International Journal of Biomathematics

Due to the unpredictable growth of tumor cells, the tumor–immune interactive dynamics continues to draw attention from both applied mathematicians and oncologists. Mathematical modeling is a powerful tool to improve our understanding of the complicated biological system for tumor growth. With this goal, we report a mathematical model which describes how tumor cells evolve and survive the brief encounter with the immune system mediated by immune effector cells and host cells which includes discrete time delay. We analyze the basic mathematical properties of the considered model such as positivity of the system and the boundedness of the solutions. By analyzing the distribution of eigenvalues, local stability analysis of the biologically feasible equilibria and the existence of Hopf bifurcation are obtained in which discrete time delay is used as a bifurcation parameter. Based on the normal form theory and center manifold theorem, we obtain explicit expressions to determine the direction of Hopf bifurcation and the stability of Hopf bifurcating periodic solutions. Numerical simulations are carried out to illustrate the rich dynamical behavior of the delayed tumor model. Our model simulations demonstrate that the delayed tumor model exhibits regular and irregular periodic oscillations or chaotic behaviors, which indicate the scenario of long-term tumor relapse.

In this paper, we analyze the dynamical behavior of fish farm model related to Atangana–Baleanu derivative of arbitrary order. The model is constituted with the group of nonlinear differential equations having nutrients, fish and mussel. We have included discrete kind gestational delay of fish. The solution of fish farm model is determined by employing homotopy analysis transforms method (HATM). Existence of and uniqueness of solution are studied through Picard–Lindelof approach. The influence of order of new non-integer order derivative on nutrients, fish and mussel is discussed. The complete study reveals that the outer food supplies manage the behavior of the model. Moreover, to show the outcomes of the study, some numerical results are demonstrated through graphs.

This research paper implements the fractional homotopy analysis transform technique to compute the approximate analytical solution of the nonlinear three-species food chain model with time-fractional derivatives. The offered technique is a fantastic blend of homotopy analysis method (HAM) and Laplace transform (LT) operator and has been used fruitfully in the numerical computation of various fractional differential equations (FDEs). This paper involves the fractional derivatives of Caputo style. The numerical solutions of this selected fractional-order food chain model are evaluated by making use of the associated initial conditions. It is revealed by the adopting procedure that the more desirable estimation of the solution can be easily acquired through the calculation of some number of iteration terms only — a fact which authenticates the easiness and soundness of the suggested hybrid scheme. The variations of fractional order of time derivative on the solutions for different specific cases have been depicted through graphical presentations. The outcomes demonstrated through the graphs expound that the adopted scheme is very fantastic and accurate.

In this paper, a novel stochastic two-species competitive system with saturation effect is formulated, in which there exist two noise resources and their coupling mode is relatively complex and every noise source has effect on the intrinsic growth rates of both species. With the help of some suitable Lyapunov functions, sufficient conditions for stochastic permanence are established as exponential extinction, extinction, permanence in time average and asymptotic pathwise estimation of system. The effect of coupling noise on the asymptotic behaviors of the populations is shown.

In this paper, we are concerned with a class of fractional-order Lasota–Wazewska red blood cell models. By applying a fixed point theorem on a normal cone, we first obtain the sufficient conditions for the existence of a unique almost periodic positive solution of the considered models. Then, considering that all of the red blood cells in animals survive in a finite-time interval, we study the finite-time stability of the almost periodic positive solution by using some inequality techniques. Our results and methods of this paper are new. Finally, we give numerical examples to show the feasibility of the obtained results.

In this paper, allowing for general transmission and recovery times distributions, we proposed an edge-based age-structured-like compartmental model for STIs (EBACMS) in a coupled network. We considered sexual transmissions between men with also heterosexual contacts. Mathematically, we gave the general approach of proving the nonnegativity of solutions for the system coupling ordinary and partial differential equations, which can be applied to all edge-based compartment models. We then analyzed the epidemic threshold ${{ℛ}}_0$ with different distributions which couples the thresholds of the single-layer and bipartite networks in the percolation theory. We also studied the global stability of disease-free equilibrium with ${{ℛ}}_0<1$ and the final epidemic size $F$ (the proportion of the population experiencing infection during the epidemic) with ${{ℛ}}_0>1$. In addition, numerical simulations indicated that given a fixed exponential transmission distribution, a higher variance (with same mean) in general recovery distribution gives smaller ${{ℛ}}_0$ and $F$. Sensitivity analysis on ${{ℛ}}_0$ and $F$ in terms of the parameters illustrated that male-to-male transmission routes have a greater impact on ${{ℛ}}_0$ and $F$ than the heterosexual transmission routes for the Markovian transmission process and arbitrary recovery process. The results provide a good theoretical guideline to consider the distributions of real-world STIs.