International Journal of Biomathematics

A class of time-varying delay impulsive reaction–diffusion tree–grass–water–nitrogen system driven by Lévy jump process is considered. First, we prove the existence and uniqueness of the global positive solution of the model by constructing the Lyapunov function. Secondly, several sufficient conditions for finite-time stability are given by using comparison theorem and mean impulse interval method. Finally, numerical simulations are carried out to verify the effectiveness of the theoretical analysis.

In this paper, we focus on asymptotic speeds of spread for a reaction–diffusion two-group SIR epidemic model with constant recruitment, which lacks the comparison principle. More precisely, if $R_0<1$, then the solution of the system converges to the disease-free equilibrium as $t\to \infty$ and if $R_0>1$, there exists a critical speed $c^{\ast }$ such that the solution of the system is uniformly persistent with $\left|x\right|\le ct,\forall c\in [0,c^{\ast })$ and the infection dies out with $\left|x\right|\ge ct$ for any $c>c^{\ast }$. Finally, some numerical experiments are presented to modeling the propagation dynamics of the system.

$C{a}^{2+}$ signals have a crucial role in the immune system for cell functions such as proliferation, differentiation, apoptosis and gene transcription as well as the activation of Nuclear factor of activated T Cell (NFAT) which modulates the immune response of the cells. In this study, a fractional order spatiotemporal calcium dynamics model is formulated using the reaction-diffusion equation incorporating parameters like buffers, source influx, $J_{\mathrm{\text{SScyt}}}$ and $J_{\mathrm{\text{PMCA}}}$. Grünwald estimation is employed to obtain the solution and stability analysis has been performed. The analysis of numerical results provides novel insights about the role of Brownian motion, super diffusion, source influx, buffers, etc., in the regulation of calcium and NFAT concentration levels in T cell and the conditions which might lead to disordered immune responses causing diseases such as HINI, HIV and HBV.

In this paper, we consider a Leslie–Gower type reaction–diffusion predator–prey system with an increasing functional response. We mainly study the effect of three different types of diffusion on the stability of this system. The main results are as follows: (1) in the absence of prey diffusion, diffusion-driven instability can occur; (2) in the absence of predator diffusion, diffusion-driven instability does not occur and the non-constant stationary solution exists and is unstable; (3) in the presence of both prey diffusion and predator diffusion, the system can occur diffusion-driven instability and Turing patterns. At the same time, we also get the existence conditions of the Hopf bifurcation and the Turing–Hopf bifurcation, along with the normal form for the Turing–Hopf bifurcation. In addition, we conduct numerical simulations for all three cases to support the results of our theoretical analysis.

Hybrid neural network models are effective in analyzing time-series data by combining the strengths of neural networks and differential equation models. Although most studies have focused on linear hybrid models, few have examined nonlinear problems. This work explores the potential of a hybrid nonlinear epidemic neural network in predicting the correct infection function of an epidemic model. We design a novel loss function by combining bifurcation theory and mean-squared error loss to ensure the trainability of the hybrid model. Additionally, we identify unique existence conditions that support ordinary differential equations for estimating the correct infection function. Moreover, numerical experiments using the Runge–Kutta method confirm our proposed model’s soundness both on our synthetic data and the real COVID-19 data.

This paper examines a three-dimensional delayed discrete neural network model analytically and numerically to determine the existence of different types of bifurcations of the involved fixed points. The model exhibits different bifurcations such as pitchforks, flips, Neimark–Sackers, and flip-Neimark–Sackers. The critical coefficients are used to determine the structure of each bifurcation. The curves are calculated and plotted for each bifurcation when the parameters are changed. Further, these bifurcations are theoretically analyzed and numerically verified. From the obtained results, we observed that by drawing the curves associated with each bifurcation, the numerical simulations are consistent with the analytical results.

A four-dimensional mathematical model is formulated to explore the fear effect exerted by large carnivore in the grassland ecosystem. The model depicts the interactions among herbage, domestic herbivore, wild herbivore and large carnivore, which incorporates both direct predation and anti-predator mechanisms. The dynamic properties of the model are analytically investigated, including the dissipativity of solutions, and the existence and stability of different equilibria. Some numerical simulations are also presented to exhibit rich dynamical behaviors, such as various types of bistabilities, periodic oscillation and chaotic oscillation. The study reveals that the appropriate level of fear factors can stabilize the system and increase the density of herbage and domestic herbivore. The fear effect plays an important role in maintaining the balance of the grassland ecosystem and promoting the economy of human society.

A new viral infection model with cell-to-cell infection and periodic antiretroviral therapy was proposed and analyzed. With the help of defined basic reproduction number ${\Re }_0$, we proved that if ${\Re }_0<1$, then the infection-free steady state is globally attractive. If ${\Re }_0>1$, then the infection is uniformly persistent. Furthermore, the globally asymptotical stability of infection-free steady state has been established for the critical case of ${\Re }_0=1$. Also, the global asymptotic stability of the infection steady state for a special case of the model has been established by applying the method of Lyapunov. Numerical simulations show that increasing the drug efficacy for blocking virus-to-cell infection, cell-to-cell infection and producing non-infectious virus contributes to weakening the severity of viral infection.

Malaria is known globally as the foremost cause of death in children and adults. Several intervention strategies and controls have been implemented and proposed, among which is Long Lasting Insecticide Treated Nets (LLINs). Malaria cases have been reported to reduce by $50\%$ due to the use of LLINs. However, the laid-back behaviors of humans negatively impact its effective use through improper handling and exposure to direct sunlight. To this end, a mathematical model is formulated to investigate the influence of individual response to information on the transmission dynamics of malaria infection in both human and mosquito populations. The threshold for the effective reduction of the prevalence of malaria through a behavioral change in the use of LLINs in response to the created awareness is determined in this study. The point of departure of this study from other studies is that it incorporates information/awareness as a dynamic variable to investigate the impact of LLINs on malaria. The equilibrium points of the model are analyzed using stability theory and the associated basic reproduction number $R_0$. The implicit function theorem is used to determine the direction of bifurcation. We use Latin hypercube sampling (LHS) to test the most sensitive parameters in the basic reproduction number. The correlation between the $R_0$ and parameters is displayed using Scatter plots. The mathematical analysis shows that the direction of bifurcation is supercritical, which suggests that we only need to bring $R_0$ below unity for disease eradication. The study also revealed that an increase in awareness of the use of LLINs has an impact on reducing malaria transmission.

In this paper, we study a network-based SIR epidemic model with a saturated treatment function in which a parameter $\alpha$ is introduced to measure the extent of the effect of the infected being delayed for treatment. Our aim is to present a global analysis and to investigate how the parameter $\alpha$ affects the spreading of diseases. Our main results are as follows: (1) In the case of the threshold value $R_0<1$, there exist two values of $\alpha$: ${\alpha }_c$ and ${\alpha }_0$, such that the disease-free equilibrium is globally asymptotically stable when $\alpha \le {\alpha }_c$ and multiple endemic equilibria exist when $\alpha \ge {\alpha }_0$. This means that the parameter $\alpha$ has an essential influence on the spreading of the disease. (2) In the case of the threshold value $R_0>1$, if the model has only one endemic equilibrium, then the unique endemic equilibrium is globally attractive. In this case, it is also proved that if $\alpha \le {\alpha }_c$, then the endemic equilibrium has only one, so is globally attractive. In addition, numerical simulation is performed to illustrate our theoretical results.