International Journal of Biomathematics

This work is concerned with the global dynamics of a Susceptible–Vaccinated–Infected–Isolated–Recovered model, denoted by SVIQR, with vaccination-age. Moreover, the considered model contains a relapse term and a general form of the incidence function. This model is formulated to show the effect of sanitary isolation and vaccination on the disease prevalence when the relapse phenomena occurs. First, we show that the model is mathematically well posed by using the integrated semigroup theory. Second, we prove the existence of equilibrium points based on the basic reproduction number ${{ℛ}}_0^{\omega ,\xi }$. Moreover, by developing appropriate Lyapunov functionals and applying the LaSalles invariance principle, we established the global dynamics of the model, especially, we show that if ${{ℛ}}_0^{\omega ,\xi }\le 1$ the the disease-free equilibrium is globally asymptotically stable and, if ${{ℛ}}_0^{\omega ,\xi }>1$ the endemic equilibrium is globally asymptotically stable. Finally, using the finite difference method, numerical simulations are made to reinforce the theoretical findings and to prove the contribution of the vaccination and the isolation on the disease disappearing.

In this paper, incorporating memory-based diffusion and delay, we propose a partly diffusive plant–pollinator model under Neumann boundary condition. Then, we investigate the effects of memory-based diffusion and delay on the dynamics of this model through discussing the corresponding characteristic equation. And we find that Turing bifurcations and Hopf bifurcations can be induced by memory-based diffusion and delay, respectively. By performing some numerical simulations, stable spatially homogeneous periodic solutions and inhomogeneous steady state solutions are obtained, which illustrates and expands our results in this paper. Moreover, the unstable spatially inhomogeneous periodic solutions occur only for some time, then they eventually converge to a spatially homogeneous periodic solutions.

This paper is devoted to analyzing the influence of space-time white noise on the dynamics of biological mathematical models in spatiotemporal scenarios based on stochastic partial differential equations (SPDEs). Here, we propose a stochastic reaction–diffusion three-species food chain model with various functional response functions. The motivation of the SPDEs model construction is mainly twofold: (1) when species migrate in space, space-time white noise is a factor that cannot be ignored; (2) functional response function is an essential indicator for directly measuring the relationship between species in biological mathematical models. To discuss the long-term behaviors of the SPDEs model in spatial heterogeneous environment, we first analyze the well-posedness of mild and strong solutions, respectively. Then, the critical conditions for determining the permanence and extinction of populations are given, which reveal that high-intensity white noise can lead to the extinction of predator populations. The validity of the theoretical results is also performed by numerical simulation. In particular, this new theoretical method may be extended to the study of other stochastic biological mathematical models in spatiotemporal scenarios, such as eco-epidemiological models, high-dimensional population models and epidemic models.

To study the influence of the moving front of the infected interval and the spatial movement of individuals on the spreading or vanishing of infectious disease, we consider a nonlocal susceptible–infected–susceptible (SIS) reaction–diffusion model with media coverage, hospital bed numbers and free boundaries. The principal eigenvalue of the integral operator is defined, and the impacts of the diffusion rate of infected individuals and interval length on the principal eigenvalue are analyzed. Furthermore, sufficient conditions for spreading and vanishing of the disease are derived. Our results show that large media coverage and hospital bed numbers are beneficial to the prevention and control of disease. The difference between the model with nonlocal diffusion and that with local diffusion is also discussed and nonlocal diffusion leads to more possibilities.

In this study, we explore the dynamics of a plankton–fish system in which external toxic substances have adverse impacts on the plankton populations as well as the fish communities. Fish population is assumed to grow in a logistic fashion due to food sources other than zooplankton, and is being harvested at a nonlinear rate. Mathematically, we analyze the system for the feasibility and stability of the ecologically meaningful equilibria. We also obtain criteria for the optimal harvesting policy of the fishery resources. We modify the considered model to encapsulate the effect of negative feedback time delay involved in the growth of the fish population. By considering this time delay as a bifurcation parameter, we derive conditions for the existence of Hopf bifurcation; direction and stability of the bifurcating periodic solutions are also discussed. Extensive numerical simulations are done for the systems with and without time delay. The environmental toxins are found to affect the equilibrium abundances of plankton and fish population in the aquatic system. In the absence of time delay, we observe that the growths in fish population due to uptake of zooplankton and other food sources have potential to suppress the chaotic disorder and bring forth stability in the ecosystem. On the other hand, the excessive growth of phytoplankton due to overloading of nutrients and the increasing carrying capacity of the ecosystem for the fish population are found to exhibit destabilizing roles on the system’s dynamics. Finally, we observe that the larger magnitude of negative feedback time delay in the growth of fish population drives the system to a chaotic zone.

