International Journal of Biomathematics

In this paper, a new delayed fractional-order model including susceptible migratory birds, infected migratory birds and predators is proposed to discuss the spread of diseases among migratory birds. Fear of predators is considered in the model, as fear can reduce the reproduction rate and disease transmission rate among prey. First, some basic mathematical results of the proposed model are discussed. Then, time delay is regarded as a bifurcation parameter, and the delay-induced bifurcation conditions for such an uncontrolled system are established. A novel periodic pulse feedback controller is proposed to suppress the bifurcation phenomenon. It is found that the control scheme can successfully suppress the bifurcation behavior of the system, and the pulse width can be arbitrarily selected on the premise of ensuring the control effect. Compared with the traditional time-delay feedback controller, the control scheme proposed in this paper has more advantages in practical application, which not only embodies the advantages of low control cost and easy operation but also caters to the periodic changes of the environment. The proposed control scheme, in particular, remains effective even after the system has been disrupted by a constant. Numerical simulation verifies the correctness of the theoretical results.

It is beyond dispute that cytotoxic T-lymphocytes (CTLs) exert a vital function in the host’s antiviral defense mechanism. With the idea of the above factor and the logistic proliferation of CD4${}^{+}$ T-cells, we establish a HTLV-I (human T-cell leukemia virus type-I) mathematical model. First, two threshold parameters ${{ℛ}}_0$ and ${{ℛ}}_c$ (the basic reproduction numbers for viral infection and CTL immune response, respectively) are obtained. Second, sufficient criteria for local and global asymptotic stabilities of the feasible equilibria of the model are deduced, respectively. Third, the sensitivity analyses of ${{ℛ}}_0$ and ${{ℛ}}_c$ are performed to better understand the effective strategies for HTLV-I infection. Finally, not only numerical simulations are given to illustrate the stability conclusions, but also the biological significance is stated.

p53 kinetics plays a key role in regulating cell fate. Based on the p53 gene regulatory network composed by the core regulatory factors ATM, Mdm2, Wip1, and PIDD, the effect of the delays in the process of transcription and translation of Mdm2 and Wip1 on the dynamics of p53 is studied theoretically and numerically. The results show that these two time delays can affect the stability of the positive equilibrium. With the increase of delays, the dynamics of p53 presents an oscillating state. Further, we also study the effects of PIDD and chemotherapeutic drug etoposide on the kinetics of p53. The model indicates that (i) PIDD low-level expression does not significantly affect p53 oscillatory behavior, but high-level expression could induce two-phase kinetics of p53; (ii) Too high and too low concentration of etoposide is not conducive to p53 oscillation. These results are in good agreement with experimental findings. Finally, we consider the influence of internal noise on the system through Binomial $\tau$-leap algorithm. Stochastic simulations reveal that high-intensity noise completely destroys p53 dynamics in the deterministic model, whereas low-intensity noise does not alter p53 dynamics. Interestingly, for the stable focus, the internal noise with appropriate intensity can induce quasi-limit cycle oscillations of the system. Our work may provide the useful insights for the development of anticancer therapy.

For food chain system with three populations, direct predation is the basic interaction between species. Different species often have different predation functional responses, so a food chain system with Holling-II response for middle predator and Beddinton–DeAngelis response for top predator is proposed. Apart from direct predation, predator population can significantly impact the survival of prey population by inducing the prey’s fear, but the impact often possesses a time delay. This paper is concentrated to explore how the fear and time delay affect the system stability and the species persistence. By use of Lyapunov functional method and bifurcation theory, the positiveness and boundedness of solutions, local and global behavior of species, the system stability around the equilibrium states and various kinds of bifurcation are investigated. Numerically, some simulations are carried out to validate the main findings and the critical values of the bifurcation parameters of fear and conversion rate are obtained. It is observed that fear and delay can not only stabilize, but also destabilize the system, which depends on the magnitude of the fear and delay. The system varies from unstable to stable due to the continuous increase of the prey’s fear by middle predator. Small fear induced by top predator or small delay of the prey’s fear can stabilize the system, while they are sufficiently large, the system stability is to be destroyed. Simultaneously, the conversion rate can also change the stability and even make the species to be extinct. Some rich dynamics like multiple stabilities and various types of bistability behaviors are also exhibited, which results in the convergence of the species from one stable equilibrium to another.

In this paper, we aim to minimize the epidemic size of tungiasis disease and economic costs simultaneously, with terminal demands for infected humans. A human–jigger parasite control system with four control schemes for humans and jiggers is established. We propose a multi-objective optimal control problem with terminal constraints, in which the accumulated number of infected humans and control costs are involved. By applying the modified normal boundary intersection algorithm and the interior point scheme, numerical simulations for different combinations of control schemes are carried out, and actual data in Madagascar are used. Effective combination schemes are indicated from the perspectives of disease eradication, cost saving and time saving. Once these effective combinations are properly performed, the disease can be controlled. When only minimizing the epidemic size, the combination of the optimal treatments and adulticiding efforts is the best choice in the rainy season; the combination of the optimal personal protections and treatments is the preferential option in the dry season. When only minimizing the economical cost, the combination of the optimal adulticide and larvicide is the better selection in the rainy season; the combination of the optimal personal protections, treatments and adulticiding efforts is the prior choose in the dry season. Thus, there is a trade-off between the two objectives for all the effective combinations, decision-makers may choose an appropriate one to control the disease.

