Correlated stochastic epidemic model for the dynamics of SARS-CoV-2 with vaccination

In this paper, we propose a mathematical model to describe the influence of the SARS-CoV-2 virus with correlated sources of randomness and with vaccination. The total human population is divided into three groups susceptible, infected, and recovered. Each population group of the model is assumed to be subject to various types of randomness. We develop the correlated stochastic model by considering correlated Brownian motions for the population groups. As the environmental reservoir plays a weighty role in the transmission of the SARS-CoV-2 virus, our model encompasses a fourth stochastic differential equation representing the reservoir. Moreover, the vaccination of susceptible is also considered. Once the correlated stochastic model, the existence and uniqueness of a positive solution are discussed to show the problem’s feasibility. The SARS-CoV-2 extinction, as well as persistency, are also examined, and sufficient conditions resulted from our investigation. The theoretical results are supported through numerical/graphical findings.

It is essential to discuss extinction and persistence when investigating virus spread. The aim of this analysis is to determine when the disease will end (extinct) and under which conditions will stay (persist). Finally, all analytical findings will be supported by using some graphical representation in the form of a large-scale numerical simulation by using the Euler-Maruyama scheme. It will be performed via coding the proposed problem with the help of MATLAB and we will show the analytical finding graphically.

Formulation of the model with fundamental analysis
Let us assume a filtered probability space (�, We classify the total human population into three human population groups and one class of reservoir. The three population groups are susceptible, SARS-CoV-2 virus infected and recovered, which are symbolized by s(t), i(t) and r(t) respectively, while the reservoir class is denoted by w(t). The quantity w is the environmental reservoir which is an important element in the study of our epidemic model. It represents the concentration of the coronavirus in the environmental reservoir and it includes rates of the infected individuals contributing the coronavirus to the environmental reservoir and the removal rate of the virus from the environment. All the population groups and the reservoir is distributed by different Brownian motions. The schematic diagram for distribution process of the various population groups is given in Fig. 1. Thus we suggest a correlated stochastic epidemic model by the following system: The above-proposed model is a generalization of standard epidemic deterministic models. It allows the different quantity of the model to vary stochastically, which mean that the variations are not only time-dependent but also subject to haphazard fluctuations. The random noise detected from real data is considered in the above stochastic model but neglected in deterministic models. In Eq. (1) the various parameters are characterized as: the newborn rate is symbolized with , and β i , i = 1, 2 , are routes of disease transmission from the infected human as well as from the reservoir. Moreover, v is the vaccination of the susceptible population and µ is the natural death rate while death from the disease is described with d 1 . We also symbolize the recovery rate by σ and a rate contributed to the virus to the environment by α . The removing SARS-CoV virus rate is denoted by η . If k1 = 1 for k = 1, 2, 3, 4 , and ki = 0 otherwise, then B 1 = B 2 = B 3 = B 4 and the model is reduced to the stochastic model studied in Khan et al., 23 . Also, it could be clearly noted that the above system (1) will reduce to the deterministic form, whenever η 1 = η 2 = η 3 = η 4 = 0 . It can be seen also an extension of 1 . In addition (1) To move towards the endemic equilibrium, we will calculate the basic reproductive number first, which is defined to be the average number of secondary infectious produced an infective whenever reached to a totally non-infected population. We assume X = (i, w) T and p 2 = σ + µ + d 1 , then the deterministic version of the model (1) yields The basic reproductive number is then the spectral radius of ρ(FV −1 ) and consequently looks like We use this quantity, to find the components of the endemic equilibrium which may take the form Sensitivity analysis. In every disease the role of the threshold parameter (basic reproductive number) is very important and the disease spreads whenever the value of this quantity is more than one and the disease dies out if its value is less than unity. We will discuss the sensitivity of threshold parameter to find the relation between basic reproductive number and model parameter. We also calculate the sensitivity indexes that which parameters is how much sensitive to disease control and transmission. Generally the sensitivity index of a parameter say φ is denoted by ϒ φ and define as φ R 0 ∂R 0 ∂φ . By following this formula we calculate the sensitivity indices of model parameters as: ϒ β 1 = 0.9937106918 , ϒ β 2 = 0.006289308180 , ϒ α = 0.006289308180 , ϒ ν = −0.8333333334 and ϒ σ = −0.6862610536 , where the parameters value are taken to be = 2 , β 1 = 0.079 , β 2 = 0.001 , µ = 0.16 , The biological interpretation of these analyses investigate that the epidemic parameters β 1 , β 2 and α have a positive influence on the threshold quantity while there is a negative influence with the parameters ν and σ . This shows that decreasing in the value of β 1 , β 2 and α , and increasing in the value of ν and σ will decrease the value of the basic reproductive number, which is significant in disease elimination. It could be also noted that β 1 and ν got the highest sensitivity index and so are the most sensitive parameters to the disease transmission and control. We observed that increasing the value of β 1 say by 10% would significantly increase the value of R 0 by 9.9% as depicted in Fig. 2, while increasing the value of ν say by 10% decreasing the value of R 0 by 8.3% as shown in Fig. 5. Similarly, β 2 and α collectively effect R 0 by 0.1257861636 whenever these parameters are increased or decreased by 10% as depicted by Figs. 3 and 4. The relation between σ and R 0 is also an inverse as increased σ by 10% would decrease the threshold quantity by 6.86% is given in Fig. 5.
Existence and uniqueness analysis. In this portion of the manuscript the existence of the solution and uniqueness with the positivity of Eq. (1) will be discussed.
It is worth mentioning that the Itô formula is one of the most useful formulas in stochastic calculus. It is utilized, among others, to solve stochastic differential equations. Here, we describe a Multidimensional Itô formula for getting our results by following the book of stochastic calculus 24 .   www.nature.com/scientificreports/ Lemma 2.1 Let a = (α 1 , . . . , α n ) and b = (β 1 , . . . , β n ) represent the adapted processes with square-integrable n-dimensional. We consider X = (X 1 , . . . , X n ) , where X k is driven by the stochastic differential equation and k ∈ {1, . . . , n} , thus Let F is a given twice continuously differentiable function f : R n → R , then we have We use the Lyapunov theory and the virtue of the Itô formula to prove that the solution of Eq. (1) exists globally and is positive. Define The result that discusses the existing analysis of the problem is given by the following theorem.

