How does within‐host dynamics affect population‐level dynamics? Insights from an immuno‐epidemiological model of malaria

Malaria is one of the most common mosquito‐borne diseases widespread in the tropical and subtropical regions. Few models coupling the within‐host malaria dynamics with the between‐host mosquito‐human dynamics have been developed. In this paper, by adopting the nested approach, a malaria transmission model with immune response of the host is formulated. Applying age‐structured partial differential equations for the between‐host dynamics, we describe the asymptomatic and symptomatic infectious host population for malaria transmission. The basic reproduction numbers for the within‐host model and for the coupled system are derived, respectively. The existence and stability of the equilibria of the coupled model are analyzed. We show numerically that the within‐host model can exhibit complex dynamical behavior, possibly even chaos. In contrast, equilibria in the immuno‐epidemiological model are globally stable and their stabilities are determined by the reproduction number. Increasing the activation rate of the within‐host immune response “dampens” the sensitivity of the population level reproduction number and prevalence to the increase of the within‐host reproduction of the pathogen. From public health perspective this means that treatment in a population with higher immunity has less impact on the population‐level reproduction number and prevalence than in a population with less immunity.     

Optimal pulse fishing policy in stage-structured models with birth pulses

In this paper, we propose exploited models with stage structure for the dynamics in a fish population for which periodic birth pulse and pulse fishing occur at different fixed time. Using the stroboscopic map, we obtain an exact cycle of system, and obtain the threshold conditions for its stability. Bifurcation diagrams are constructed with the birth rate (or pulse fishing time or harvesting effort) as the bifurcation parameter, and these are observed to display complex dynamic behaviors, including chaotic bands with period windows, period-doubling, multi-period-halving and incomplete period-doubling bifurcation, pitch-fork and tangent bifurcation, non-unique dynamics (meaning that several attractors or attractor and chaos coexist) and attractor crisis. This suggests that birth pulse and pulse fishing provide a natural period or cyclicity that make the dynamical behaviors more complex. Moreover, we show that the pulse fishing has a strong impact on the persistence of the fish population, on the volume of mature fish stock and on the maximum annual-sustainable yield. An interesting result is obtained that, after the birth pulse, the population can sustain much higher harvesting effort if the mature fish is removed as early as possible.     

Period-doubling and Neimark–Sacker bifurcations in a larch budmoth population model

We investigate a discrete consumer-resource system based on a model originally proposed for studying the cyclic dynamics of the larch budmoth population in the Swiss Alps. It is shown that the moth population can persist indefinitely for all of the biologically feasible parameter values. Using intrinsic growth rate of the consumer population as a bifurcation parameter, we prove that the system can either undergo a period-doubling or a Neimark–Sacker bifurcation when the unique interior steady state loses its stability.     

Analysis of a malaria model with mosquito-dependent transmission coefficient for humans

In this paper, we discuss an ordinary differential equation mathematical model for the spread of malaria in human and mosquito population. We suppose the human population to act as a reservoir. Both the species follow a logistic population model. The transmission coefficient or the interaction coefficient of humans is considered to be dependent on the mosquito population. It is seen that as the factors governing the transmission coefficient of humans increase, so does the number of infected humans. Further, it is observed that as the immigration constant increases, it leads to a rise in infected humans, giving an endemic shape to the disease.     

An extended discrete Ricker population model with Allee effects

Based on the classical discrete Ricker population model, we incorporate Allee effects by assuming rectangular hyperbola, or Holling-II type functional form, for the birth or growth function and formulate an extended Ricker model. We explore the dynamics features of the extended Ricker model. We obtain domains of attraction for the trivial fixed point. We determine conditions for the existence and stability of positive fixed points and find regions where there exist no positive fixed points, two positive fixed points one of which is stable and two positive fixed points both of which are unstable. We demonstrate that the model exhibits period-doubling bifurcations and investigate the existence and stability of the cycles. We also confirm that Allee effects have stabilization effects, by different measures, through numerical simulations.     

具有常数迁入率和非线性传染率βI~pS~q的SI模型分析

对一种具有种群动力和非线性传染率的传染病模型进行了研究,建立了具有常数迁入率和非线性传染率βI~pS~q的SI模型.与以往的具有非线性传染率的传染病模型相比,这种模型引入了种群动力,也就是种群的总数不再为常数,因此,该类模型更精确地描述了传染病传播的规律.还讨论了模型的正不变集,运用微分方程稳定性理论分析了模型平衡点的存在性及稳定性,得出了疾病消除平衡点和地方病平衡点的全局渐进稳定的充分条件.进一步的,得出了在某些参数范围内会出现Hopf分支现象,并对上述模型进行了生物学讨论.     

