Dynamics of COVID-19 mathematical model with stochastic perturbation

Acknowledging many effects on humans, which are ignored in deterministic models for COVID-19, in this paper, we consider stochastic mathematical model for COVID-19. Firstly, the formulation of a stochastic susceptible–infected–recovered model is presented. Secondly, we devote with full strength our concentrated attention to sufficient conditions for extinction and persistence. Thirdly, we examine the threshold of the proposed stochastic COVID-19 model, when noise is small or large. Finally, we show the numerical simulations graphically using MATLAB.

SIR model for control of this pandemic. But there is no one until now who could control this virus. If we make the contact rates very small it will show the best effect on the further spreading of COVID-19, so for this purpose all governments take action for in terms of the household effect. For the estimation of the final size of the coronavirus epidemic, Batista [3] presented the logistic growth regression model. Many researchers discussed this COVID-19 in different models in integer and in fractional order, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17], because of many applications of fractional calculus, stochastic modeling and bifurcation analysis [18][19][20][21][22][23][24][25][26]. For the more realistic models, several authors studied the stochastic models by introducing white noise [27][28][29][30][31]. The effects of the environment in the AIDS model were studied by Dalal et al. [27] using the method of parameter perturbation. Stochastic models will likely produce results different from deterministic models every time the model is run for the same parameters. Stochastic models possess some inherent randomness. The same set of parameter values and initial conditions for deterministic models will lead to an ensemble of different outputs. Tornatore et al. [28][29][30] studied the stochastic epidemic models with vaccination. In this work, they proved the existence, uniqueness, and positivity of the solution. A stochastic SIS epidemic model containing vaccination is discussed by Zhu et al. [31]. They obtained the condition of the disease extinction and persistence according to noise and threshold of the deterministic system. Similarly, several authors discussed the same conditions for stochastic models; see [32][33][34][35][36][37][38][39].
To study the effects of the environment on spreading of COVID-19 and make the research more realistic, first we formulate a stochastic mathematical COVID-19 model. Then sufficient conditions for extinction and persistence are examined. Furthermore, the threshold of the proposed stochastic COVID-19 model is determined. It plays an important role in mathematical models as a backbone, when there is small or large noise. Finally, we show the numerical simulations graphically with the aid of MATLAB.
The rest of the paper is organized as follows: Sect. 2 is concerned with the COVID-19 model with random perturbation formulation. Section 3 is related to the unique positive solution of proposed model. Furthermore, we investigate the exponential stability of the proposed model in Sect. 4. The persistent conditions are shown in Sect. 5. Finally, we conclude with the results and outcomes of the paper in Sect. 6.

Model formulation
In this section, a COVID-19 mathematical model with random perturbation is formulated as follows: where the description of parameters and variables are given in Table 1. In deterministic form the model (1) is given by where Also, we have So, the solution has a positivity property. For stability analysis of model (2), we have the reproductive number, which is If R 0 < 1, then system (2) will be locally stable and unstable if R 0 ≥ 1. Similarly for = 0, the system (2) will be globally asymptotically stable.

Existence and uniqueness of the positive solution
Here, we first make the following assumptions: satisfies the usual conditions. Generally, consider a stochastic differential equation of n-dimensions as with initial value y(t 0 ) = y 0 ∈ R d . By defining the differential operator L with Eq. (6) If the operator L acts on a function V = (R d ×R + ;R + ), then , and solution will be left in R 3 + , with probability 1.
Proof Since the coefficient of the differential equations of system (1) are locally Lipschitz continuous for (S(0), where τ e is the time for noise caused by an explosion (see [6]). For demonstrating the solution to be global, it is sufficient that τ e = ∞ a.s. Suppose that k 0 ≥ 0 is sufficiently large so that (S(0), where we set inf φ(empty set) = ∞ throughout the paper. For k → ∞, τ k is clearly increasing. Set τ ∞ = lim k→∞ τ k whither τ ∞ ≤ τ e . If we can show that τ ∞ = ∞ a.s, then τ e = ∞. If false, then there are a pair of constants T > 0 and ∈ (0, 1) such that So there is an integer k 1 ≥ k 0 , which satisfies By applying the Itô formula, we obtain where LV : R 3 + →R + is defined by By choosing c = γ +μ β , it follows that Further proof follows from Ji et al. [31].

Extinction
In this section, we investigate the condition for extinction of the spread of the coronavirus.
Here, we define A useful lemma concerned with this work is as follows. Then In addition Proof Performing the integration of system (1) Then we have
Now, from third equation of system (1), it follows that By applying the L'Hospital's rule to the previous result, we have From Eq. (4), it follows that Hence, we have completed the proof.

Persistence
This section concerns the persistence of system (1).
Using Eq. (17) we have . Furthermore, By applying the limit t → ∞, we have .
Hence, the proof is complete.

Numerical simulation
For the illustration of our obtained results, we use the values of the parameters and the variables given in Table 2.   Now for the numerical simulation, we use Milstein's higher order method [40]. The results obtained through this method are shown graphically in Fig. 1 for both deterministic and stochastic forms.

Conclusion
In this work, a formulation of a stochastic COVID-19 mathematical model is presented. The sufficient conditions are determined for extinction and persistence. Furthermore, we discussed the threshold of proposed stochastic model when there is small or large noise. Finally, we showed numerical simulations graphically with the help of software MATLAB. The conclusions obtained are that the spread of COVID-19 will be under control ifR < 1 and ρ 2 ≤ βμ Λ means that white noise is not large and the value ofR > 1 will lead to the prevailing of COVID-19.