be the initial classes and assumed to be in
Proof We use the procedure as adopted in 25 and so in the light of this the local Lipschitz continuity property holds for system (1), therefore the solution symbolized by (s, i, r, w) of the proposed problem in [0, τ e ) subject to initial conditions in R 4 + is unique and local for the explosion time τ e . Moreover, we investigate that τ e = ∞ a.s as to show the solution globalization. It is assumed that κ 0 ≥ 0 is sufficiently large and 1 0)) . We define the stopping time for every κ ≥ κ 0 as: Further, let φ is empty set and inf φ = ∞ . Since τ k depend on k and whenever k increasing τ k also increasing as k increases without bound i.e., tend to ∞ .
We now only need to show that τ e = ∞ . For this, we use the assumption that for any two constants, T > 0 and ε ∈ (0, 1) , we have So k 1 ≥ k 0 is an integer that Let H is twice continuously differentiable function i.e., H ∈ C 2 and H : Clearly, H ≥ 0 , so for 0 ≤ T and k 0 ≤ k , and by the application of the Itô formula leads to the assertion In Eq. (10), LH is defined as Simplifying and re-writing the above equation may lead to the following inequality It could be noted from the fact that s + i + r + w ≤ 1 , so the last inequality gives Plugging Eq. (13) in Eq. (10) we may arrive The integration of both sides reveals that Setting a notion of � k = T ≥ τ k for all k ≥ k 1 . The use of Eq. (7) gives that P(� k ) ≥ ǫ . Noted that there is at least one s(ω, τ k ) or i(ω, τ k ) or r(ω, τ k ) or w(ω, τ k ) equal 1/k or k for all ω ∈ � k . Since 1 k + log k − 1 or − log k + k − 1 . Hence So Eqs. (7) and (15) gives implies that where 1 �k(ω) is a function known indicator function for � k (ω) . Let k → ∞ we ultimately obtain ∞ > H N(0) + KT = ∞ , which contradicts, therefore ∞ = τ ∞ a.s.

Remark 1
The uniqueness as well as the existence reveals that for any initial compartments (N(0)) ∈ R 4 + , the unique solution with global axiom (s, i, r, w) ∈ R 4 + almost surly (a.s) exists for the proposed problem under consideration as reported by Eq. (1). The previous result can be also proved by the next theorem. Here k = 4 , m = 4 , kj = kj , γ kj = 0 (for k, j = 1, 2, 3, 4 ) and (14) (17) and (18) show clearly that the solution of our model (1) exists and it is unique and positive if we impose the positivity of the deterministic integral. This ends the proof.
Extinction and persistence. In this section, the extinction and persistence analysis of the stochastic model (1) are discussed. We derive the various conditions in the form of some expressions to show permanence and extinction. These expressions containing the model parameters and intensities of noises. Before the formal analysis we define that Now it could be described that the persistence of novel coronavirus SARS-CoV-2 is subjected to lim inf i(t) and lim inf w(t) whenever are positive as t increases without bound i.e., to ∞ . Moreover, the stochastic reproductive number of corona dynamical system represented by Eq. (1) is symbolized by R S 0 and define as Similarly, if and holds, the epidemic problem represented by Eq. (1) states that the disease will persist. Thus for the extinction analysis of the proposed problem we state the following subsequent result.

