,\) \({R}_{0}^{s}={R}_{0}^{d}-\frac{{\sigma }^{2}}{2\left(\beta +\mu +\delta -b\right)}<1\)

$${lim}_{t\to \infty }sup\frac{I(t)}{t}\le 0.$$

\({lim}_{t\to \infty }I\left(t\right)=0\) as desired.

Numerical analysis

In this section, we will analyze some standard and non-standard numerical methodologies of the solution of the stochastic model and present some numerical results. It is important to note that this method has the benefits of offering limited computational cost and extremely precise estimates. Furthermore, it is important to note that applying the non-standard method may ensure the preservation of the number of intriguing aspects of appropriate solutions.

Stochastic non-standard computational method

Our parametric perturbation model of Eq. (16) can be expressed in the non-standard computational method which is named as stochastic non-standard finite difference method.

$$\frac{dS}{dt}=\left[a-cSI\left(1+\gamma I\right){e}^{-\mu \tau }-\mu S+\alpha R\right]-\sigma c[SI\left(1+\gamma I\right]\frac{dB}{dt}.$$
(26)

Equation (16) decomposed in the stochastic non-standard finite difference method as shown:

$$\frac{{S}^{n+1}-{S}^{n}}{h}=\left[a-c{S}^{n+1}{I}^{n}\left(1+\gamma {I}^{n}\right){e}^{-\mu \tau }-\mu {S}^{n+1}+\alpha {R}^{n}\right]dt-\sigma c\left[{S}^{n+1}{I}^{n}\left(1+\gamma {I}^{n}\right){e}^{-\mu \tau }\right]\Delta {B}_{n}.$$
(27)

Same as Eq. (27), which is in the stochastic NSFD process, we write Eq. (16) in this way accordingly.

$${S}^{n+1}=\frac{{S}^{n}+ha+h\alpha {R}^{n}}{1+hc{I}^{n}\left(1+\gamma {I}^{n}\right){e}^{-\mu \tau }+h\mu +\sigma hc{I}^{n}\left(1+\gamma {I}^{n}\right){e}^{-\mu \tau }\Delta {B}_{n}},$$
(28)
$${I}^{n+1}=\frac{{I}^{n}+hc{{S}^{n}I}^{n}\left(1+\gamma {I}^{n}\right){e}^{-\mu \tau }+hb{I}^{n}+\sigma hc{{S}^{n}I}^{n}\left(1+\gamma {I}^{n}\right){e}^{-\mu \tau }\Delta {B}_{n}}{1+h(\beta +\mu +\delta -b)},$$
(29)
$${R}^{n+1}=\frac{{R}^{n}+h\beta {I}^{n}}{1+h(\alpha +\mu )},$$
(30)

where, \(n=\text{0,1},2,\dots\) and \(\Delta {\text{B}}_{\text{n}}=\Delta {\text{B}}_{{t}_{n+1}}-\Delta {\text{B}}_{{t}_{n}}\) is standardized distribution. i.e. \(\Delta {\text{B}}_{\text{n}}\sim \text{N}(0, 1\)).

Convergence analysis

In this section, we present some theorems related to convergence analysis.

Theorem 10

The region \(\Gamma =\{{S}^{n}\ge 0,{I}^{n}\ge 0,{R}^{n}\ge 0; {S}^{n}+{I}^{n}+{R}^{n}\le \frac{a}{\mu }\}\) is feasible for Eq. (30–32), \(\forall n\ge 0\), where n is the positive function.

Proof

The Eq. (27–30) decayed as shown below.

$$\frac{{S}^{n+1}-{S}^{n}}{\text{h}}=(\left[a-c{{S}^{n}I}^{n}\left(1+\gamma {I}^{n}\right){e}^{-\mu \tau }-\mu {S}^{n}+\alpha {R}^{n}\right]dt-\sigma c[{S}^{n}{I}^{n}\left(1+\gamma {I}^{n}\right){e}^{-\mu \tau }]\Delta {B}_{n}$$

Same as for \(I \,and \,R\),

$$\frac{{I}^{n+1}-{I}^{n}}{\text{h}}=(\left[c{{I}^{n}S}^{n}\left(1+\gamma {I}^{n}\right){e}^{-\mu \tau }-(\beta +\mu +\delta -b){I}^{n}\right]dt-\sigma c[{S}^{n}{I}^{n}\left(1+\gamma {I}^{n}\right){e}^{-\mu \tau }]\Delta {B}_{n}$$
$$\frac{{R}^{n+1}-{R}^{n}}{\text{h}}=(\beta {I}^{n}-\left(\alpha +\mu \right){R}^{n}).$$

These equations are added up then we get,

$$\frac{{(S}^{n+1}+{I}^{n+1}+{R}^{n+1})-({S}^{n}+ {I}^{n}+{R}^{n})}{h}=a-\mu \left({S}^{n}+ {I}^{n}+{R}^{n}\right)-(\delta -b){I}^{n}$$
$$\frac{{(S}^{n+1}+{I}^{n+1}+{R}^{n+1})-({S}^{n}+ {I}^{n}+{R}^{n})}{h}\le a-\mu \left({S}^{n}+ {I}^{n}+{R}^{n}\right)$$
$${(S}^{n+1}+{I}^{n+1}+{R}^{n+1})\le \frac{a}{\mu }.$$

The result is bounded \(\forall n\ge 0\).

