The coronavirus infection is a highly contagious viral disease that has effect and spread to almost every country in the world. It is caused by a virus called Severe Acute Respiratory Syndrome coronavirus 2 (SARS-CoV-2) and that was firstly identified in December 2019. In March 2020, the World Health Organization (WHO) declared COVID-19 a pandemic due to its rapid spread and impact on millions of people worldwide [1]. After certain time period COVID-19 has affected more people than previous outbreaks of MERS and SARS. The initial outbreak of COVID-19 cases was detected in Wuhan, China. Many individuals infected with proposed virus may not show any symptoms or only experience mild symptoms. However, those with symptoms typically experience a muscle or joint pain, dry cough, fever and in many active cases, specially shortness of breath which was more painful stage for effected people. COVID-19 poses a significant risk to individuals with pre-existing health conditions such as diabetes, kidney disease, cardiovascular and obesity disease. Most of the people with chronic respiratory diseases are especially vulnerable to severe complications and death from COVID-19. The primary mode of transmission is through direct contact with attenuated objects or surfaces, as well as consumption of respiratory aerosol from infected individuals, whether they show symptoms or not [2].

The COVID-19 pandemic continuously to show mannerism a significant threat to people worldwide, particularly due to the increasing incidence of infection in certain countries. Consequently, it has become crucial to detect, assess, and understand the exempt feedback to SARS-CoV-2 disease. While diseases like Polio can be controlled by regular vaccination to the children below a certain 5 years of age and most of medicines and antibiotics can help to manage the outbreaks of infection, some infectious diseases such as TB, Cholera, and Malaria still cause fatalities globally [3]. However, recently some effective and safe vaccines have been developed by many countries still rely on non pharmaceutical mediation like social distancing, lock downs, and face masks to contest the infection. Nonetheless, vaccination remains an effective and simple strategy in controlling infectious diseases, as demonstrated by the successful eradication of smallpox through vaccination [4]. Consequently, scientists worldwide are agreeing in development of effective COVID-19 vaccines, recognizing their potential to safe countless lives as a new tool in the battle against the virus [4].

In order to gain a deeper comprehension of how epidemic illnesses work and suggest more effective approaches to prevent their spread, mathematical models can be highly valuable. These models serve as helpful tools for predicting the patterns and progression of diseases, allowing us to implement strategies that can reduce the rate of infection and mortality. Recent advancements have introduced new mathematical models specifically designed to forecast disease dynamics and flatten the curves of infection and death [5,6]. Moreover, numerous compartmental mathematical models have been developed to explore the influence and significance of vaccination in combating various infectious diseases, as detailed in a range of studies [7,8,9,10].

Based on the existing literature, the focus of this study is to develop a new stochastic model that can analyzed the dynamics of COVID-19, taking into account the impact of vaccination. The proposed model incorporates a stochastic term to enhance its accuracy. The first objective of current research is to examine the transmission patterns of COVID-19, specifically in relation to Non-Pharmaceutical Interventions (NPIs) and the vaccination process. To establish the parameters of the model, actual pandemic cases in Pakistan are used for estimation. The study also presents the basic reproduction number and equilibrium points, along with conducting various fundamental analysis. The research findings from this study can be valuable for government authorities and public health agencies in formulating effective strategies to minimize the spread of future outbreaks.

This research paper is organized into different sections to provide a comprehensive analysis of the stochastic COVID-19 vaccination model. In Section 2, we present the mathematical modeling of this model, which involves describing the various factors and variables that influence the spread of the virus and the effectiveness of vaccinations. To further enhance our understanding, Section 3 reviews the application of Legendre polynomials and the spectral method in this context, as these mathematical tools have proven valuable in analyzing complex systems. In Section 4, we briefly discuss the basic reproduction number, an important metric in epidemiology, and conduct a stability analysis to assess the robustness of our model. In Section 5, we delve into the numerical results obtained from our simulations and discuss their implications for COVID-19 vaccination strategies. Finally, we conclude our research paper by summarizing our findings and highlighting the key takeaways in the last section.

In this section, we focus on modeling a stochastic COVID-19 vaccination model to investigate the impact of certain non-pharmaceutical interventions (NPIs), specifically social distancing in combination with vaccination. Although the COVID-19 vaccine is currently in an experimental phase and only available to a limited extent, it plays a crucial role in controlling the pandemic alongside other pharmaceutical mediation. To construct mathematical model, we divide the total population into distinct classes: susceptible individuals (S(t)), vaccinated individuals (V(t)), exposed individuals (E(t)), asymptomatic infected individuals ($I_A\left(t\right)$), the symptomatic infected individuals (I(t)) and recovered from infection is denoted by (R(t)). Where, total population (N(t)) is the sum of all above classes.

