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Prishchepov [16] proved that all equations of length at most six over torsion-free groups are solvable. A different proof was given by Ivanov and Klyachko in [12]. This supports the conjecture stated by Levin [15] that any equation over a torsion-free group is solvable. Here it is shown that all equations of length seven over torsion-free groups are solvable.

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This writing is an attempt to explain a reliable numerical treatment for stochastic computer virus model. We are comparing the solutions of stochastic and deterministic computer virus models. This paper reveals that a stochastic computer virus paradigm is pragmatic in contrast to the deterministic computer virus model. Outcomes of threshold number C * hold in stochastic computer virus model. If C * < 1 then in such a condition virus controlled in the computer population while C * > 1 shows virus persists in the computer population. Unfortunately, stochastic numerical methods fail to cope with large step sizes of time. The suggested structure of the stochastic non-standard finite difference scheme (SNSFD) maintains all diverse characteristics such as dynamical consistency, boundedness and positivity as defined by Mickens. The numerical treatment for the stochastic computer virus model manifested that increasing the antivirus ability ultimates small virus dominance in a computer community.

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.