Narrow Results

In this paper, we propose a Zika transmission model which considers human-to-human sexual transmission, the extrinsic incubation period of mosquitoes, and the vector-bias effect. Firstly, the explicit expression of the basic reproduction number $R_0$ is given by using the next-generation operator method, and the global dynamics of the model are established by taking $R_0$ as the threshold condition, that is, if $R_0\le 1$, the disease-free equilibrium is globally asymptotically stable, if $R_0>1$, the model has a unique endemic equilibrium that is locally asymptotically stable and the disease persists. And when we ignore the vector-bias effect, the global asymptotic stability of the endemic equilibrium is proved by constructing a Lyapunov function. Then, we select the reported epidemic data from Brazil for fitting, which verifies the obtained theoretical results. Meanwhile, we study the impact of human-to-human sexual transmission rate and mosquito-to-human transmission rate on the spread and prevalence of Zika. In addition, we calculate the sensitivity indices of $R_0$ to the model parameters and provide effective measures to control Zika transmission. The simulation results indicate that extending the extrinsic incubation period of mosquitoes is beneficial for disease control while ignoring the vector-bias effect will underestimate the risk of Zika transmission.

In this paper, we proposed and analyzed a five-dimensional system of ordinary differential equations modeling the competition of two competing bacteria in a chemostat under the influence of the leachate recirculation and in the presence of a pathogen associated only with the bacteria 1. We suppose that the nutriment is present into two forms, soluble and insoluble nutriment, and both two forms are continuously added to the chemostat. The proposed model takes the form of an “SI” epidemic model and uses general increasing growth functions and general increasing incidence rate. It admits multiple equilibria that we give the conditions under which we assure both the existence and the local stability of each equilibrium point. The possibility of periodic trajectory was excluded, and the uniform persistence of both types of bacteria was proved. Finally, several numerical examples confirming the theoretical findings are given.

Seasonality is repetitive in the ecological, biological and human systems. Seasonal factors affect both pathogen survival in the environment and host behavior. In this study, we considered a five-dimensional system of ordinary differential equations modeling an epidemic in a seasonal environment with a general incidence rate. We started by studying the autonomous system by investigating the global stability of steady states. Later, we proved the existence, uniqueness, positivity and boundedness of a periodic orbit in a non-autonomous system. We demonstrate that the global dynamics are determined using the basic reproduction number ${{ℛ}}_0$ which is defined by the spectral radius of a linear integral operator. We showed that if ${{ℛ}}_0<1$, then the disease-free periodic solution is globally asymptotically stable and if ${{ℛ}}_0>1$, then the trajectories converge to a limit cycle reflecting the persistence of the disease. Finally, we present a numerical investigation that support our results.

In this study, we consider a nonlocal almost periodic reaction–diffusion–advection model to study the global dynamics of a single phytoplankton population under the assumption that nutrients are abundant and their metabolism is only affected by light intensity. First, we prove that the single phytoplankton species model is strongly monotone with respect to the order induced by cone $X$. Second, we characterize the upper Lyapunov exponent ${\lambda }^{\ast }$ for a class of almost periodic reaction–diffusion–advection equations, and provide a numerical method to compute it. On this basis, we prove that ${\lambda }^{\ast }$ is the threshold parameter for studying the global dynamic behavior of the population model. Our results show that if ${\lambda }^{\ast }<0$, phytoplankton species will become extinct, and if ${\lambda }^{\ast }>0$, phytoplankton species will be uniformly persistent. Finally, we verified the above results using numerical simulations.

In this paper, we propose a diffusive SIR model with general incidence rate, saturated treatment rate and spatially heterogeneous diffusion coefficients. We first prove the global existence of bounded solutions for the model and compute the basic reproduction number. We study the local and global stabilities of the disease-free equilibrium and the uniform persistence. In the case when the diffusion rate of infected individuals is constant, we carry out a bifurcation analysis of equilibria by considering the maximal treatment rate as the bifurcation parameter. Finally, we perform some numerical simulations, which show that the solutions to our model present periodic oscillations for certain values of the parameters.

In this paper, we analyze a class of three-dimensional eco-epidemiological models where prey is subject to Allee effects and infection. We first establish the existence, uniqueness, positivity and uniform ultimate boundedness of the solutions for the proposed system in the positive octant. For three subsystems, we investigate the existence of their respective trivial and positive equilibria and determine the conditions for some bifurcations (Hopf bifurcation, Bogdanov–Takens bifurcation of codimension-2 and saddle-node bifurcation) to occur. We find that the Allee effect, nonmonotonic functional response and intra-class competition in susceptible preys enable the S–I and S–P subsystems to have richer dynamics. For example, the S–I subsystem can have up to three positive equilibria, the S–P subsystem with nonmonotonic functional response can have two positive equilibria while it is impossible in monotonic situation, and high intra-class competition in susceptible preys may lead to the extinction of the predator population, etc. We show that the strong Allee effect can create a separatrix curve (or surface), leading to multistability. Then, we study the uniform persistence of the full system and identify an interior periodic orbit by applying Poincaré map and bifurcation theory. Our analysis reveals that the introduction of the infection or predation may act as a biological control to save the population from extinction and the interaction between these two factors yields a diverse array of biologically relevant behaviors. Finally, some numerical simulations are performed to support and supplement our analytical findings.