Theorem 11

(for stability) Our computational analysis method is stable if eigenvalues lie in the unit circle for any \(n\ge 0\) and \(\Delta {B}_{n}=0.\)30

Proof

Let, \(A=\frac{S+ha+h\alpha R}{1+hcSI\left(1+\gamma I\right){e}^{-\mu \tau }+k\mu }, B=\frac{1+hcSI(1+\gamma I){e}^{-\mu \tau }}{1+h\left(\beta +\delta +\mu -b\right)I}, C=\frac{R+h\beta I}{1+h(\alpha +\mu )}\).

The system (27–30) converges to equilibrium points of the model, it satisfies the condition \(\rho (J)<1\), where \(\rho (J)\) is spectral radius of the Jacobean. If \(\rho (J)>1\), it is unstable then \(\rho \left(J\right)=1\) is neutrally stable. The Jacobian matrix as

$$J=\left[\begin{array}{ccc}\frac{\partial A}{\partial S}& \frac{\partial A}{\partial I}& \frac{\partial A}{\partial R}\\ \frac{\partial B}{\partial S}& \frac{\partial B}{\partial I}& \frac{\partial B}{\partial R}\\ \frac{\partial C}{\partial S}& \frac{\partial C}{\partial I}& \frac{\partial C}{\partial R}\end{array}\right]$$
$$\frac{\partial A}{\partial S}=\frac{\partial }{\partial S}\left(\frac{S+ha+h\alpha R}{1+hcSI\left(1+\gamma I\right){e}^{-\mu \tau }+h\mu }\right), \frac{\partial A}{\partial S}=\frac{1}{1+hcI\left(1+\gamma I\right){e}^{-\mu \tau }+k\mu }$$
$$\frac{\partial A}{\partial I}=\frac{\partial }{\partial I}\left(\frac{S+ha+h\alpha R}{1+hcSI\left(1+\gamma I\right){e}^{-\mu \tau }+h\mu }\right), \frac{\partial A}{\partial I}=\frac{(S+ha+h\alpha R)(hc+2hc\gamma I{e}^{-\mu \tau })}{{(1+hcI\left(1+\gamma I\right){e}^{-\mu \tau })}^{2}}$$
$$\frac{\partial A}{\partial R}=\frac{\partial }{\partial R}\left(\frac{S+ha+h\alpha R}{1+hcSI\left(1+\gamma I\right){e}^{-\mu \tau }+h\mu }\right), \frac{\partial A}{\partial R}=\frac{h\alpha }{1+hcI(1+\gamma I)}$$
$$\frac{\partial B}{\partial S}=\frac{hcI(1+\gamma I){e}^{-\mu \tau }}{1+h(\beta +\mu +\delta -b)},\frac{\partial B}{\partial I}=\frac{1+hcS+2hcS\gamma I+hb}{1+h(\beta +\mu +\delta -b)}, \frac{\partial B}{\partial R}=0$$
$$\frac{\partial C}{\partial S}=0,\frac{\partial C}{\partial I}=\frac{h\beta }{1+h(\alpha +\beta )},\frac{\partial C}{\partial R}=\frac{1}{1+h(\alpha +\beta )}.$$

At corona free equilibrium point \({E}_{1}=(\frac{a}{\mu },\text{0,0})\). The Jacobian matrix is

$$J({E}_{1})=\left[\begin{array}{ccc}\frac{1}{1+h\mu }& -\left(\frac{a}{\mu }+ha\right)hc& h\alpha \\ 0& \frac{1+\frac{hac}{\mu }+hb}{1+h(\beta +\mu +\delta -b)}& 0\\ 0& \frac{h\beta }{1+h(\alpha +\beta )}& \frac{1}{1+h(\alpha +\beta )}\end{array}\right]$$

The eigenvalues are \({\lambda }_{1}=\frac{1}{1+h\mu }<1\), \({\lambda }_{2}=\frac{\mu +hac\mu +h\mu b}{\mu (1+h\left(\beta +\mu +\delta -b\right))}<1,\) \({\lambda }_{3}=\frac{1}{1+h(\alpha +\beta )}<1.\)

