This article concerns establishing a system of fractional-order differential equations (FDEs) to model a plant–herbivore interaction. Firstly, we show that the model has non-negative solutions, and then we study the existence and stability analysis of the constructed model. To investigate the case according to a low population density of the plant population, we incorporate the Allee function into the model. Considering the center manifold theorem and bifurcation theory, we show that the model shows flip bifurcation. Finally, the simulation results agree with the theoretical studies.

We consider a system of fractional-order differential equations to analyze breast cancer growth in the immune-chemotherapeutic treatment process under some control parameters: ketogenic diet, immune booster, and anti-cancer drugs. The established model assumes the growth of the tumor density under chemotherapy treatment and the immune response during the interaction between the normal cells and tumor cells. For the local stability of the critical points (tumor-free critical point, dead critical point, and co-existing critical point), we used the Routh-Hurwitz criteria to show the necessary effect of the immune booster; moreover, we addressed the ketogenic rate in the treatment process. Our theoretical and numerical studies pointed out that on early detection of the tumor density (with weak Allee effect) the treatment should be supported by ketogenic nutrition. Several examples are shown to present our theoretical findings.

In this study, we classify the genera of COVID-19 and provide brief information about the root of the spread and the transmission from animal (natural host) to humans. We establish a model of fractional-order differential equations to discuss the spread of the infection from the natural host to the intermediate one, and from the intermediate one to the human host. At the same time, we focus on the potential spillover of bat-borne coronaviruses. We consider the local stability of the co-existing critical point of the model by using the Routh–Hurwitz Criteria. Moreover, we analyze the existence and uniqueness of the constructed initial value problem. We focus on the control parameters to decrease the outbreak from pandemic form to the epidemic by using both strong and weak Allee Effect at time t. Furthermore, the discretization process shows that the system undergoes Neimark–Sacker Bifurcation under specific conditions. Finally, we conduct a series of numerical simulations to enhance the theoretical findings.

In this paper, we established a fractional-order logistic model for a monoclonal brain tumor growth known as Glioblastoma (GB). We show at first that the model possesses non-negative solutions. Furthermore, we studied the stability, existence, and uniqueness of the constructed model. To investigate the case for the extinction of the tumor population, we consider the Allee threshold. By using the center manifold theorem and bifurcation theory, we show that the model undergoes flip bifurcation. The numerical results support the theoretical study.