International Journal of Biomathematics

Bacillus subtilis with competent states absorbs DNA and may improve the growth of bacteria by integrating new genetic material. Therefore, it is important to clarify how the genes interact in the circuit so that cells enter into a competent state or return to a vegetative state. The gene regulatory circuit consists of two positive feedback loops and one negative feedback loop. In this paper, a mathematical model is developed by considering transcription time delays to further study dynamic behavior of the B. subtilis competent gene regulatory network. Combined with theoretical calculation and numerical simulation, it is verified that the time delay in indirect transcription inhibition indeed has the effect of inducing the periodic oscillation of the B. subtilis competent system. In addition, some important chemical reaction rates can also regulate system dynamic behavior. However, under the control of time delay, the effects of the important chemical reaction rates have changed significantly. In particular, the time delay can advance critical value of the important chemical reaction rates where vibration occurs and can also weaken or even eliminate the effect of the important chemical reaction rates. These results will help us to analyze the competent state of B. subtilis.

Zika virus infection, which is closely related to dengue, is becoming a global threat to human society. The transmission of the Zika virus from one human to another is spread by bites of Aedes mosquitoes. Recent studies also reveal the fact that the Zika virus can be transmitted by sexual interaction. In this paper, we use the fractional derivative with Mittag–Leffler non-singular kernel to study Zika virus transmission dynamics. This fractional is also known as the Atangana–Baleanu Caputo (ABC) derivative which is employed for the resulting system of ordinary differential equations. We investigate the proposed Zika virus model by using the Legendre spectral method. The model parameters are estimated and validated numerically by investigating the effect of fractional order exponent on various cases such as Susceptible human, infected human, asymptomatic carrier, exposed human, and recovered human. Numerical results obtained with the proposed method have been compared with exact solutions, showing in all parameters a very satisfactory agreement.

In this paper, a novel numerical technique, the first-order Smooth Composite Chebyshev Finite Difference method, is presented. Imposing a first-order smoothness of the approximation polynomial at the ends of each subinterval is originality of the method. Both round-off and truncation error analyses of the method are performed beside the convergence analysis. Diffusion of oxygen in a spherical cell including nonlinear uptake kinetics is solved by using the method. The obtained results are compared with the existing methods in the literature and it is observed that the proposed method gives more reliable results.

This paper is concerned with the globally exponential stability of traveling wave fronts for a class of population dynamics model with quiescent stage and delay. First, we establish the comparison principle of solutions for the population dynamics model. Then, by the weighted energy method combining comparison principle, the globally exponential stability of traveling wave fronts of the population dynamics model under the quasi-monotonicity conditions is established, which extends the results of nonlocal scalar equations to the system.

In this paper, a mathematical model for solid avascular tumor growth with a time delay in regulatory apoptosis is studied. In the model, two types of cell apoptoses are considered, one is natural apoptosis and the other is regulatory apoptosis. The process of regulatory apoptosis is delayed compared to the processes of proliferation and natural apoptosis. The existence and uniqueness of a solution to the model are proved. The long-time asymptotic behavior of the solutions is studied. The results show that the dynamical behavior of solutions to this mathematical model is similar to that of the corresponding quasi-stationary problem for some special parameter values. Numerical simulations of some special parameter values are also given to verify our results.

In this paper, we present a fractional model for the HIV/AIDS epidemic and incorporate into the model control parameters of pre-exposure prophylaxis (PrEP), behavioral change and antiretroviral therapy (ART) aimed at controlling the spread of diseases. We prove the local and global asymptotic stability of disease-free and endemic equilibria of the model. We present a general fractional optimal control problem (FOCP) with free terminal time and develop the Adapted Forward-Backward Sweep method for numerical solving of the FOCP. Necessary conditions for a state/control/terminal time triplet to be optimal are obtained. The results show that the use of all controls increases the life expectancy of HIV-treated patients with ART and remarkably increases the number of people undergoing PrEP and changing their sexual habits. Also, when the derivative order $\alpha$ ($0.45\le \alpha \le 1$) limits to 1, the value of optimal terminal time increases while the value of objective functional decreases.