At endemic equilibrium point \({E}_{2 }=({S}^{*},{I}^{*},{R}^{*})\). The Jacobian matrix is

$$J({E}_{2})=\left[\begin{array}{ccc}\frac{1}{1+hc{I}^{*}\left(1+\gamma {I}^{*}\right){e}^{-\mu \tau }+k\mu }& \frac{({S}^{*}+ha+h\alpha {R}^{*})(hc+2hc\gamma {I}^{*}{e}^{-\mu \tau })}{{(1+hc{I}^{*}\left(1+\gamma {I}^{*}\right){e}^{-\mu \tau })}^{2}}& \frac{h\alpha }{1+hc{I}^{*}(1+\gamma {I}^{*})}\\ \frac{hc{I}^{*}(1+\gamma {I}^{*}){e}^{-\mu \tau }}{1+h(\beta +\mu +\delta -b)}& \frac{1+hc{S}^{*}+2hc{S}^{*}\gamma {I}^{*}+hb}{1+h(\beta +\mu +\delta -b)}& 0\\ 0& \frac{h\beta }{1+h(\alpha +\beta )}& \frac{1}{1+h(\alpha +\beta )}\end{array}\right]$$

By using MATLAB and the values of parameters given in Table 2, verified that the maximum eigenvalues for both Jacobian \(J({E}_{1})\) and \(J({E}_{2})\) are less than one, as desired.

Table 2 Description of model parameters.
Full size table

Results

In this section, we concentrate on recognizing the essential computational characteristics of the nonlinear computational model (27–30). Specifically, we aim to demonstrate the dynamical properties like positivity, boundedness, and dynamical consistency through computational methods around the steady state of the stochastic delayed coronavirus model. Furthermore, by analyzing the moduli of the discrete system's eigenvalues, we can show the stability of the discrete model. We show some results with the help of graphs of coronavirus existing equilibrium points. These comparison graphs are plotted with computational methods like Stochastic Euler, Stochastic Runge Kutta, and stochastic NSFD method under tiny changes of time step size. Also, the effect of delay on susceptible and infected compartments has been observed. The infected human rate converges to zero, when the value \(\tau =0.7\), \(\tau\) is the delay effect, and the susceptible rate increases. Figure 2a, c, e observed that the results of the system merge at the desired equilibrium points with a small time step size through the numerical methods like stochastic Euler, stochastic Runge Kutta and stochastic NSFD. Time is defined in days and in Fig. 2b, d, we obtain unbounded and negative results with very small changes in time step size. Its means that stochastic Euler and stochastic Runge Kutta violate the dynamical properties of the model and depend on time step size. But Fig. 2f, shows that the results from the stochastic NSFD is consistent with biological nature of the model and independent of time size method. In Fig. 3a, b, (delay effect on infected person), we conclude that for the different values of \(\tau\), the rate of infected persons converges to zero. It means that coronavirus-controlled, susceptible rate increases. Figure 4, admits a comparison of the delay effect and reproduction number as desired.

Figure 2
Figure 2
Full size image

Comparison between computational methods of corona existing equilibrium (a) the behavior of sub-populations for coronavirus existing equilibrium by using stochastic Euler method at \(h=0.01\). (b) The stochastic Euler method shows divergent behavior at \(h=1\), (c) represent the behaviour of CVEE by using stochastic Rung Kutta method at \(h=0.01\) (d). The stochastic Rung-Kutta shows unbounded result at \(h=2\), (e) the behaviour of sub-populations for coronavirus equlibrium by using stochastic NSFD method at \(h=0.01\) (f). The stochastic NSFD method shows the convergent behavior in at \(h=100.\)

Figure 3
Figure 3
Full size image

Graph of delay effect on susceptible and infected humans. (a) Infected results for different values of delay. (b) Susceptible results for different values of delay.

Figure 4
Figure 4
Full size image

Comparison of reproduction number and delay effect with time step size.

Conclusion

In the present study, a stochastic delayed mathematical model examined in this work may be used to simulate the spread of infectious illness. Specifically, the stochastic delayed mathematical model examined in this paper might be able to explain how the coronavirus spreads. In a deterministic system, the mathematical model takes into consideration the existence of several subpopulations and their nonlinear interactions. A generic expansion of this system in the form of a stochastic delay differential equation was developed. This study examines an extension that takes into consideration different states and transition probabilities with parametric perturbation; the most generic version of the extension is provided here. The positivity, boundedness, extinction, and persistence of the stochastic delayed mathematical model have been studied rigorously. Also, some standard numerical methods are used to simulate its results like Euler Maruyama, stochastic Euler, and stochastic Runge Kutta. Unfortunately, these methods did not restore the dynamical properties of the model. Then the newly developed method like nonstandard finite difference in the sense of stochastic with delay for the particular model has been designed. The proposed approach restores the feasible region of the model without any constraints. Also, the delay effect has been observed for both susceptible and infected classes. The increase in delay means that the susceptibility of humans increases in other words the infectivity of humans decreases and eventually moves to zero and vice versa. In the future, we shall extend our results to other types of modeling of real-world problems.