In this paper, a modified Leslie–Gower predator-prey model with Beddington–DeAngelis functional response and double Allee effect in the growth rate of a predator population is proposed. In order to consider memory effect on the proposed model, we employ the Caputo fractional-order derivative. We investigate the dynamic behaviors of the proposed model for both strong and weak Allee effect cases. The existence, uniqueness, non-negativity, and boundedness of the solution are discussed. Then, we determine the existing condition and local stability analysis of all possible equilibrium points. Necessary conditions for the existence of the Hopf bifurcation driven by the order of the fractional derivative are also determined analytically. Furthermore, by choosing a suitable Lyapunov function, we derive the sufficient conditions to ensure the global asymptotic stability for the predator extinction point for the strong Allee effect case as well as for the prey extinction point and the interior point for the weak Allee effect case. Finally, numerical simulations are shown to confirm the theoretical results and can explore more dynamical behaviors of the system, such as the bi-stability and forward bifurcation.

In this paper, a fractional order of a modified Leslie–Gower predator-prey model with disease and the double Allee effect in predator population is proposed. Then, we analyze the important mathematical features of the proposed model such as the existence and uniqueness as well as the non-negativity and boundedness of solutions to the fractional-order system. Moreover, the local and global asymptotic stability conditions of all possible equilibrium points are investigated using Matignon’s condition and by constructing a suitable Lyapunov function, respectively. Finally, numerical simulations are presented to verify the theoretical results. We show numerically the occurrence of two limit cycles simultaneously driven by the order of the derivative, the bistability phenomenon for both the weak and strong Allee effect cases, and more dynamic behaviors such as the forward, backward, and saddle-node bifurcations which are driven by the transmission rate. We have found that the risk of extinction for the predator with a strong Allee effect is much higher when the spread of disease is relatively high.

In this paper, we propose a fractional-order COVID-19 epidemic model with a quarantine and standard incidence rate using the Caputo fractional-order derivative. The model consists of six classes: susceptible (S), exposed (E), infected (I), quarantined (Q), recovered (R), and deceased (M). In our proposed model, we simultaneously consider the recovery rate and quarantine rate of infected individuals, which has not been considered in other fractional-order COVID-19 epidemic models. Furthermore, we consider the standard incidence rate in the model. For our proposed model, we prove the existence, uniqueness, non-negativity, and boundedness of the solution. The model has two equilibrium points: disease-free equilibrium and endemic equilibrium. Implementing the spectral radius of the next-generation matrix, we obtain the basic reproduction number (${\mathcal{R}}_0$). The disease-free equilibrium always exists and is locally and globally asymptotically stable only if ${\mathcal{R}}_0<1$. On the other hand, endemic equilibrium exists and is globally asymptotically stable if ${\mathcal{R}}_0>1$. Our numerical simulation confirms the stability properties of the equilibrium. The smaller the order of the derivative, the slower the convergence of the solution of the model. Both the recovery rate and quarantine rate of the infected class are important parameters determining the stability of the equilibrium point. Based on parameter estimation from COVID-19 data in Indonesia, the fractional-order model has better performance than the first-order model for both the calibration and 20-day forecasting of confirmed daily active cases of COVID-19.

Virotherapy is one of the most promising therapies in the treatment of tumors which may be further combined with chemotherapy to accelerate the healing rate. In this article, we propose a mathematical model for the treatment of tumors using oncolytic virus and chemotherapy. This model takes the form of nonlinear ordinary differential equations describing the interactions between uninfected tumor cells, infected tumor cells, an oncolytic virus, and chemotherapy. It is assumed that the rate of infection between uninfected tumor cells and infected tumor cells is in a saturated form. The saturation effect takes into account the fact that the number of contacts between them reaches the maximum value when the immune system works to stop the virus. The dynamical analysis, which includes the existence of equilibrium points, and its stability analysis is investigated. The analysis result shows that the system has three equilibrium points: tumor-free equilibrium point, virus-free equilibrium point and endemic equilibrium point. It is proven that these equilibrium points are conditionally stable. The numerical simulations show the successful combination of chemotherapy and virotherapy using an oncolytic virus in eliminating the tumor cells.

Schistosomiasis is a parasitic disease caused by the schistosoma worm. A snail can act as the intermediate host for the parasite. Snail-population control is considered to be an effective way to control schistosomiasis spread. In this paper, we discuss the schistosomiasis model incorporating a snail predator as a biological control agent. We prove that the solutions of the model are non-negative and bounded. The existence condition of equilibrium points is investigated. We determine the basic reproduction number when the predator goes to extinction and when the predator survives. The local stability condition of disease-free equilibrium point is proved using linearization, and the Lienard–Chipart and Routh–Hurwitz criteria. We use center-manifold theory to prove the local stability condition of the endemic equilibrium points. Furthermore, we constructed a Lyapunov function to investigate the global stability condition of the disease-free equilibrium points. To support the analytical results, we presented some numerical simulation results. Our findings suggest that a snail predator as a biological control agent can reduce schistosomiasis prevalence. Moreover, the snail-predator birth rate plays an essential role in controlling schistosomiasis spread.