International Journal of Biomathematics

In this paper, we investigate a two-dimensional avian influenza model with Allee effect and stochasticity. We first show that a unique global positive solution always exists to the stochastic system for any positive initial value. Then, under certain conditions, this solution is proved to be stochastically ultimately bounded. Furthermore, by constructing a suitable Lyapunov function, we obtain sufficient conditions for the existence of stationary distribution with ergodicity. The conditions for the extinction of infected avian population are also analytically studied. These theoretical results are conformed by computational simulations. We numerically show that the environmental noise can bring different dynamical outcomes to the stochastic model. By scanning different noise intensities, we observe that large noise can cause extinction of infected avian population, which suggests the repression of noise on the spread of avian virus.

This contribution probes into ergodic stationary distribution for two stochastic SVELIT (susceptible-vaccinated-early latent-late latent-infective-treated) tuberculosis (TB) models to observe the impact of white noises and color noises on TB control in random environments. We first investigate the existence and uniqueness of ergodic stationary distribution (EUESD) for the autonomous SVELIT model subject to white noises via the proper Lyapunov functions, and sufficient conditions on the extinction of disease are acquired. Next, sufficient conditions for the EUESD and the extinction of disease for the SVELIT model with Markov switching are also established. Eventually, some numerical examples validate the theoretical findings. What’s more, it has been observed that higher amplitude noises may lead to the eradication of TB, which is conducive to TB control.

This paper considers the production of biomass of two interconnected chemostats in series with biomass mortality and a growth kinetic of the biomass described by an increasing function. A comparison is made with the productivity of a single chemostat with the same mortality rate and with volume equal to the sum of the volumes of the two chemostats. We determine the operating conditions under which the productivity of the serial configuration is greater than the productivity of the single chemostat. Moreover, the differences and similarities in the results corresponding to the case with mortality and the one without mortality, are highlighted. The mortality leads to surprising results where the productivity of a steady state where the bacteria are washed out in the first chemostat is greater than the one where the bacteria are present in both chemostats.

In this paper, a class of brucellosis transmission model with seasonal alternation, density-dependent growth, stage-structure, maturation delay, time-varying incubation is established. The basic reproduction number ${\mathcal{ℛ}}_0$ is derived, by which we find that the brucellosis is uniformly persistent if ${\mathcal{ℛ}}_0>1$, while the disease-free periodic solution is globally attractive if ${\mathcal{ℛ}}_0<1$. The theoretical results are illustrated by numerical simulation, from which we find that the brucellosis transmission would be overestimated (or underestimated) if we ignore the influence of time-varying incubation or maturation delay. If density-dependent growth of animals is ignored, the risk of brucellosis may be far underestimated, the extinction of brucellosis can be obtained by numerical simulation under the same conditions. Seasonality significantly affects the long-term dynamic behavior of brucellosis, and the inconsistency of parameter periods results in complex dynamic behavior.

In this paper, we formulate the transmission phenomena of Hand–Foot–Mouth Disease (HFMD) through non-integer derivative. We interrogate the biological meaningful results of the recommended system of HFMD. The basic reproduction number is determined through next generation method and the impact of different parameters on the reproduction number is examined with the help of partial rank correlation coefficient (PRCC) technique. In addition, we concentrated on qualitative analysis and dynamical behavior of HFMD dynamics. Banach’s and Schaefer’s fixed-point theorems are used to analyze the uniqueness and existence of the solution of the proposed HFMD model. The HFMD system’s Ulam–Hyers stability has been confirmed to be sufficient. To highlight the impact of the parameters on the dynamics of HFMD, we performed several simulations through numerical scheme to conceptualize the transmission route of the infection. To be more specific, numerical simulations are used to visualize the effect of input parameters on the systems dynamics. We have shown the key input parameters of the system for the control of infection in the society.

In this paper, we study a stochastic predator–prey model with Beddington–DeAngelis functional response and time-periodic coefficients. By analyzing the stability of the solution on the boundary and some stochastic estimates, the threshold conditions for the time-average persistence in probability and extinction of each population are established. Furthermore, the existence of a unique periodic measure of the model is also presented under the condition of the time-average persistence in probability of the model. Several numerical simulations are given to verify the effectiveness of the theoretical results and to illustrate the effects of the white noises on the persistence and periodic measure of the model.

In this paper, we formulate and study a fractional-order network model with four neurons, bidirectional ring structure and self-delay feedback. For the scenario of non-identical neurons, we develop a new algebraic technique to deal with the characteristic equation with ${\mathrm{\text{e}}}^{-4s\tau }$ ($\tau$ is the self-feedback delay) term and thus establish the easy-to-check criteria to determine the Hopf bifurcation point of self-feedback delay by fixing communication delay in its stable interval. For the scenario of identical neurons, we apply the crossing curves method to the fractional functional equations and thus procure the Hopf bifurcation curve. The obtained results accommodate the fact that the model cannot preserve its stability behavior when the self-feedback delay crosses the Hopf bifurcation point in the positive direction. Finally, we deliberate on the correctness of our methodology through two demonstration examples.

The aim of this paper is to study the dynamical behaviors of a piecewise smooth predator–prey model with predator harvesting. We consider a harvesting strategy that allows constant catches if the population size is above a certain threshold value (to obtain predictable yield) and no catches if the population size is below the threshold (to protect the population). It is shown that boundary equilibrium bifurcation and sliding–grazing bifurcation can happen as the threshold value varies. We provide analytical analysis to prove the existence of sliding limit cycles and sliding homoclinic cycles, the coexistence of them with standard limit cycles. Some numerical simulations are given to demonstrate our results.

In this paper, we propose a deterministic model to study the transmission dynamics of anthrax disease, which includes live animals, carcasses, spores in the environment and vectors. We derive three biologically plausible and insightful quantities (reproduction numbers) that determine the stability of the equilibria. We carry out rigorous mathematical analysis on the model dynamics, the global stability of the disease-free and vector-free equilibrium, the disease-free equilibrium and the vector-free disease equilibrium is proved. The global stability of the endemic equilibrium as the basic reproduction number is greater than one is derived in the special case in which the disease-related death rate is zero. The possibility of backward bifurcation is briefly discussed. Numerical analyses are carried out to understand the transmission dynamics of anthrax and investigate effective control strategies for the outbreaks of the disease. Our studies suggest that the larval vector control measure should be taken as early as possible to control the vector population size, a vaccination policy and an animal carcass removal policy are useful methods to control the prevalence of the diseases in infected animal populations, the adult vector control measure is also necessary to prevent the transmission of anthrax.

This study focuses on the stability and local bifurcations of a discrete-time SIR epidemic model with logistic growth of the susceptible individuals analytically, and numerically. The analytical results are obtained using thenormal form technique and numerical results are obtained using the numerical continuation method. For this model, a number of bifurcations are studied, including the transcritical (pitchfork) and flip bifurcations, the Neimark–Sacker (NS) bifurcations, and the strong resonance bifurcations. We especially determine the dynamical behaviors of the model for higher iterations up to fourth-order. Numerical simulation is employed to present a closed invariant curve emerging about an NS point, and its breaking down to several closed invariant curves and eventuality giving rise to a chaotic strange attractor by increasing the bifurcation parameter.