This paper deals with a kind of reaction–diffusion–advection model which depicts a predator–prey ecosystem in rivers or streams. We obtain a complete classification on the dynamical behavior of the system in the parameter space of the predator’s mortality rate $\mu$ and the prey’s intrinsic growth rate $r$. More precisely, both species fail to survive when $r$ is small. With the increase of $r$, there exist two critical values $r^{\ast }$ and ${\sigma }^{\ast }$ such that both species can coexist in the long run when $r>r^{\ast }$ and $0<\mu <{\sigma }^{\ast }$, otherwise the prey survives alone. Finally, with the aid of numerical simulations, we investigate the effects of diffusion and advection on the global dynamics of the system by computing these two critical values. Numerical simulations illustrate that the diffusion of both predators and prey would benefit the invasion of the predators, and the advection would be unfavorable to the survival of the predators. Moreover, numerical simulations also suggest that the unique positive steady state is globally asymptotically stable among all non-negative and nontrivial initial data.

In this paper, we formulate a new-age structured heroin transmission model with respect to the age of vaccination which structures the vaccine wanes rate $\alpha (a)$ and infection ratio of vaccination individuals $\sigma (a)$. The well-posedness and the basic reproduction number ${{ℛ}}_0$ of our model are first presented. If ${{ℛ}}_0<1$, the drug-free steady state ${{ℰ}}^0$ is locally stable and there will be multiple positive steady states due to the imperfect vaccine. If ${{ℛ}}_0>1$, there is a unique drug spread steady state, and our model is uniformly persistent. To reveal the dynamics of our model in detail, we carry out a further analysis in some special cases, including the backward and forward bifurcation results of our model when $\alpha (a)=\alpha$ and $\sigma (a)=\sigma$, and the unique drug-spread steady state’s stability when ${{ℛ}}_0>1$. Finally, a brief conclusion and discussion are presented.

The current series of three papers is concerned with the asymptotic dynamics in the following parabolic–elliptic chemotaxis system with space- and time-dependent logistic source:

A simple discrete SIS model with vaccination is proposed. Its dynamics depend on a lumped parameter R_vac. The model exhibits the classical threshold behavior when vaccination is totally ineffective. When vaccination is partially effective, a backward transcritical bifurcation may occur at R_vac = 1. In this case, the model also undergoes a saddle–node bifurcation at certain parameter values when R_vac < 1. The disease can persist for R_vac > 1 and can be eradicated for R_vac < 1 if a forward transcritical bifurcation occurs at R_vac = 1. However, the disease may persist even when R_vac < 1 if a backward bifurcation occurs at R_vac = 1.

We propose predator-prey-parasite models to study the effects of parasites upon the predator-prey interaction. There are two parameters that are used to model the effectiveness of the infected prey and infected predator. For the spatial homogeneous system, the asymptotic dynamics depend on the reproductive number of the parasite. The parasite can persist in the population if this reproductive number is larger than one. Numerical simulations suggest that less competitiveness of the infected predator can make the predator-prey interaction less stable. The dynamics may move from coexisting steady state to oscillations. For the spatial heterogeneous system, diffusion may destabilize the homogeneous interior steady state for a particular set of diffusion coefficients. However, both systems do not exhibit complicated dynamical behavior.

Infection caused by antibiotic-resistant pathogens is one of global public health problems. Many factors contribute to the emergence and spread of these pathogens. A model which describes the transmission dynamics of susceptible and resistant bacteria in a pregnant woman and the fetus is presented. Detailed qualitative analysis about positivity, boundedness, global stability and uniform persistence of the model is carried out. Numerical simulation and sensitivity analysis show that antibiotic input has potential impact for neonatal drug resistance. Our results show that the resistant bacteria in baby mainly come from antibiotics which are wrongly-used during gestational period, or foods containing antibiotic residues.

We propose deterministic discrete-time, discrete stage-structured population models with harvesting to investigate population persistence and extinction. The mathematical analysis is centered around the inherent net reproductive number of the population. We apply data of the red snapper in the Gulf of Mexico to simulate the models. We use the models to test for a hypothesis proposed by marine biologists by adding a stochastic component to the deterministic systems that simulating pulses of early juveniles from other populations. We conclude that although pulses of early juveniles from other populations can contribute to the stock size of the early juveniles in the Gulf of Mexico, this contribution is insignificant for the adult population. Therefore, the population may still be in danger of extinction since the ocean environment is unpredictable. Other control strategies are needed in order for the population to be harvested annually.

