Fractional stochastic modeling: New approach to capture more heterogeneity.

To further capture holding complexities of nature that arise in many fields of science, technology, and engineering, we suggested in this paper a novel approach of modeling. The novel approach is a coupling of fractional differential and integral operators with the stochastic approach. The approach is texted using systems of chaotic problems. The numerical simulation impulsively shows that the new approach is able to capture hiding behaviors that could not be captured by fractional differential and integral operators and the stochastic approach only. We believe that this approach is the future way to model complex problems.

A fractional model for predator-prey with omnivore.

We consider the model of interaction of predator and prey with omnivore using three different waiting time distributions. The first waiting time is induced by the power law distribution which is the generator of Pareto statistics. The second waiting time is induced by exponential decay law with a particular property of Delta Dirac distribution when the fractional order tends to 1, this distribution is link to the Poison distribution. While the last waiting distribution, induced by the Mittag-Leffler distribution, presents a crossover from exponential to power law. For each model, we presented the conditions under which the existence of unique set of exact solutions is reached using the fixed-point Picard's method. Making use of a recent suggested numerical scheme, we solved each model numerically and some numerical simulations were generated for different values of fractional orders. We noticed a new attractor which can be considered as a combination of the Brownian motion and power law distribution in the model with the Atangana-Baleanu fractional derivative. With the aim to capture more attractors, we modified the model and presented also some numerical simulations. Our new model provides more attractors than the existing one even for fractional differential cases. We presented finally the Maximal Lyapunov exponent and the bifurcation diagrams. The comparative study shows that modeling with non-local and non-singular kernel fractional derivative leads to more attractors as this kernel is able to capture more physical problems.

Modelling the effects of heavy alcohol consumption on the transmission dynamics of gonorrhea with optimal control.

Alcoholism has become a global threat and has a serious health consequence in the society. In this paper, a deterministic alcohol model is formulated, analyzed and the basic properties established. The reproduction number R0 of system is determined. The steady states examined and local stability is found to be both locally and globally stable. The endemic state exhibit three equilibra solutions. Furthermore, time dependent control is incorporated into the system in order to establish the best strategy in controlling the alcohol consumption and gonorrhea dynamics, using Pontryagin's Maximum Principle. The numerical results depict that the best strategy to controlling gonorrhea is the application of the three controls at the same time.

Mathematical modeling and stability analysis of Pine Wilt Disease with optimal control.

This paper presents and examine a mathematical system of equations which describes the dynamics of pine wilt disease (PWD). Firstly, we examine the model with constant controls. Here, we investigate the disease equilibria and calculate the basic reproduction number of the disease. Secondly, we incorporate time dependent controls into the model and then analyze the conditions that are necessary for the disease to be controlled optimally. Finally, the numerical results for the model are presented.