This paper studies the minimal wave speed of traveling wave solutions in predator–prey models, in which there are several groups of predators that compete among different groups. We investigate the existence and nonexistence of traveling wave solutions modeling the invasion of predators and coexistence of these species. When the positive solution of the corresponding kinetic system converges to the unique positive steady state, a threshold that is the minimal wave speed of traveling wave solutions is obtained. To finish the proof, we construct contracting rectangles and upper–lower solutions and apply the asymptotic spreading theory of scalar equations. Moreover, multiple propagation thresholds in the corresponding initial value problem are presented by numerical examples, and one threshold may be the minimal wave speed of traveling wave solutions.

Brucellosis is a major worldwide problem in public health. In this paper, we established and analyzed the Susceptible-Exposed-Infectious-Vaccinated for small ruminant–Susceptible-Exposed-Infectious-Vaccinated for cattle–Susceptible-Infectious-Recovered for human (SEIV–SEIV–SIR) model of Brucellosis with an incubation period which considered that human was infected through direct contact with infected humans, infected small ruminant, infected cattle and indirect contact with Brucella in the environment. The basic reproduction number $R_0$ of the model is computed by the spectral radius of the next-generation matrix. We proved that the disease-free equilibrium exists and we analyzed that the disease-free equilibrium point and endemic equilibrium point are global stability by constructing the $Lyapunov$ function and applying the $Lasalle$ invariant principle. Numerical solutions and analytical reveal that the prevention and control measures for Brucella should be to detect infected livestock in time, reduce contact with infected livestock and increase the cure rate of infected humans.

Based on the population model with an Allee effect, the corresponding delayed model is built up. According to the qualitative theory of differential equations, the model is analyzed completely to describe various intriguing dynamic properties as singular point of high order, Hopf bifurcation, etc. Next, combined impulse control theory with the ecological principles, the control models with impulse effects are proposed and the corresponding controllers are designed not only to avoid the extinction of the populations but also to prevent outbreak of the populations. The study would not only reveal the complexity dynamic behavior of ecosystems, but also provide the theory reference for the management of populations.

To define a vaccination policy and antiviral treatment against the spreading of viral infections a nonlinear optimal (H-infinity) control approach is proposed. Actually, because of the scarcity of the resources for treating infectious diseases in terms of vaccines, antiviral drugs and other medical facilities, there is need to implement optimal control against the epidemics deployment. In this approach, the state-space model of the epidemics dynamics undergoes first approximate linearization around a temporary operating point which is recomputed at each time-step of the control method. The linearization is based on Taylor series expansion and on the computation of the associated Jacobian matrices. Next, an optimal (H-infinity) feedback controller is developed for the approximately linearized model of the epidemics. To compute the controller’s feedback gains an algebraic Riccati equation is solved at each iteration of the control algorithm. Furthermore, the global asymptotic stability properties of the control scheme are proven through Lyapunov stability analysis. This paper’s results confirm that optimal control of the infectious disease dynamics allows for eliminating its spreading while also keeping moderate the consumption of the related medication, that is vaccines and antiviral drugs.

In this paper, the numerical solution of stochastic FitzHugh–Nagumo equation (SFNE) has been obtained by Chebyshev spectral collocation. Semi-implicit Euler–Maruyama scheme has been used for temporal variable to discretize the stochastic FitzHugh–Nagumo equation. A detailed stability analysis for stochastic FitzHugh–Nagumo equation has also been discussed. Graphical representations of obtained results of the stochastic FitzHugh–Nagumo equation have been discussed to provide a clear idea about the behavior of the solutions.

As the source of replenishment, rainfall has an extensive impact because its variability shapes biologically efficient pulses of soil moisture recharge across layers from rainfall events. In this paper, a mathematical model is proposed to explore the importance of transpiration from agricultural crops and aerosols on the pattern of rainfall. For the system without seeding, the simulation results show destabilizing roles of parameters related to formation of cloud drops due to transpiration of agricultural crops, formation of raindrops due to cloud drops and growth of agricultural crops due to rain. The model without seeding is extended to its stochastic counterpart to encapsulate the uncertainty observed in some important parameters. We observe the variability in the system’s variables and found their distributions at certain fixed times, which explore the importance of stochasticity in the system. Our findings show that transpiration through agricultural crops plays an important role in cloud formation, and thus, affects the effectiveness of different rainfall events. Moreover, the combined actions of transpiration and seeding are much more beneficial in producing rain. Finally, we see the behavior of system by considering seasonal variations of some rate parameters.