International Journal of Biomathematics

This paper deals with the global dynamics of a tuberculosis (TB) model with age-structure and delay. We perform some rigorous analyses for the model, including presenting an explicit formula for the basic reproduction number of the model, addressing the persistence of the solution semi-flow and the existence of the global attractor. Based on these analyses, we establish some results on stability and instability of equilibrium of the system. Finally, some numerical examples are provided to illustrate our obtained results.

In this paper, we investigate a reaction–diffusion–advection model with expanding fronts, which models the spatial transmission of West Nile virus (WNv) in a heterogeneous environment. A free boundary problem is formulated and the global existence and uniqueness of the solution is presented. In addition to a classical basic reproduction number, the spatial-temporal basic reproduction number for the model with null Dirichlet boundary condition is introduced and the risk index associated with the virus in spatial setting is defined, and their properties are discussed. Sufficient conditions for the WNv to vanish or spread are given, and the asymptotic behavior of the solution to the free boundary problem when the spreading occurs is established. Our results show that the initial number of infected populations and the expanding capability of the expanding fronts exhibit important impacts on the extinction or persistence of the virus.

This paper displays an efficient numerical technique of realizing mathematical models for an adiabatic tubular chemical reactor which forms an irreversible exothermic chemical reaction. At a steady-state solution for an adiabatic rounded reactor, the model can be diminished to a conventional nonlinear differential equation which converts into a system of the nonlinear equation that can proceed numerically utilizing Newton’s iterative method. An operational matrix of coordination is derived and is utilized to decrease the model for an adiabatic tubular chemical reactor to an arrangement of algebraic equations. Simple execution, basic activities, and precise arrangements are the fundamental highlights of the proposed wavelet technique. The numerical solutions attained by the present technique have been contrasted and compared with other techniques.

In this paper, certain delayed virus dynamical models with cell-to-cell infection and density-dependent diffusion are investigated. For the viral model with a single strain, we have proved the well-posedness and studied the global stabilities of equilibria by defining the basic reproductive number $R_0$ and structuring proper Lyapunov functional. Moreover, we found that the infection-free equilibrium is globally asymptotically stable if $R_0<1$, and the infection equilibrium is globally asymptotically stable if $R_0>1$. For the multi-strain model, we found that all viral strains coexist if the corresponding basic reproductive number $R_j^{𝜖}>1$, while virus will extinct if $R_j^{𝜖}<1$. As a result, we found that delay and the density-dependent diffusion does not influence the global stability of the model with cell-to-cell infection and homogeneous Neumann boundary conditions.

The citrate secreted by the rice (Oryza sativa L.) roots will promote the absorption of phosphate, and this process is described by the Kirk model. In our work, the Kirk model is divided into citrate sub-model and phosphate sub-model. In the citrate sub-model, we obtain the analytical solution of citrate with the Laplace transform, inverse Laplace transform and convolution theorem. The citrate solution is substituted into the phosphate sub-model, and the analytical solution of phosphate is obtained by the separation variable method. The existence of the solutions can be proved by the comparison test, the Weierstrass M-test and the Abel discriminating method.

A deterministic model for the transmission dynamics of melioidosis disease in human population is designed and analyzed. The model is shown to exhibit the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium when the basic reproduction number ${{ℛ}}_0$ is less than one. It is further shown that the backward bifurcation dynamics is caused by the reinfection of individuals who recovered from the disease and relapse. The existence of backward bifurcation implies that bringing down ${{ℛ}}_0$ to less than unity is not enough for disease eradication. In the absence of backward bifurcation, the global asymptotic stability of the disease-free equilibrium is shown whenever ${{ℛ}}_0<1$. For ${{ℛ}}_0>1$, the existence of at least one locally asymptotically stable endemic equilibrium is shown. Sensitivity analysis of the model, using the parameters relevant to the transmission dynamics of the melioidosis disease, is discussed. Numerical experiments are presented to support the theoretical analysis of the model. In the numerical experimentations, it has been observed that screening and treating individuals in the exposed class has a significant impact on the disease dynamics.