Theorem 2.4
The SARS-CoV-2 virus will die out exponentially whenever the stochastic reproductive number parameter ( R S 0 ) is less then unity i.e.,

Also
Proof To prove the result, we integrate the system (1) on both sides which lead to implies that It is very much clear from Eq. (5) that s + i + r + w ≤ 1 , thus we noted that s(t)w(t) therefore the above assertion leads to the inequality given by Using the value of s(t) with some algebraic manipulation and following the well-known strong law of large number 27 i.e., lim sup ξ 2 B 2 t = 0 a.s as t → ∞ we obtain implies that whenever the condition R S 0 < 1 holds, then lim i(t) = 0 and so lim i(t) = 0 a.s., as t → ∞ . Moreover, the last equation of system (25) implies that Since the limiting value of i(t) is zero then w(t) = 0 whenever t → ∞ , thus the first equation of the system (25) looks like gives that if t → ∞ , lim s(t) = �/p 1 . We conclude that the novel disease extinct continuously depends on the value of R S 0 , and ultimately whenever R S 0 < 1 , it will extinct. (26) (34) www.nature.com/scientificreports/ We have seen from the previous theorem that the virus will die out exponentially if R S 0 < 1 . The next theorem discusses the case when the stochastic reproductive number parameter R S 0 > 1 is greater than one. b. Putting X 0 := X 0 and applying X k+1 by following the formula given below for B k = B k+1 − B k and k = 0, . . . , N − 1. The stochastic Euler Maruyama technique will be applied for the numerical simulation of the system reported by Eq. (1) which takes the form which implies that Using Matlab software and coding the above algorithm to solve the proposed system. To run our model for largescale numerical findings we use feasible parameters value with time units of 0 to 400 days. Once we execute the algorithm the following graphs are generated as given by Figs. 6, 7, 8, 9, 10, 10, 11, 12 and 13. This may verify our analytical findings. Moreover, Figs. 6, 7, 8 and 9 demonstrate the temporal dynamics of the susceptible, infected, recovered, and the reservoir respectively, which theoretically investigate that there will be always susceptible and recovered population while the SARS-CoV-2 virus infected population and reservoir will vanishes. This may verify the results of our extinction analysis. Since the disease extinct continuously depends on the basic reproductive parameter and whenever R S 0 < 1 the disease could be easily eliminated. So from the biological point of view, it is very important to keep this quantity low as much as possible to eliminate the disease. On the other hand Figs. 10, 11, 12 and 13 visualize the persistence analysis of the proposed problem. We noted that in this the trajectories of susceptible s(t), SARS-CoV-2 virus infected (i(t)), recovered (r(t)) and reservoir (w(t)) reveals that the the disease will persist and all these compartments reach to their endemic stage whenever the value of R S 0 > 1 . So special attention is required to make a control mechanism. Since the sensitivity analysis reveals that the disease transmission co-efficient has the highest sensitivity index and a great influence on the threshold parameter therefore minimization of this parameter would significantly decrease the value of the threshold parameter. It could be also noted from the sensitivity index of the vaccination parameter that vaccination has also a strong influence and so increasing the vaccination would strongly decrease the value of basic reproductive number. Finally, we also noted a relationship between the noise intensity with disease extinction and persistence s k+1 = s k + � − β 1 s k i k − β 2 s k w k − p 1 s k �t + η 1 s k �B 1k , i k+1 = i k + β 1 s k i k + β 2 s k i k − p 2 i k �t + η 2 i k �B 2k , r k+1 = r k + [σ i k + νs k − µr k ]�t + η 3 r k �B 3k , w k+1 = w k + [αi k − ηw k ]�t + η 4 r k �B 4k . www.nature.com/scientificreports/ i.e., there is a direct relation between the intensity of white noise and extinction while inverse relation between the intensity of white noise and persistence.

Conclusion
We developed a correlated stochastic epidemic model to discuss the temporal dynamics of the SARS-CoV-2 virus keeping in view the various source of randomness and vaccination of susceptible individuals. We proved the existence and positivity of the solutions which guarantees the well-posedness of the model. In addition, conditions of SARS-CoV-2 extinction analysis and persistence were obtained. A detailed sensitivity analysis has been performed and showed that the disease transmission coefficient and vaccination parameters are the highest sensitive parameters to disease transmission and control. This suggests that the vaccination has a major impact on the dynamics of the SARS-CoV-2. We observed that a rise in this parameter's value would significantly increase disease extinction. Conversely, the disease persistence reduction is subjected to speedy vaccination, and therefore there is a need for a fast vaccination immunization. Numerical findings were conducted and support the analytical results. Results of this study permit supplementary discussion, such as increasing the impact of the noise. We would encourage researchers to investigate adding jumps to our model.     17-20 (1988).