The primary varicella-zoster virus (VZV) infection that causes chickenpox (also known as varicella), spreads quickly among people and, in severe circumstances, can cause to fever and encephalitis. In this paper, the Mittag-Leffler fractional operator is used to examine the mathematical representation of the VZV. Five fractional-order differential equations are created in terms of the disease’s dynamical analysis such as S: Susceptible, V: Vaccinated, E: Exposed, I: Infectious and R: Recovered. We derive the existence criterion, positive solution, Hyers–Ulam stability, and boundedness of results in order to examine the suggested fractional-order model’s wellposedness. Finally, some numerical examples for the VZV model of various fractional orders are shown with the aid of the generalized Adams–Bashforth–Moulton approach to show the viability of the obtained results.

Many discrete systems have more distinctive dynamical behaviors compared to continuous ones, which has led lots of researchers to investigate them. The discrete predator–prey model with two different functional responses (Holling type I and II functional responses) is discussed in this paper, which depicts a complex population relationship. The local dynamical behaviors of the interior fixed point of this system are studied. The detailed analysis reveals this system undergoes flip bifurcation and Neimark–Sacker bifurcation. Especially, we prove the existence of Marotto’s chaos by analytical method. In addition, the hybrid control method is applied to control the chaos of this system. Numerical simulations are presented to support our research and demonstrate new dynamical behaviors, such as period-10, 19, 29, 39, 48 orbits and chaos in the sense of Li–Yorke.

Self-diffusion prerequisite is obtained as the spreading approach of biological populations. Cooperative hunting is a common behavior in predator populations that promotes predation and the coexistence of the prey–predator system. On the other side, the Allee effect among prey may cause the system to become unstable. In this paper, a diffusive prey–predator system with cooperative hunting and the weak Allee effect in prey populations is discussed. The linear stability and Hopf-bifurcation analysis had been used to examine the system’s stability. From the spatial stability of the system, the conditions for Turing instability have been derived. The multiple-scale analysis has been used to derive the amplitude equations of the system. The stability analysis of these amplitude equations leads to the formation of Turing patterns. Finally, numerical simulations are used to analyze spatial patterns forming in 1-D and 2-D. The studies indicate that the model can generate a complex pattern structure and that self-diffusion has a drastic impact on species distribution.

This paper investigates the global dynamics of a reaction–diffusion–advection Leslie–Gower predator–prey model in open advective environments. We find that there exist critical advection rates, intrinsic growth rates, diffusion rates and length of the domain, which classify the global dynamics of the Leslie–Gower predator–prey system into three scenarios: coexistence, persistence of prey only and extinction of both species. The results reveal some significant differences with the classical specialist and generalist predator–prey systems. In particular, it is found that the critical advection rates of prey and predator are independent of each other and the parameters about predation rate have no influence on the dynamics of system. The theoretical results provide some interesting highlights in ecological protection in streams or rivers.

In this paper, we mainly investigate the dynamics of traveling wavefronts for a model describing host tissue degradation by bacteria. We first establish the existence of spreading speed, and show that the spreading speed coincides with the minimal wave speed of traveling wavefronts. Moreover, a lower bound estimate of the spreading speed is given. Then, we prove that the traveling wavefronts with large speeds are globally exponentially stable, when the initial perturbation around the traveling wavefronts decays exponentially as $x\to -\infty$, but the initial perturbation can be arbitrarily large in other locations. The adopted methods are the weighted energy and the squeezing technique.

This paper develops a mathematical model to investigate the Human Immunodeficiency Virus (HIV) infection dynamics. The model includes two transmission modes (cell-to-cell and cell-free), two adaptive immune responses (cytotoxic T-lymphocyte (CTL) and antibody), a saturated CTL immune response, and latent HIV infection. The existence and local stability of equilibria are fully characterized by four reproduction numbers. Through sensitivity analyses, we assess the partial rank correlation coefficients of these reproduction numbers and identify that the infection rate via cell-to-cell transmission, the number of new viruses produced by each infected cell during its life cycle, the clearance rate of free virions, and immune parameters have the greatest impact on the reproduction numbers. Additionally, we compare the effects of immune stimulation and cell-to-cell spread on the model’s dynamics. The findings highlight the significance of adaptive immune responses in increasing the population of uninfected cells and reducing the numbers of latent cells, infected cells, and viruses. Furthermore, cell-to-cell transmission is identified as a facilitator of HIV transmission. The analytical and numerical results presented in this study contribute to a better understanding of HIV dynamics and can potentially aid in improving HIV management strategies.