We propose and investigate a discrete-time predator–prey system with cooperative hunting in the predator population. The model is constructed from the classical Nicholson–Bailey host-parasitoid system with density dependent growth rate. A sufficient condition based on the model parameters for which both populations can coexist is derived, namely that the predator’s maximal reproductive number exceeds one. We study existence of interior steady states and their stability in certain parameter regimes. It is shown that the system behaves asymptotically similar to the model with no cooperative hunting if the degree of cooperation is small. Large cooperative hunting, however, may promote persistence of the predator for which the predator would otherwise go extinct if there were no cooperation.

In this paper, we investigate the stochastic avian influenza model with human-to-human transmission, which is disturbed by both white and telegraph noises. First, we show that the solution of the stochastic system is positive and global. Furthermore, by using stochastic Lyapunov functions, we establish sufficient conditions for the existence of a unique ergodic stationary distribution. Then we obtain the conditions for extinction. Finally, numerical simulations are employed to demonstrate the analytical results.

In this study, we investigate the optimal control problem for a diffusion eco-epidemiological predator–prey model. We applied two controllers to this model. One is the separation control, which separates the uninfected prey from the infected prey population, and the other is used as a treatment control to decrease the mortality caused by the disease. Then, we propose an optimal problem to minimize the infected prey population at the final time and the cost cause by the controls. To do this, by the operator semigroup theory we prove the existence of the solution to the controlled system. Furthermore, we prove the existence of the optimal controls and obtain the first-order necessary optimality condition for the optimal controls. Finally, some numerical simulations are carried out to support the theoretical results.

In this paper, stochastic properties of solution for a chemostat model with a distributed delay and random disturbance are studied, and we use distribution delay to simulate the delay in nutrient conversion. By the linear chain technique, we transform the stochastic chemostat model with weak kernel into an equivalent degenerate system which contains three equations. First, we state that this model has a unique global positive solution for any initial value, which is helpful to explore its stochastic properties. Furthermore, we prove the stochastic ultimate boundness of the solution of system. Then sufficient conditions for solution of the system tending toward the boundary equilibrium point at exponential rate are established, which means the microorganism will be extinct. Moreover, we also obtain some sufficient conditions for ergodicity of solution of this system by constructing some suitable stochastic Lyapunov functions. Finally, we provide some numerical examples to illustrate theoretical results, and some conclusions and analysis are given.

In this paper, a fractional-order delayed SIQR model with nonlinear incidence rate is investigated. Two time delays are incorporated in the model to describe the incubation period and the time caused by the healing cycle. By analyzing the associated characteristic equations, the stability of the endemic equilibrium and the existence of Hopf bifurcation are obtained in three different cases. Besides, the critical values of time delays at which a Hopf bifurcation occurs are obtained, and the influence of the fractional order on the dynamics behavior of the system is also investigated. Numerically, it has been shown that when the endemic equilibrium is locally stable, the convergence rate of the system becomes slower with the increase of the fractional order. Besides, our studies also imply that the decline of the fractional order may convert a oscillatory system into a stable one. Furthermore, we find in all these three cases, the bifurcation values are very sensitive to the change of the fractional order, and they decrease with the increase of the order, which means the Hopf bifurcation gradually occurs in advance.

In this paper, we study the Hopf bifurcation of predator–prey system with two delays and disease transmission. Furthermore, the global existence of bifurcated periodic solution was studied, the influence of disease transmission is given. At last, some simulations are given to support our result.

The aim of this paper is to investigate a stochastic threshold for a delayed epidemic model driven by Lévy noise with a nonlinear incidence and vaccination. Mainly, we derive a stochastic threshold ${{ℛ}}_s$ which depends on model parameters and stochastic coefficients for a better understanding of the dynamical spreading of the disease. First, we prove the well posedness of the model. Then, we study the extinction and the persistence of the disease according to the values of ${{ℛ}}_s$. Furthermore, using different scenarios of Tuberculosis disease in Morocco, we perform some numerical simulations to support the analytical results.

In this paper, we are concerned with the stability for a model in the form of system of integro-partial differential equations, which governs the evolution of two competing age-structured populations. The age-specified environment is incorporated in the vital rates, which displays the hierarchy of ages. By a non-zero fixed-point result, we show the existence of positive equilibria. Some conditions for the stability of steady states are derived by means of semigroup method. Furthermore, numerical experiments are also presented.