In this paper, an SVIR epidemiological model with infection age (time elapsed since the infection) and nonlinear incidence is studied. In the model, in order to reflect the dependence of disease progress on the infection age, the infected individual is structured by the infection age, and transmission and removal rates are assumed to depend on the infection age. By analyzing corresponding characteristic equations, the local stability of each of steady states of the model is established. It is proved that the semi-flow generated by this system is asymptotically smooth, and if the basic reproduction number is greater than unity, the system is uniformly persistent. By using Lyapunov functional and LaSalle’s invariance principle, the global dynamics of the model is investigated. It is shown that if the basic reproduction number is less than unity, the disease-free steady state is globally asymptotically stable and hence the disease dies out; and if the basic reproduction number is greater than unity, the endemic steady state is globally asymptotically stable and the disease persists. Numerical simulations are carried out to illustrate the main analytic results.

This paper presents a generalized predator–prey system and considers the effect of habitat complexity on the dynamical consequences. The results show that habitat complexity has a major impact on the dynamical consequences of the considered system. On the one hand, habitat complexity has a stabilizing impact under certain conditions. A numerical simulation in our study and in experiments conducted in the published studies elaborate on this stabilizing effect. On the other hand, the most interesting and open issue is that a destabilizing effect of habitat complexity is found theoretically. All results are explained and illustrated from the ecological viewpoint.

In this paper, we investigate a nonautonomous and an autonomous model of schistosomiasis transmission with a general incidence function. Firstly, we formulate the nonautonomous model by taking into account the effect of climate change on the transmission. Through rigorous analysis via theories and methods of dynamical systems, we show that the nonautonomous model has a globally asymptotically stable disease-free periodic equilibrium when the associated basic reproduction ratio ${{ℛ}}_0$ is less than unity. Otherwise, the system admits at least one positive periodic solutions if ${{ℛ}}_0$ is greater than unity. Secondly, using the average of periodic functions, we further derive the autonomous model associated with the nonautonomous model. Therefore, we show that the disease-free equilibrium of the autonomous model is locally and globally asymptotically stable when the associated reproduction ratio ${{ℛ}}_1$ is less than unity. When ${{ℛ}}_1$ is greater than unity, the existence and global asymptotic stability of the endemic equilibrium is established under certain conditions. Finally, using linear and nonlinear specific incidence function, we perform some numerical simulations to illustrate our theoretical results.

A class of deterministic predator–prey systems, where the prey population is subject to an infectious disease, is studied. The disease can be transmitted both horizontally and vertically within the host population but cannot be spread between the two trophic levels. Using the mathematical tools of uniform persistence, we derive sufficient conditions for which the interacting populations can coexist. Criteria based on model parameters for which either only the infected prey or healthy prey persists are also provided. It is found through numerical investigations that predation can change competition outcomes between healthy and infected prey populations. Depending on the parameter regimes and initial conditions, predation can either eradicate or promote the disease. As infected prey provides a resource for the predators, disease may promote persistence of the predators. However, infected prey may dominate the predator–prey community if disease is very infectious. In addition, disease in the prey population can mediate a hydra effect in predators and may induce chaotic interactions.

An epidemic model is proposed to comprehend the disease dynamics between humans and animals and back to humans with a culling intervention strategy. The proposed model is separated into two cases with two different culling rates: (1) at a per-capita constant rate and (2) constant population being culled. The global asymptotic stability of equilibria is determined in terms of the basic reproduction numbers. Further, we find that the culling rate (2) considered in the model could change the dynamics by having multiple positive equilibria. Sensitivity analysis recommends developing a strategy that promotes animals’ natural and disease-related death rates. By ranking the efficacies of various intervention strategies, we obtain that vaccination in the human population, isolation and public awareness are the largely effective control interventions. Our general theory raises concerns about both human and animal populations becoming reservoirs of the disease and affecting each other dynamically.

In the proposed study, a nonlinear model is developed to explore the interactive dynamics between cattle and invertebrates when they coexist in a grassland system and compete with one another for the same resource — the grass biomass. The constructed model is theoretically investigated using the qualitative theory of differential equations to show the system’s rich dynamical properties, which are crucial for maintaining the ecosystem’s balance in grasslands. The qualitative findings show that, depending on the parameter combinations, the system not only displays stability of many equilibrium states but also experiences transcritical and Hopf bifurcations. The model results support the idea that inter-specific competition between cattle and invertebrates does not always produce regular dynamic patterns but may also produce periodic and destabilizing patterns. The model’s outputs may assist in striking a balance between pasture and natural grass biomass in grassland with the invertebrates.

This paper describes the synergistic interaction between the growth of malignant gliomas and the immune system interactions using a system of coupled ordinary differential equations (ODEs). The proposed mathematical model comprises the interaction of glioma cells, macrophages, activated Cytotoxic T-Lymphocytes (CTLs), the immunosuppressive factor TGF-$\beta$ and the immuno-stimulatory factor IFN-$\gamma$. The dynamical behavior of the proposed system both analytically and numerically is investigated from the point of view of stability. By constructing Lyapunov functions, the global behavior of the glioma-free and the interior equilibrium point have been analyzed under some assumptions. Finally, we perform numerical simulations in order to illustrate our analytical findings by varying the system parameters.