The proposed model for the stochastic COVID-19 vaccine can be represented as a set of stochastic differential equations, which are nonlinear in nature. Mathematically, this model describes the relationship between various factors involved in the vaccine’s effectiveness and its impact on the spread of the disease. By using these equations, we can better understand and predict how the vaccine will interact with the virus and the human immune system, providing valuable insights into its potential efficacy. with positive initial values ${\lambda }_i;i=1,2,...,6$, respectively for each class.

In the considered system described by Eqs. 1, the parameter $\Lambda$ represents the assumed birth rate, whereas $\mu$ denotes the natural death rate within each class of the model. The rate at which susceptible individuals receive vaccinations is represented by ${\omega }_1$, and those who have been vaccinated become susceptible again at a rate of $\psi$. Latent individuals become infected at a rate of $\kappa (1-r)$, where the remaining individuals (proportion r) join the group of asymptomatic infection. Where parameter ${\mu }_1$ shows the COVID-19 convinced mortality rate, while ${\varphi }_1$ and ${\varphi }_2$ indicate the rate at which both symptomatically and asymptomatically infected individuals recover. The transmission of the disease is characterized by $\beta$, and the transmissibility coefficient of asymptomatic individuals is denoted by ${\beta }_A$.

It is expected that all the parameters have positive values.

In present research work, we implement the spectral method for the numerical solution of COVID-19 system. This method were initially applied for differential and integral system [11,12,13]. Moreover, for numerical solutions of different diseases models the proposed method are applied by authors [14,15,16,17].

Since infectious diseases are subject to randomness in terms of the nature of transmission, the deterministic COVID-19 model given in Eqs. 1 is perturbed by white noise to obtain a stochastic model given in form:

In this study, we are focusing on the spectral method for obtaining the solution to a stochastic COVID-19 model described by Eqs. 2. In this model, $B\left(t\right)$ represents Brownian motion, which is a random process that exhibits erratic behavior over time. The parameter $\varphi$ is referred the intensity of Brownian motion, which determines the degree of randomness or volatility in the system. By developing an approximate solution to this stochastic model, we aim to better understand and predict the dynamics of COVID-19.

Before apply spectral method, to provide an preview of Legendre polynomials given in [18]. The $n^{th}$ order Legendre polynomials denoted by $P_n\left({\tau }_a\right)$. Where the function $u\left({\tau }_a\right)$ is approximated by:

$u_i$ indicates Legendre coefficients, ${\tau }_{ai},i=0,...,n$ are collocation nodes and $P_n\left({\tau }_a\right)$ denotes $n^{th}$-order. Where the Legendre polynomials are:

Spectral method procedure we considered the Legendre-Gauss-Lobatto points ${\left\{t_j\right\}}_{j=0}^N$.

For this, taking integral on Eq. 2 from $[0,t]$. where $S\left(0\right),E\left(0\right),I\left(0\right),I_A\left(0\right),V\left(0\right),R\left(0\right)$ are initial conditions on the functions of each class respectively. To convert the present interval to $[-1,1]$ interval, we transform s like: $s=\frac{t}{2}(1+\varpi )$, then Eqs. 5 becomes: where the semi discretized spectral system Eqs. 6 are: where the weight function refer [12,13], for Eq. 1 is given by

L is the Lagrange polynomials.

Also the weight function for Eq. 2 is given by

Now the spectral solution of each $S,E,I,I_A,V$ and R by using the above Eq. 5

Legendre coefficients of each of functions $S\left(t\right)$, $E\left(t\right)$, $I\left(t\right)$, $I_A\left(t\right)$, $V\left(t\right)$ and $R\left(t\right)$ are in the form: $S_n,E_n,I_n,{\left(I_A\right)}_n,V_n,R_n$, respectively. Now using the solution Eqs. 9 becomes:

We take ${\zeta }_k=\left(\frac{t(1+\varpi )}{2}\right)$. The proposed system Eqs. 9 gives of $6N+6$ of unknowns in which $6N$ nonlinear equations. Using initial conditions say:

Eqs. 10 and Eqs. 9 of a system of $(6N+6)$ equations. Therefore, the solving the above two systems gives the solution of corresponding all unknowns. In last using the unknown values in Eqs. 8, and get a solution to the model given in Eqs. 2.

Stability analysis is a crucial tool for investigating the behavior of a dynamical system. In this particular section, we delve into the examination of the stability of equilibrium solution. These equilibrium solutions are associated with both deterministic and stochastic systems, which are represented by equations labeled as Eqs. 1 and Eqs. 2 respectively. By studying the stability of these solutions, we gain insights into how the system’s dynamics unfold over time, accounting for both deterministic and random influences.

The COVID-19 model described by Eqs. 1 can have up to two equilibrium solutions. The first one, called the infection-free equilibrium $E_0$, represents a steady state solution where there is no disease in the population. The second equilibrium solution is known as the endemic equilibrium, which occurs when the disease persists within the population.