The dynamics of an epidemic model for pest control with impulsive effect

In pest control, there are only a few papers on mathematical models of the dynamics of microbial diseases. In this paper a model concerning biologically-based impulsive control strategy for pest control is formulated and analyzed. The paper shows that there exists a globally stable susceptible pest eradication periodic solution when the impulsive period is less than some critical value. Further, the conditions for the permanence of the system are given. In addition, there exists a unique positive periodic solution via bifurcation theory, which implies both the susceptible pest and the infective pest populations oscillate with a positive amplitude. In this case, the susceptible pest population is infected to the maximum extent while the infective pest population has little effect on the crops. When the unique positive periodic solution loses its stability, numerical simulation shows there is a characteristic sequence of bifurcations, leading to a chaotic dynamic, which implies that this model has more complex dynamics, including period-doubling bifurcation, chaos and strange attractors.     

Bifurcation analysis and chaos control in a discrete‐time plant quality and larch budmoth interaction model with Ricker equation

We investigate the dynamics of two‐dimensional discrete‐time model of leaf quality and larch budmoth interaction with Ricker equation. More precisely, the qualitative behavior of larch budmoth model is discussed in which the effect of food source upon the moth population is through intrinsic growth rate. We find the parametric conditions for local asymptotic stability of the unique positive fixed point. It is also proved that under certain parametric conditions, the system undergoes period‐doubling bifurcation with the help of center manifold theory. The parametric conditions for existence and direction of Neimark‐Sacker bifurcation at positive fixed point is investigated with the help of standard mathematical techniques of bifurcation theory. The chaos control in the system is discussed through implementation of hybrid control methodology. Finally, numerical simulations are provided to illustrate theoretical results. These results of numerical simulations demonstrate chaotic long‐term behavior over a broad range of parameters. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the system.     

Global stability and bifurcations of perturbed Gumowski–Mira difference equation

A generalized convergence theorem for higher order difference equations is established by quasi-Lyapunov function method. From this stability result we deduce the existence of global asymptotically stable fixed point and attractive two-periodic solution of the perturbed Gumowski–Mira difference equation. We also study global bifurcations of this system as the parameters vary. For instance we show that as the recombination coefficient moves through a critical curve, a fixed point loses its asymptotic stability and an attractive cycle of period 2 emerges near the fixed point due to a period-doubling bifurcation. The associated existence regions are also located.     

The effect of vaccines on backward bifurcation in a fractional order HIV model

In this paper, a homogeneous-mixing population fractional model for human immunodeficiency virus (HIV) transmission, which incorporates anti-HIV preventive vaccines, is proposed. The dynamics of the model indicate that the basic reproduction number being the unity is a strict threshold for disease eradication when there is no vaccine. However, it has been shown that when the efficacy or dosage of vaccines is low, the model exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium point (DFE) coexists with a stable endemic equilibrium point (EE) when the associated reproduction number is less than unity. Therefore, driving the basic reproduction number below the unity is not enough to eradicate the disease. A new critical value at the turning point should be deduced as a new threshold of disease eradication. We have generalized the integer LaSalle invariant set theorem into fractional system and given some sufficient conditions for the disease-free equilibrium point being globally asymptotical stability. Mathematical results in this paper suggest that improving the efficiency and dosage of vaccines are all valid methods for the control of disease.     

Stability and Neimark-Sacker bifurcation of a semi-discrete population model

In this paper, a semi-discrete model is derived for a nonlinear simple population model, and its stability and bifurcation are investigated by invoking a key lemma we present. Our results display that a Neimark-Sacker bifurcation occurs in the positive fixed point of this system under certain parametric conditions. By using the Center Manifold Theorem and bifurcation theory, the stability of invariant closed orbits bifurcated is also obtained. The numerical simulation results not only show the correctness of our theoretical analysis, but also exhibit new and interesting dynamics of this system, which do not exist in its corresponding continuous version.     

The chaos and control of a food chain model supplying additional food to top-predator

The control and management of chaotic population is one of the main objectives for constructing mathematical model in ecology today. In this paper, we apply a technique of controlling chaotic predator–prey population dynamics by supplying additional food to top-predator. We formulate a three species predator–prey model supplying additional food to top-predator. Existence conditions and local stability criteria of equilibrium points are determined analytically. Persistence conditions for the system are derived. Global stability conditions of interior equilibrium point is calculated. Theoretical results are verified through numerical simulations. Phase diagram is presented for various quality and quantity of additional food. One parameter bifurcation analysis is done with respect to quality and quantity of additional food separately keeping one of them fixed. Using MATCONT package, we derive the bifurcation scenarios when both the parameters quality and quantity of additional food vary together. We predict the existence of Hopf point (H), limit point (LP) and branch point (BP) in the model for suitable supply of additional food. We have computed the regions of different dynamical behaviour in the quantity–quality parametric plane. From our study we conclude that chaotic population dynamics of predator prey system can be controlled to obtain regular population dynamics only by supplying additional food to top predator. This study is aimed to introduce a new non-chemical chaos control mechanism in a predator–prey system with the applications in fishery management and biological conservation of prey predator species.     

