Nonlinear stochastic modeling plays a significant role in disciplines such as psychology, finance, physical sciences, engineering, econometrics, and biological sciences. Dynamical consistency, positivity, and boundedness are fundamental properties of stochastic modeling. A stochastic coronavirus model is studied with techniques of transition probabilities and parametric perturbation. Well-known explicit methods such as Euler Maruyama, stochastic Euler, and stochastic Runge–Kutta are investigated for the stochastic model. Regrettably, the above essential properties are not restored by existing methods. Hence, there is a need to construct essential properties preserving the computational method. The non-standard approach of finite difference is examined to maintain the above basic features of the stochastic model. The comparison of the results of deterministic and stochastic models is also presented. Our proposed efficient computational method well preserves the essential properties of the model. Comparison and convergence analyses of the method are presented.
Chen et al. [
The dynamics of COVID-19 are considered as follows. For any arbitrary time
The above state variables exhibit the nonnegative solution for
Thus solutions exist for given initial conditions that are eventually bounded on every finite time interval.
Therefore, the set
There are three steady states of
where
Note that
Let us consider the vector
The expectation and variance of the stochastic COVID-19 pandemic model are defined as
Expectation
Var =
The general form of SDEs is
Stochastic drift
Stochastic diffusion =
The SDE of
with initial conditions
Raza et al. [
where
Allen et al. [
The Brownian motion is denoted by
This method can be applied to
where the time step is
This method can be applied to
Stage 1:
Stage 2:
Stage 3:
Stage 4:
Final stage:
where the time step is
This method can be applied to
where the time step is
For essential properties, we shall satisfy the following theorems for positivity, boundedness, consistency, and stability.
Trivial equilibrium: (T.E) =
Corona-free equilibrium (
Corona-present equilibrium (
where
almost surely.
The Jacobian matrix is defined as
Now, we want to linearize the model about the steady-state of the model for corona-free equilibrium
The given Jacobian is
The eigenvalues of the Jacobian matrix are
(i).
(ii).
(iii).
are always satisfied if
are always satisfied if
This guarantees that all eigenvalues of the Jacobian lie in the unit circle. So,
For the present corona equilibrium, we plot the largest eigenvalue by using fitted values of parameters and MATLAB, as presented in
Hence, the largest eigenvalue for corona-present equilibrium is less than one. The remaining eigenvalues will be less than one.
The numerical solution is in good agreement with the dynamical behavior of the model using different values of the parameters. Khan et al. [
We plot each compartment of the model for corona-present equilibrium in
We plot the solution of
In comparison to the model’s, we have to say stochastic analysis of the most practical and actual model. The explicit numerical methods are conditionally convergent and depend on the time-step size. The stochastic NSFD method is independent of the time-step size. This method is unconditionally convergent as compared to other explicit numerical methods. This method preserves all essential properties of stochastic models, such as consistency, stability, positivity, and boundedness [
We always warmly, thanks to anonymous referees. We are also grateful to Vice-Chancellor, Air University, Islamabad, for providing an excellent research environment and facilities. The first and fourth author also thanks Prince Sultan University for funding this work through the COVID-19 Emergency Research Program.