In this paper, we study the effects of the nonlocal competition and double Allee effect in prey on a diffusive predator–prey model. We investigate the local stability of coexistence equilibrium in the predator–prey model by analyzing the eigenvalue spectrum. We study the consequence of double Allee effect on the prey population. Also, we discuss the existence of Hopf bifurcation under different parameters by using the gestation time delay of predators as a bifurcation parameter and analyzing the distribution of eigenvalues. By utilizing the normal form method and center manifold theorem, we have given some conditions that could determine the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions. Through our research, we obtain that the double Allee effect can affect the coexistence of prey and predator and induce periodic oscillations of the densities of prey and predator. In addition, the nonlocal competition can also affect the stability and homogeneity of the solutions, but may receive the consequences of the Allee effect as well as time delay. In the numerical simulation part, we further demonstrate the correctness of this conclusion by comparison.

COVID-19 is a worldwide pandemic that poses a great threat to people’s health and has a huge impact on economic and social life. The epidemic is currently at a low level, but is still prevalent worldwide. Therefore, it is still important to study the prevention and control of the spread of the COVID-19 epidemic. We collected the numbers of COVID-19 infections in China during the past three years, and obtained the regional distribution of outbreaks through analysis. To assess the capacity of medical resources in these regions in response to the outbreaks of COVID-19, we developed a model of COVID-19 transmission dynamics including mild and severe cases. The simulation showed that 11 regions, Beijing, Shanghai, Heilongjiang, Yunnan, Xinjiang, Inner Mongolia, Liaoning, Jilin, Hainan, Guangdong and Fujian, all have a serious shortage of medical beds in reserve. When facing the next new outbreak, these areas will have insufficient medical resources. We simulated the effects of drug effectiveness and the proportion of asymptomatic infected individuals on the spread of the epidemic, and obtained that an accurate determination of the proportion of asymptomatic patients in the population is crucial to the impact on healthcare resources, while improving the effectiveness of drugs can significantly reduce the burden on healthcare resources. Regions should invest more in healthcare and increase more medical resources, including medical beds. Government should accelerate the development of effective COVID-19 drugs to effectively control the scale of the outbreak. The proportion of asymptomatic patients in the population has a significant impact on the allocation of healthcare resources. The monitoring of antibody levels in the population should be strengthened to better control the spread of the epidemic.

In this paper, we study the pulsating traveling fronts for a spatially discrete periodic reaction–diffusion system with a quiescent stage. It is known that there exists a critical number $c_{\ast }>0$ (called minimal wave speed) such that a pulsating traveling front exists if and only if its speed is above $c_{\ast }$. In this paper, we derive a uniqueness theorem for supercritical pulsating traveling fronts. Further, we show that all supercritical pulsating traveling fronts are exponentially stable.

Considering the existence of magnetic induction effect with different intensities in the process of subthreshold and suprathreshold oscillations of bioelectrical activities, a non-smooth feedback strategy for memristive current with time delay is proposed, and then a four-dimensional Filippov Hindmarsh–Rose (HR) neuron model is established. The local stability and bifurcation patterns of delayed subsystems are qualitatively analyzed. Accordingly, the discriminant formula for the direction and stability of periodic solutions generated by Hopf bifurcation is obtained on the center manifold. Importantly, the stability of subsystems has switching behavior, which is accompanied by abundant hidden electrical activities under the effect of time delay. The theoretical analysis clarifies that the proposed feedback strategy leads to complex sliding mode dynamics, including sliding segments, various equilibrium points and sliding bifurcations. Meanwhile, the analytical conditions for motions of grazing, sliding, and crossing are developed and verified based on the flow switching theory. Moreover, the mechanism and evolutive rule of the self-excited and hidden sliding electrical activities are revealed by the fast-slow variable dissection method. Finally, it is verified that the time delay can not only induce bistable structures composed of the quiescent state and periodic bursting, but also eliminate the hidden sliding dynamics.

Drug resistance is one of the most intractable issues associated with cancer treatment in clinical practice. Mathematical models provide an analytic framework for facilitating the understanding of resistance evolution dynamics and the design of cancer clinical trial. In this paper, we develop an elementary, compartmental mathematical model for absolute drug resistance, focusing on the effects of point mutations in genetic drivers of malignancy. A set of ordinary differential equations (ODEs) is used to describe the dynamics of competing heterogeneous cancer cell populations while taking account of pharmacokinetics. All possible equilibria and their local geometric properties are analyzed, with the result suggests that the system exhibits bistable dynamics. The existence of optimal treatment time is discussed. To identify the critical parameters which influence cellular dynamics, we also perform parameter sensitivity analysis. Finally, numerical simulations are presented to verify the feasibilities of our analytical results and to find that the pre-existence of resistant cell phenotypes contributes more than resistant mutants generated during the treatment phase.