Dynamic complexities in a parasitoid-host-parasitoid ecological model

Chaotic dynamics have been observed in a wide range of population models. In this study, the complex dynamics in a discrete-time ecological model of parasitoid-host-parasitoid are presented. The model shows that the superiority coefficient not only stabilizes the dynamics, but may strongly destabilize them as well. Many forms of complex dynamics were observed, including pitchfork bifurcation with quasi-periodicity, period-doubling cascade, chaotic crisis, chaotic bands with narrow or wide periodic window, intermittent chaos, and supertransient behavior. Furthermore, computation of the largest Lyapunov exponent demonstrated the chaotic dynamic behavior of the model.     

Mathematical Analysis of SIR Epidemic Model with Piecewise Infection Rate and Control Strategies

The limitation of contact between susceptible and infected individuals plays an important role in decreasing the transmission of infectious diseases. Prevention and control strategies contribute to minimizing the transmission rate. In this paper, we propose SIR epidemic model with delayed control strategies, in which delay describes the response and effect time. We study the dynamic properties of the epidemic model from three aspects: steady states, stability and bifurcation. By eliminating the existence of limit cycles, we establish the global stability of the endemic equilibrium, when the delay is ignored. Further, we find that the delayed effect on the infection rate does not affect the stability of the disease-free equilibrium, but it can destabilize the endemic equilibrium and bring Hopf bifurcation. Theoretical results show that the prevention and control strategies can effectively reduce the final number of infected individuals in the population. Numerical results corroborate the theoretical ones.     

Dynamics of stage-structured discrete mosquito population models

We formulate discrete-time stage-structured models, based on systems of difference equations, for mosquito populations. We include the four distinct mosquito metamorphic stages, egg, pupa, larva, and adult, in the models. We derive a formula for the inherent net reproductive number, and investigate existence and stability of fixed points. We also show that the models, by means of numerical simulations, exhibit richer dynamics.     

On the hyperbolicity of the period-doubling fixed point

We give a new proof of the hyperbolicity of the fixed point for the period-doubling renormalization operator using the local dynamics near a semi-attractive fixed point (in a Banach space) and the theory of holomorphic motions. We also give a new proof of the exponential contraction of the Feigenbaum renormalization operator in the hybrid class of the period-doubling fixed point: our proof uses the non-existence of invariant line fields in the period-doubling tower (C. McMullen), the topological convergence (D. Sullivan), and a new infinitesimal argument.

     

An SIRS model with a nonlinear incidence rate

The global dynamics of an SIRS model with a nonlinear incidence rate is investigated. We establish a threshold for a disease to be extinct or endemic, analyze the existence and asymptotic stability of equilibria, and verify the existence of bistable states, i.e., a stable disease free equilibrium and a stable endemic equilibrium or a stable limit cycle. In particular, we find that the model admits stability switches as a parameter changes. We also investigate the backward bifurcation, the Hopf bifurcation and Bogdanov–Takens bifurcation and obtain the Hopf bifurcation criteria and Bogdanov–Takens bifurcation curves, which are important for making strategies for controlling a disease.     

Existence of positive solutions for a prey-predator model with refuge and diffusion

We are concerned with a prey-predator reaction-diffusion model with monotonic functional response and specific refuge size. We discuss both existence and nonexistence of positive solutions of the model by using fixed point index theory and bifurcation theory as the main argument tools. We also discuss bifurcation solutions of which stability is established by using spectrum analysis methods. Moreover, we analyze the effects of refuge size on the dynamics of the model.     

Stochastically perturbed vector-borne disease models with direct transmission

In this paper we study a stochastic epidemic model of vector-borne diseases with direct mode of transmission and its delay modification. More precisely, we extend the deterministic epidemic models by introducing random perturbations around the endemic equilibrium state. By using suitable Lyapunov functions and functionals, we obtain stability conditions for the considered models and study the effect of the delay on the stability of the endemic equilibrium. Finally, numerical simulations for the stochastic model of malaria disease transmission are presented to illustrate our mathematical findings.