Nonlinear stochastic modelling plays an important character in the different fields of sciences such as environmental, material, engineering, chemistry, physics, biomedical engineering, and many more. In the current study, we studied the computational dynamics of the stochastic dengue model with the real material of the model. Positivity, boundedness, and dynamical consistency are essential features of stochastic modelling. Our focus is to design the computational method which preserves essential features of the model. The stochastic non-standard finite difference technique is most efficient as compared to other techniques used in literature. Analysis and comparison were explored in favour of convergence. Also, we address the comparison between the stochastic and deterministic models.

Dengue disease was first described in 1780 when dengue cases are observed in the Philadelphia epidemic (Rush 1789). In 1906, dengue transmission was established by the Aedes mosquito. In 1953, dengue hemorrhagic was first apprehend in the Philippines and 1955 it was recognized in Thailand. Dengue virus is one of the most common viruses. Four dengue (DENV-1 to DENV-4) serotypes viruses that can spread to feminine mosquitoes causes dengue fever (DF). A DENV serotype induces permanent immunity to serotypes but only partial temporary immunity to three different serotypes. Annually there are about 390 million dengue infections, of which 96 million are registered on a clinical basis (any severity of the disease). DENV infection is at risk for 128 countries. This is because of a mosquito-borne virus. A mosquito bites a human and repeats this process to another human, and this cycle continues. The rainy season is a suitable season for this virus [

A GBM or (otherwise called exponential Brownian motion) is a ceaseless time stochastic procedure wherein the logarithm of the haphazardly fluctuating amount pursues a Brownian movement (additionally known as Wiener procedure) with the float. It is a significant cause of stochastic procedures fulfilling a stochastic differential condition (SDE). Specifically, it is utilized in the scientific fund to show stock costs operating at a profit Scholes model [

with initial values

In this section, we considered the dynamics of the deterministic model, as presented in [

The transmission rates are explained as _{H}_{H}_{V}_{V}

therefore,

The stabilized form of model

There are two regions for the system

The equilibria of system

The construction of a stochastic material of the model from the deterministic model has presented in [

Put

where “B” is the geometric Brownian motion.

This scheme is constructed for the model

where ‘

Rates | DFE | EE |
---|---|---|

0.5 | 0.5 | |

0.1 | 10.1 | |

0.5 | 0.5 | |

0.2 | 10.2 | |

0.9 | 0.9 | |

0.8 | 0.8 | |

0.5 | 0.5 | |

0.5 | 0.5 | |

0.1 | 0.1 | |

0.5 | 0.5 |

The pseudo-code for stochastic Runge Kutta for the system

Begin:

Declare all constants

Set the step size ‘

Declare arrays for

Put initial values for

For

Calculate stage 1 equations

Calculate stage 2 equations

Calculate stage 3 equations

Calculate stage 4 equations

Calculate final stage equations

End For

Plot required data

end program

We make the simulation of discussed technique by using parameters values prearranged in [

This scheme is constructed for the model

We make the simulation of the above-discussed technique by using parameters values prearranged in [

For this, we shall satisfy the following theorems as follows:

Proof: Since all rates and state variables of the system are non-negative. So, the proof is straightforward.

Proof: Now, we rewrite the system

Also, we rewrite the system

More precisely,

Proof: Considering the functions F, G, H, I, J and K from the system

The Jacobi matrix

By simplify, we get the following eigenvalues as follows:

Thus, the system is stable at the rates of the material of the model.

In this part, we are going to discuss the comparison of existing stochastic techniques and proposed technique as follows:

The correlation coefficient has obtained among the compartments of the model, and results are described in

Sub-populations | Correlation coefficient |
Relationship |
---|---|---|

−0.3595 | Inverse | |

−0.7083 | Inverse | |

−0.3930 | Inverse | |

−0.2906 | Inverse | |

−0.8331 | Inverse | |

−0.2824 | Inverse |

In this study, we must claim the most effective and real stochastic analysis in comparison with the deterministic analysis of the model. Also, the construction of the stochastic model has presented with the rates of the material of the model. Unfortunately, numerical research is presented in all disciplines of science because of the non-differentiability of Brown’s motion. The standard stochastic methods employed here were the usual stochastic Euler method and the fourth-order stochastic Runge–Kutta method. Numerical properties of those methods are well known like positivity, boundedness and dynamical consistency. Various simulations were produced using different step sizes, and comparisons were shown graphically. The results showed that the methodology proposed in this work is capable of preserving the essential features of the relevant solutions of the mathematical model, as expected. On the other hand, the graphical results showed that the standard methods were not able to guarantee the preservation of some of those essential features. In particular, we proved computationally that those standard stochastic methods are incapable of preserving the essential features of the model, as desired. In that sense, the stochastic NSFD method designed in this work is a more reliable technique. Moreover, we shall extend our work in all discipline as bio-economics, biophysics, biochemistry, and many more sub-branches of material sciences. Also, the given essential features preserving analysis could be extended in the fractional-order models [

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 author is also thankful to Prince Sultan University for funding this work.